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/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | generalize hg : (fun n1 : ℕ ↦ w1 ^ n1 • h.series2CoeffN0Sum n1 w0) = g | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | generalize ha' : ∑' n, f n = a' | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | have gs : HasSum g a := by rw [← hg, ← ha]; exact cauchy2_hasSum_n1n0 h w0m w1m | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | have fs : ∀ n1 : ℕ, HasSum (fun n0 ↦ f ⟨n1, n0⟩) (g n1) := by
intro n1; rw [← hf, ← hg]; simp only
simp_rw [smul_comm (w0 ^ _) _]; apply HasSum.const_smul; exact cauchy2_hasSum_n0 h w0m n1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | have fs' : HasSum f a' := by rw [← ha']; exact sf.hasSum | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | have gs' := HasSum.prod_fiberwise fs' fs | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | rwa [HasSum.unique gs gs'] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | rw [← hg, ← ha] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | exact cauchy2_hasSum_n1n0 h w0m w1m | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | intro n1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | rw [← hf, ← hg] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | simp only | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | simp_rw [smul_comm (w0 ^ _) _] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | apply HasSum.const_smul | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coef... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | exact cauchy2_hasSum_n0 h w0m n1 | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coef... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | intro n | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | rw [← hf] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | rw [norm_smul, norm_smul, mul_assoc] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | rw [Complex.norm_eq_abs, Complex.norm_eq_abs, ← mul_assoc] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | trans abs w0 ^ n.snd * abs w1 ^ n.fst * (b * r⁻¹ ^ (n.snd + n.fst)) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | bound [series2Coeff_bound h n.snd n.fst] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | rw [pow_add, div_eq_mul_inv, div_eq_mul_inv, inv_pow, inv_pow] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | ring_nf | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | rfl | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | simp only [Metric.mem_ball, dist_zero_right, Complex.norm_eq_abs] at w0m w1m | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 => w1 ^ n1... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | refine .of_norm_bounded _ ?_ fb | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 => w1 ^ n1... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 => w1 ^ n1... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | simp_rw [mul_assoc] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 => w1 ^ n1... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 => w1 ^ n1... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | apply Summable.mul_left | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 => w1 ^ n1... | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 =>... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | simp_rw [mul_comm ((abs w0 / r) ^ _) _] | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 =>... | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 =>... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | apply Summable.mul_of_nonneg | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 =>... | case hf.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | exact summable_geometric_of_lt_one (by bound) ((div_lt_one h.rp).mpr w1m) | case hf.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | bound | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 => w1 ^ n1... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | exact summable_geometric_of_lt_one (by bound) ((div_lt_one h.rp).mpr w0m) | case hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hg
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | bound | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n1 => w1 ^ n1... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | intro n | case hf.hf'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n... | case hf.hf'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 •... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | simp only [Pi.zero_apply, div_pow] | case hf.hf'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n... | case hf.hf'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 •... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | bound | case hf.hf'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 •... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | intro n | case hf.hg'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n... | case hf.hg'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hg'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 •... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | simp only [Pi.zero_apply, div_pow] | case hf.hg'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n... | case hf.hg'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hg'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 •... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | bound | case hf.hg'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.1) = f
