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/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | have h2 : ‖(f x - F i x) * g x‖ < ε / 2 := by
rw [norm_mul]
by_cases h : g x = 0
case pos => simp [h, half_pos hε]
case neg =>
convert mul_lt_mul (hF x hx) (h1 x hx) (norm_pos_iff.mpr h) (by positivity) using 1
simp only [div_mul, mul_div_cancel, hMg.ne.symm, Ne.def, not_false_iff] | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
⊢ ‖(f * g) x - (F * G) i x‖ < ε | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
⊢ ‖(f * g) x - (F * G) i x‖ < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
⊢ ‖(f * g) x - (F * G) i x‖ < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | have h3 : ‖F i x * (g x - G i x)‖ < ε / 2 := by
rw [norm_mul]
by_cases h : F i x = 0
case pos => simp [h, half_pos hε]
case neg =>
convert mul_lt_mul' (hf x hx) (hG x hx) (norm_nonneg _) hMf using 1
field_simp [hMf.ne.symm]; ring | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
⊢ ‖(f * g) x - (F * G) i x‖ < ε | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h3 : ‖F i x * (g x - G i x)‖ < ε / 2
⊢ ‖(f * g) x - (F * G) i x‖ < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
⊢ ‖(f * g) x - (F * G) i x‖ < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | simp_rw [Pi.mul_apply, lxyab] | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h3 : ‖F i x * (g x - G i x)‖ < ε / 2
⊢ ‖(f * g) x - (F * G) i x‖ < ε | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h3 : ‖F i x * (g x - G i x)‖ < ε / 2
⊢ ‖(f x - F i x) * g x + F i x * (g x - G i x)‖ < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h3 : ‖F i x * (g x - G i x)‖ < ε / 2
⊢ ‖(f * g) x - (F * G) i x‖ < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | exact (norm_add_le _ _).trans_lt (add_halves' ε ▸ add_lt_add h2 h3) | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h3 : ‖F i x * (g x - G i x)‖ < ε / 2
⊢ ‖(f x - F i x) * g x + F i x * (g x - G i x)‖ < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h3 : ‖F i x * (g x - G i x)‖ < ε / 2
⊢ ‖(f x - F i x) * g x + F i x * (g x - G i x)‖ < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | positivity | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
⊢ 0 < Mf | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
⊢ 0 < Mf
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | positivity | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
⊢ 0 < Mg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
⊢ 0 < Mg
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | filter_upwards [hf] with i hF x hx using (hF x hx).trans ((le_abs_self mf).trans (lt_add_one _).le) | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | filter_upwards [hg] with i hG x hx using (hG x hx).trans ((le_abs_self mg).trans (lt_add_one _).le) | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | intro x hx | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
⊢ ∀ x ∈ s, ‖g x‖ ≤ Mg | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
x : α
hx : x ∈ s
⊢ ‖g x‖ ≤ Mg | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
⊢ ∀ x ∈ s, ‖g x‖ ≤ Mg
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | refine le_of_tendsto ((continuous_norm.tendsto (g x)).comp (hG.tendsto_at hx)) ?_ | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
x : α
hx : x ∈ s
⊢ ‖g x‖ ≤ Mg | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
x : α
hx : x ∈ s
⊢ ∀ᶠ (c : ι) in p, ((fun a => ‖a‖) ∘ fun n => G n x) c ≤ Mg | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
x : α
hx : x ∈ s
⊢ ‖g x‖ ≤ Mg
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | filter_upwards [hg] with i hg using hg x hx | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
x : α
hx : x ∈ s
⊢ ∀ᶠ (c : ι) in p, ((fun a => ‖a‖) ∘ fun n => G n x) c ≤ Mg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
x : α
hx : x ∈ s
⊢ ∀ᶠ (c : ι) in p, ((fun a => ‖a‖) ∘ fun n => G n x) c ≤ Mg
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | positivity | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
⊢ ε / (2 * Mg) > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
⊢ ε / (2 * Mg) > 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | positivity | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
⊢ ε / (2 * Mf) > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
⊢ ε / (2 * Mf) > 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | rw [norm_mul] | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