g : ℕ → E
hg : (fun n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hg'
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 •... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | rw [← ha'] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum_2d | [487, 1] | [517, 29] | exact sf.hasSum | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ → E
hf : (fun n => w0 ^ n.2 • w1 ^ n.1 • h.series2Coeff n.2 n.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f✝ c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
ha : f✝ (c0 + w0, c1 + w1) = a
f : ℕ × ℕ →... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | HasSum.antidiagonal_of_2d | [520, 1] | [532, 26] | generalize hg : (fun n ↦ (Finset.range (n + 1)).sum fun n1 ↦ f (n1, n - n1)) = g | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum f a
⊢ HasSum (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) a | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum f a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
⊢ HasSum g a | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum f a
⊢ HasSum (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) a
TACTIC:... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | HasSum.antidiagonal_of_2d | [520, 1] | [532, 26] | rw [←Finset.sigmaAntidiagonalEquivProd.hasSum_iff] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum f a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
⊢ HasSum g a | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
⊢ HasSum g a | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum f a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | HasSum.antidiagonal_of_2d | [520, 1] | [532, 26] | have fg : ∀ n, HasSum (fun d : Finset.antidiagonal n ↦
(f ∘ Finset.sigmaAntidiagonalEquivProd) ⟨n, d⟩) (g n) := by
intro n; simp only [Function.comp_apply, Finset.sigmaAntidiagonalEquivProd_apply]
have fs := hasSum_fintype fun d : ↥(Finset.antidiagonal n) ↦ f ↑d
have e : (Finset.univ.sum fun d : ↥(Finset.anti... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
⊢ HasSum g a | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
fg : ∀ (n : ℕ), HasSu... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | HasSum.antidiagonal_of_2d | [520, 1] | [532, 26] | exact HasSum.sigma h fg | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
fg : ∀ (n : ℕ), HasSu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | HasSum.antidiagonal_of_2d | [520, 1] | [532, 26] | intro n | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
⊢ ∀ (n : ℕ), HasSum (... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
n : ℕ
⊢ HasSum (fun d... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | HasSum.antidiagonal_of_2d | [520, 1] | [532, 26] | simp only [Function.comp_apply, Finset.sigmaAntidiagonalEquivProd_apply] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
n : ℕ
⊢ HasSum (fun d... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
n : ℕ
⊢ HasSum (fun d... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | HasSum.antidiagonal_of_2d | [520, 1] | [532, 26] | have fs := hasSum_fintype fun d : ↥(Finset.antidiagonal n) ↦ f ↑d | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
n : ℕ
⊢ HasSum (fun d... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
n : ℕ
fs : HasSum (fu... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | HasSum.antidiagonal_of_2d | [520, 1] | [532, 26] | have e : (Finset.univ.sum fun d : ↥(Finset.antidiagonal n) ↦ f ↑d) = g n := by
rw [Finset.sum_coe_sort, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk, ← hg] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
n : ℕ
fs : HasSum (fu... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
n : ℕ
fs : HasSum (fu... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | HasSum.antidiagonal_of_2d | [520, 1] | [532, 26] | rwa [← e] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
n : ℕ
fs : HasSum (fu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | HasSum.antidiagonal_of_2d | [520, 1] | [532, 26] | rw [Finset.sum_coe_sort, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk, ← hg] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n + 1)).sum fun n1 => f (n1, n - n1)) = g
n : ℕ
fs : HasSum (fu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℕ × ℕ → E
a : E
h : HasSum (f ∘ ⇑Finset.sigmaAntidiagonalEquivProd) a
g : ℕ → E
hg : (fun n => (Finset.range (n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | have sum := (cauchy2_hasSum_2d h w0m w1m).antidiagonal_of_2d | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f (c0 + w0, c1 + w1)) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
sum :
HasSum
(fun n =>
(Finset.range (n + 1)).sum fun n1 =>
w0 ^ (n1, n - n1).2 • w1 ^ (n1,... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | simp only [ge_iff_le] at sum | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
sum :
HasSum
(fun n =>
(Finset.range (n + 1)).sum fun n1 =>
w0 ^ (n1, n - n1).2 • w1 ^ (n1,... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
sum :
HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x) x)
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
sum :
HasSum
(fun n =>
(Finset.range (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | generalize ha : f (c0 + w0, c1 + w1) = a | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
sum :
HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x) x)
... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
sum :
HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x) x)
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
sum :
HasSum (fun n => (Finset.range (n + 1)).su... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | rw [ha] at sum | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
sum :
HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x) x)
... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x)... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
sum :
HasSum (fun n => (Finset.range (n + 1)).su... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | clear ha | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x)... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x)... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | have e : (fun n ↦
(Finset.range (n + 1)).sum fun n1 ↦ w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1) =
fun n ↦ series2 h n fun _ : Fin n ↦ (w0, w1) := by
clear sum; funext n
rw [series2]; simp only [ge_iff_le, ContinuousMultilinearMap.sum_apply]
simp_rw [termCmmap_apply]
nth_rw 1 [← Finset.sum_ra... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x)... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x)... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | rwa [←e] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | clear sum | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)).sum fun x => w0 ^ (n - x) • w1 ^ x • h.series2Coeff (n - x)... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
⊢ (fun n => (Finset.range (n + 1)).sum fun n1 => w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1) = ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
sum : HasSum (fun n => (Finset.range (n + 1)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | funext n | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
⊢ (fun n => (Finset.range (n + 1)).sum fun n1 => w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1) = ... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun n1 => w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
⊢ (fun n => (Finset.range (n + 1)).sum fun n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | rw [series2] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun n1 => w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun n1 => w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | simp only [ge_iff_le, ContinuousMultilinearMap.sum_apply] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun n1 => w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun n1 => w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | simp_rw [termCmmap_apply] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun n1 => w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun n1 => w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | nth_rw 1 [← Finset.sum_range_reflect] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun n1 => w0 ^ (n - n1) • w1 ^ n1 • h.series2Coeff (n - n1) n1... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun j =>
w0 ^ (n - (n + 1 - 1 - j)) • w1 ^ (n + 1 - 1 - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | simp | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun j =>
w0 ^ (n - (n + 1 - 1 - j)) • w1 ^ (n + 1 - 1 - ... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun x => w0 ^ (n - (n - x)) • w1 ^ (n - x) • h.series2Coeff (n... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | apply Finset.sum_congr rfl | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum fun x => w0 ^ (n - (n - x)) • w1 ^ (n - x) • h.series2Coeff (n... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ∀ x ∈ Finset.range (n + 1),
w0 ^ (n - (n - x)) • w1 ^ (n - x) • h.series2Coeff (n - (n... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ((Finset.range (n + 1)).sum f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | intro n0 n0n' | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ∀ x ∈ Finset.range (n + 1),
w0 ^ (n - (n - x)) • w1 ^ (n - x) • h.series2Coeff (n - (n... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n n0 : ℕ
n0n' : n0 ∈ Finset.range (n + 1)
⊢ w0 ^ (n - (n - n0)) • w1 ^ (n - n0) • h.series2Coeff (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n : ℕ
⊢ ∀ x ∈ Finset.range (n + 1),
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | simp only [Finset.mem_range] at n0n' | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n n0 : ℕ
n0n' : n0 ∈ Finset.range (n + 1)
⊢ w0 ^ (n - (n - n0)) • w1 ^ (n - n0) • h.series2Coeff (... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n n0 : ℕ
n0n' : n0 < n + 1
⊢ w0 ^ (n - (n - n0)) • w1 ^ (n - n0) • h.series2Coeff (n - (n - n0)) (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n n0 : ℕ
n0n' : n0 ∈ Finset.range (n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | have n0n := Nat.le_of_lt_succ n0n' | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n n0 : ℕ
n0n' : n0 < n + 1
⊢ w0 ^ (n - (n - n0)) • w1 ^ (n - n0) • h.series2Coeff (n - (n - n0)) (... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n n0 : ℕ
n0n' : n0 < n + 1
n0n : n0 ≤ n
⊢ w0 ^ (n - (n - n0)) • w1 ^ (n - n0) • h.series2Coeff (n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n n0 : ℕ
n0n' : n0 < n + 1