⊢ ‖(f x - F i x) * g x‖ < ε / 2 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
⊢ ‖(f x - F i x) * g x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | by_cases h : g x = 0 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2 | case pos
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : g x = 0
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2
case neg
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : ¬g x = 0
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | case pos => simp [h, half_pos hε] | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : g x = 0
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : g x = 0
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | case neg =>
convert mul_lt_mul (hF x hx) (h1 x hx) (norm_pos_iff.mpr h) (by positivity) using 1
simp only [div_mul, mul_div_cancel, hMg.ne.symm, Ne.def, not_false_iff] | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : ¬g x = 0
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : ¬g x = 0
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | simp [h, half_pos hε] | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : g x = 0
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : g x = 0
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | convert mul_lt_mul (hF x hx) (h1 x hx) (norm_pos_iff.mpr h) (by positivity) using 1 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : ¬g x = 0
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2 | case h.e'_4
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : ¬g x = 0
⊢ ε / 2 = ε / (2 * Mg) * Mg | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : ¬g x = 0
⊢ ‖f x - F i x‖ * ‖g x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | simp only [div_mul, mul_div_cancel, hMg.ne.symm, Ne.def, not_false_iff] | case h.e'_4
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : ¬g x = 0
⊢ ε / 2 = ε / (2 * Mg) * Mg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : ¬g x = 0
⊢ ε / 2 = ε / (2 * Mg) * Mg
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | positivity | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : ¬g x = 0
⊢ 0 ≤ ε / (2 * Mg) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h : ¬g x = 0
⊢ 0 ≤ ε / (2 * Mg)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | rw [norm_mul] | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
⊢ ‖F i x * (g x - G i x)‖ < ε / 2 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
⊢ ‖F i x * (g x - G i x)‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | by_cases h : F i x = 0 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2 | case pos
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : F i x = 0
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2
case neg
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | case pos => simp [h, half_pos hε] | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : F i x = 0
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : F i x = 0
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | case neg =>
convert mul_lt_mul' (hf x hx) (hG x hx) (norm_nonneg _) hMf using 1
field_simp [hMf.ne.symm]; ring | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | simp [h, half_pos hε] | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : F i x = 0
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : F i x = 0
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | convert mul_lt_mul' (hf x hx) (hG x hx) (norm_nonneg _) hMf using 1 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2 | case h.e'_4
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ε / 2 = Mf * (ε / (2 * Mf)) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ‖F i x‖ * ‖g x - G i x‖ < ε / 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | field_simp [hMf.ne.symm] | case h.e'_4
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ε / 2 = Mf * (ε / (2 * Mf)) | case h.e'_4
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ε * (2 * Mf) = Mf * ε * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ε / 2 = Mf * (ε / (2 * Mf))
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_le | [76, 1] | [113, 72] | ring | case h.e'_4
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ε * (2 * Mf) = Mf * ε * 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
h✝ : NeBot p
Mf : ℝ := |mf| + 1
Mg : ℝ := |mg| + 1
hMf : 0 < Mf
hMg : 0 < Mg
hf✝ : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ Mf
hg : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ Mg
h1 : ∀ x ∈ s, ‖g x‖ ≤ Mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
ε : ℝ
hε : ε > 0
i : ι
hf : ∀ x ∈ s, ‖F i x‖ ≤ Mf
hF : ∀ x ∈ s, ‖f x - F i x‖ < ε / (2 * Mg)
hG : ∀ x ∈ s, ‖g x - G i x‖ < ε / (2 * Mf)
x : α
hx : x ∈ s
h2 : ‖(f x - F i x) * g x‖ < ε / 2
h : ¬F i x = 0
⊢ ε * (2 * Mf) = Mf * ε * 2
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | have h1 : ∀ᶠ i in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1 := by
simp_rw [Metric.tendstoUniformlyOn_iff, dist_eq_norm] at hF
filter_upwards [hF 1 zero_lt_one] with i hF x hx
have : ‖F i x‖ ≤ ‖f x - F i x‖ + ‖f x‖ := by
simpa [← norm_neg (F i x - f x)] using norm_add_le (F i x - f x) (f x)
linarith [hF x hx, hf x hx] | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
⊢ TendstoUniformlyOn (F * G) (f * g) p s | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
⊢ TendstoUniformlyOn (F * G) (f * g) p s | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
⊢ TendstoUniformlyOn (F * G) (f * g) p s
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | have h2 : ∀ᶠ i in p, ∀ x ∈ s, ‖G i x‖ ≤ mg + 1 := by
simp_rw [Metric.tendstoUniformlyOn_iff, dist_eq_norm] at hG
filter_upwards [hG 1 zero_lt_one] with i hG x hx
have : ‖G i x‖ ≤ ‖g x - G i x‖ + ‖g x‖ := by
simpa [← norm_neg (G i x - g x)] using norm_add_le (G i x - g x) (g x)
linarith [hG x hx, hg x hx] | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
⊢ TendstoUniformlyOn (F * G) (f * g) p s | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
h2 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg + 1
⊢ TendstoUniformlyOn (F * G) (f * g) p s | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
⊢ TendstoUniformlyOn (F * G) (f * g) p s
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | exact hF.mul_of_le hG h1 h2 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
h2 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg + 1
⊢ TendstoUniformlyOn (F * G) (f * g) p s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
h2 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg + 1
⊢ TendstoUniformlyOn (F * G) (f * g) p s
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | simp_rw [Metric.tendstoUniformlyOn_iff, dist_eq_norm] at hF | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | filter_upwards [hF 1 zero_lt_one] with i hF x hx | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1 | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
i : ι
hF : ∀ x ∈ s, ‖f x - F i x‖ < 1
x : α
hx : x ∈ s
⊢ ‖F i x‖ ≤ mf + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | have : ‖F i x‖ ≤ ‖f x - F i x‖ + ‖f x‖ := by
simpa [← norm_neg (F i x - f x)] using norm_add_le (F i x - f x) (f x) | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
i : ι
hF : ∀ x ∈ s, ‖f x - F i x‖ < 1
x : α
hx : x ∈ s
⊢ ‖F i x‖ ≤ mf + 1 | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
i : ι
hF : ∀ x ∈ s, ‖f x - F i x‖ < 1
x : α
hx : x ∈ s
this : ‖F i x‖ ≤ ‖f x - F i x‖ + ‖f x‖
⊢ ‖F i x‖ ≤ mf + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
i : ι
hF : ∀ x ∈ s, ‖f x - F i x‖ < 1
x : α
hx : x ∈ s
⊢ ‖F i x‖ ≤ mf + 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | linarith [hF x hx, hf x hx] | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
i : ι
hF : ∀ x ∈ s, ‖f x - F i x‖ < 1
x : α
hx : x ∈ s
this : ‖F i x‖ ≤ ‖f x - F i x‖ + ‖f x‖
⊢ ‖F i x‖ ≤ mf + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
i : ι
hF : ∀ x ∈ s, ‖f x - F i x‖ < 1
x : α
hx : x ∈ s
this : ‖F i x‖ ≤ ‖f x - F i x‖ + ‖f x‖
⊢ ‖F i x‖ ≤ mf + 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | simpa [← norm_neg (F i x - f x)] using norm_add_le (F i x - f x) (f x) | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
i : ι
hF : ∀ x ∈ s, ‖f x - F i x‖ < 1
x : α
hx : x ∈ s
⊢ ‖F i x‖ ≤ ‖f x - F i x‖ + ‖f x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
hF✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖f x - F n x‖ < ε
i : ι
hF : ∀ x ∈ s, ‖f x - F i x‖ < 1
x : α
hx : x ∈ s
⊢ ‖F i x‖ ≤ ‖f x - F i x‖ + ‖f x‖