⊢ w0 ^ (n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2_hasSum | [535, 1] | [551, 11] | rw [Nat.sub_sub_self n0n, min_eq_left n0n] | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n n0 : ℕ
n0n' : n0 < n + 1
n0n : n0 ≤ n
⊢ w0 ^ (n - (n - n0)) • w1 ^ (n - n0) • h.series2Coeff (n ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
a : E
n n0 : ℕ
n0n' : n0 < n + 1
n0n : n0 ≤... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ 0 < ENNReal.ofReal r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ 0 < r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ 0 < ENNReal.ofReal r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | exact h.rp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ 0 < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ 0 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | simp only [Metric.emetric_ball, Metric.mem_ball, dist_zero_right, Prod.forall] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ ∀ {y : ℂ × ℂ}, y ∈ EMetric.ball 0 (ENNReal.ofReal r) → HasSum (fun n => (series2 h n) fun x => y) (f ((c0, c1) + y)) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ ∀ (a b_1 : ℂ), ‖(a, b_1)‖ < r → HasSum (fun n => (series2 h n) fun x => (a, b_1)) (f ((c0, c1) + (a, b_1))) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ ∀ {y : ℂ × ℂ}, y ∈ EMetric.ball 0 (ENNReal.ofReal r) → HasSum (fun n => (series2 h n) fu... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | intro w0 w1 wr | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ ∀ (a b_1 : ℂ), ‖(a, b_1)‖ < r → HasSum (fun n => (series2 h n) fun x => (a, b_1)) (f ((c0, c1) + (a, b_1))) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : ‖(w0, w1)‖ < r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f ((c0, c1) + (w0, w1))) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
⊢ ∀ (a b_1 : ℂ), ‖(a, b_1)‖ < r → HasSum (fun n => (series2 h n) fun x => (a, b_1)) (f ((c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | rw [Prod.norm_def] at wr | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : ‖(w0, w1)‖ < r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f ((c0, c1) + (w0, w1))) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : max ‖(w0, w1).1‖ ‖(w0, w1).2‖ < r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f ((c0, c1) + (w0, w1))) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : ‖(w0, w1)‖ < r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f ((c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | simp only [Complex.norm_eq_abs, ge_iff_le, max_lt_iff] at wr | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : max ‖(w0, w1).1‖ ‖(w0, w1).2‖ < r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f ((c0, c1) + (w0, w1))) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f ((c0, c1) + (w0, w1))) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : max ‖(w0, w1).1‖ ‖(w0, w1).2‖ < r
⊢ HasSum (fun n => (series2 h n) fun x ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | have w0m : w0 ∈ ball (0 : ℂ) r := by simp; exact wr.left | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f ((c0, c1) + (w0, w1))) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f ((c0, c1) + (w0, w1... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
⊢ HasSum (fun n => (series2 h n) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | have w1m : w1 ∈ ball (0 : ℂ) r := by simp; exact wr.right | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f ((c0, c1) + (w0, w1... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
⊢ HasSum (fun... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | exact cauchy2_hasSum h w0m w1m | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ball 0 r
⊢ HasSum (fun n => (series2 h n) fun x => (w0, w1)) (f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
w1m : w1 ∈ ba... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
⊢ w0 ∈ ball 0 r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
⊢ Complex.abs w0 < r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
⊢ w0 ∈ ball 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | exact wr.left | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
⊢ Complex.abs w0 < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
⊢ Complex.abs w0 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | simp | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
⊢ w1 ∈ ball 0 r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
⊢ Complex.abs w1 < r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
⊢ w1 ∈ ball 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_h | [554, 1] | [564, 39] | exact wr.right | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
⊢ Complex.abs w1 < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0✝ w1✝ : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0 w1 : ℂ
wr : Complex.abs w0 < r ∧ Complex.abs w1 < r
w0m : w0 ∈ ball 0 r
⊢ Complex.abs... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood | [570, 1] | [592, 35] | intro c cs | E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
⊢ AnalyticOn ℂ f ... | E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
c : ℂ × ℂ
cs : c ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood | [570, 1] | [592, 35] | rcases Metric.isOpen_iff.mp o c cs with ⟨r, rp, rs⟩ | E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
c : ℂ × ℂ
cs : c ... | case intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood | [570, 1] | [592, 35] | have rs : closedBall c (r / 2) ⊆ s := le_trans (Metric.closedBall_subset_ball (by linarith)) rs | case intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
... | case intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood | [570, 1] | [592, 35] | rcases ((isCompact_closedBall _ _).bddAbove_image (ContinuousOn.mono fc rs).norm).exists_ge 0
with ⟨b, bp, bh⟩ | case intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
... | case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood | [570, 1] | [592, 35] | simp only [Set.forall_mem_image] at bh | case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (... | case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood | [570, 1] | [592, 35] | have h : Separate f c.fst c.snd (r / 2) b s :=
{ rp := by linarith
so := o
rs := rs
fc := fc
fa0 := fa0 _ _
fa1 := fa1 _ _
bp := bp
fb := fun {z0 z1} z0m z1m ↦ @bh (z0, z1)
(spheres_subset_closedBall (Set.mk_mem_prod z0m z1m)) } | case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (... | case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood | [570, 1] | [592, 35] | have a := (osgood_h h).analyticAt | case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (... | case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood | [570, 1] | [592, 35] | simpa only [Prod.mk.eta] using a | case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood | [570, 1] | [592, 35] | linarith | E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
c : ℂ × ℂ
cs : c ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood | [570, 1] | [592, 35] | linarith | E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
c : ℂ × ℂ
cs : c ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
o : IsOpen s
fc : ContinuousOn f s
fa0 : ∀ (z0 z1 : ℂ), (z0, z1) ∈ s → AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
fa1 : ∀ (z0 z1 : ℂ), (z0, z1) ∈... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_at' | [596, 1] | [603, 35] | rcases eventually_nhds_iff.mp h with ⟨s, h, o, cs⟩ | E : Type
f : ℂ × ℂ → E
c : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
h :
∀ᶠ (x : ℂ × ℂ) in 𝓝 c,
ContinuousAt f x ∧ AnalyticAt ℂ (fun z => f (z, x.2)) x.1 ∧ AnalyticAt ℂ (fun z => f (x.1, z)) x.2
⊢ AnalyticAt ℂ f c | case intro.intro.intro
E : Type
f : ℂ × ℂ → E
c : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
h✝ :
∀ᶠ (x : ℂ × ℂ) in 𝓝 c,
ContinuousAt f x ∧ AnalyticAt ℂ (fun z => f (z, x.2)) x.1 ∧ AnalyticAt ℂ (fun z => f (x.1, z)) x.2
s : Set (ℂ × ℂ)
h : ∀ x ∈ s, ContinuousAt f x ∧ Ana... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
f : ℂ × ℂ → E
c : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
h :
∀ᶠ (x : ℂ × ℂ) in 𝓝 c,
ContinuousAt f x ∧ AnalyticAt ℂ (fun z => f (z, x.2)) x.1 ∧ AnalyticAt ℂ (fun z => f (x.1, z)) x.2
⊢ AnalyticAt ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | osgood_at' | [596, 1] | [603, 35] | exact osgood o (fun _ m ↦ (h _ m).1.continuousWithinAt) (fun _ _ m ↦ (h _ m).2.1)
(fun _ _ m ↦ (h _ m).2.2) c cs | case intro.intro.intro
E : Type
f : ℂ × ℂ → E
c : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
h✝ :
∀ᶠ (x : ℂ × ℂ) in 𝓝 c,
ContinuousAt f x ∧ AnalyticAt ℂ (fun z => f (z, x.2)) x.1 ∧ AnalyticAt ℂ (fun z => f (x.1, z)) x.2
s : Set (ℂ × ℂ)
h : ∀ x ∈ s, ContinuousAt f x ∧ Ana... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
E : Type
f : ℂ × ℂ → E
c : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
h✝ :
∀ᶠ (x : ℂ × ℂ) in 𝓝 c,
ContinuousAt f x ∧ AnalyticAt ℂ (fun z => f (z, x.2)) x.1 ∧ AnalyticAt ℂ (fun z => f (x.1... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Measure.lean | ae_minus_null | [31, 1] | [37, 37] | simp only [Filter.EventuallyEq, Pi.sdiff_apply, eq_iff_iff] | E : Type
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
inst✝¹⁰ : SecondCountableTopology E
F : Type
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
X : Type
inst✝⁶ : MeasureSpace X
inst✝⁵ : MetricSpace X
inst✝⁴ : BorelSpace X
Y : Type
inst✝³ : Measure... | E : Type
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
inst✝¹⁰ : SecondCountableTopology E
F : Type
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
X : Type
inst✝⁶ : MeasureSpace X
inst✝⁵ : MetricSpace X
inst✝⁴ : BorelSpace X
Y : Type
inst✝³ : Measure... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
inst✝¹⁰ : SecondCountableTopology E
F : Type
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
X : Type
inst✝⁶ : MeasureSpace X
inst✝⁵ : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Measure.lean | ae_minus_null | [31, 1] | [37, 37] | have e : ∀ x, x ∉ t → (x ∈ s ↔ x ∈ s \ t) := by
intro x h; simp only [Set.mem_diff, h, not_false_iff, and_true_iff] | E : Type
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
inst✝¹⁰ : SecondCountableTopology E
F : Type
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
X : Type
inst✝⁶ : MeasureSpace X
inst✝⁵ : MetricSpace X
inst✝⁴ : BorelSpace X
Y : Type
inst✝³ : Measure... | E : Type
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
inst✝¹⁰ : SecondCountableTopology E
F : Type
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
X : Type
inst✝⁶ : MeasureSpace X
inst✝⁵ : MetricSpace X
inst✝⁴ : BorelSpace X
Y : Type
inst✝³ : Measure... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
inst✝¹⁰ : SecondCountableTopology E
F : Type
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
X : Type
inst✝⁶ : MeasureSpace X
inst✝⁵ : ... |
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