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | simp_rw [Metric.tendstoUniformlyOn_iff, dist_eq_norm] at hG | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg + 1 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hG : TendstoUniformlyOn G g p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg + 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | filter_upwards [hG 1 zero_lt_one] with i hG x hx | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg + 1 | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
i : ι
hG : ∀ x ∈ s, ‖g x - G i x‖ < 1
x : α
hx : x ∈ s
⊢ ‖G i x‖ ≤ mg + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
⊢ ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖G i x‖ ≤ mg + 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | have : ‖G i x‖ ≤ ‖g x - G i x‖ + ‖g x‖ := by
simpa [← norm_neg (G i x - g x)] using norm_add_le (G i x - g x) (g x) | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
i : ι
hG : ∀ x ∈ s, ‖g x - G i x‖ < 1
x : α
hx : x ∈ s
⊢ ‖G i x‖ ≤ mg + 1 | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
i : ι
hG : ∀ x ∈ s, ‖g x - G i x‖ < 1
x : α
hx : x ∈ s
this : ‖G i x‖ ≤ ‖g x - G i x‖ + ‖g x‖
⊢ ‖G i x‖ ≤ mg + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
i : ι
hG : ∀ x ∈ s, ‖g x - G i x‖ < 1
x : α
hx : x ∈ s
⊢ ‖G i x‖ ≤ mg + 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | linarith [hG x hx, hg x hx] | case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
i : ι
hG : ∀ x ∈ s, ‖g x - G i x‖ < 1
x : α
hx : x ∈ s
this : ‖G i x‖ ≤ ‖g x - G i x‖ + ‖g x‖
⊢ ‖G i x‖ ≤ mg + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
i : ι
hG : ∀ x ∈ s, ‖g x - G i x‖ < 1
x : α
hx : x ∈ s
this : ‖G i x‖ ≤ ‖g x - G i x‖ + ‖g x‖
⊢ ‖G i x‖ ≤ mg + 1
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_bound | [115, 1] | [131, 30] | simpa [← norm_neg (G i x - g x)] using norm_add_le (G i x - g x) (g x) | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
i : ι
hG : ∀ x ∈ s, ‖g x - G i x‖ < 1
x : α
hx : x ∈ s
⊢ ‖G i x‖ ≤ ‖g x - G i x‖ + ‖g x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x✝ y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
hF : TendstoUniformlyOn F f p s
hf : ∀ x ∈ s, ‖f x‖ ≤ mf
hg : ∀ x ∈ s, ‖g x‖ ≤ mg
h1 : ∀ᶠ (i : ι) in p, ∀ x ∈ s, ‖F i x‖ ≤ mf + 1
hG✝ : ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, ‖g x - G n x‖ < ε
i : ι
hG : ∀ x ∈ s, ‖g x - G i x‖ < 1
x : α
hx : x ∈ s
⊢ ‖G i x‖ ≤ ‖g x - G i x‖ + ‖g x‖
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.inv_of_compact | [135, 1] | [140, 93] | apply hF.inv | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hf : ContinuousOn f K
hK : IsCompact K
hfz : ∀ x ∈ K, f x ≠ 0
⊢ TendstoUniformlyOn F⁻¹ f⁻¹ p K | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hf : ContinuousOn f K
hK : IsCompact K
hfz : ∀ x ∈ K, f x ≠ 0
⊢ 𝓟 (f '' K) ⊓ 𝓝 0 = ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hf : ContinuousOn f K
hK : IsCompact K
hfz : ∀ x ∈ K, f x ≠ 0
⊢ TendstoUniformlyOn F⁻¹ f⁻¹ p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.inv_of_compact | [135, 1] | [140, 93] | rw [inf_comm, inf_principal_eq_bot] | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hf : ContinuousOn f K
hK : IsCompact K
hfz : ∀ x ∈ K, f x ≠ 0
⊢ 𝓟 (f '' K) ⊓ 𝓝 0 = ⊥ | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hf : ContinuousOn f K
hK : IsCompact K
hfz : ∀ x ∈ K, f x ≠ 0
⊢ (f '' K)ᶜ ∈ 𝓝 0 | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hf : ContinuousOn f K
hK : IsCompact K
hfz : ∀ x ∈ K, f x ≠ 0
⊢ 𝓟 (f '' K) ⊓ 𝓝 0 = ⊥
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.inv_of_compact | [135, 1] | [140, 93] | exact (hK.image_of_continuousOn hf).isClosed.compl_mem_nhds (λ ⟨z, h1, h2⟩ => hfz z h1 h2) | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hf : ContinuousOn f K
hK : IsCompact K
hfz : ∀ x ∈ K, f x ≠ 0
⊢ (f '' K)ᶜ ∈ 𝓝 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hf : ContinuousOn f K
hK : IsCompact K
hfz : ∀ x ∈ K, f x ≠ 0
⊢ (f '' K)ᶜ ∈ 𝓝 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_compact | [142, 1] | [154, 35] | by_cases h : K = ∅ | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
⊢ TendstoUniformlyOn (F * G) (f * g) p K | case pos
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : K = ∅
⊢ TendstoUniformlyOn (F * G) (f * g) p K
case neg
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : ¬K = ∅
⊢ TendstoUniformlyOn (F * G) (f * g) p K | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
⊢ TendstoUniformlyOn (F * G) (f * g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_compact | [142, 1] | [154, 35] | case pos => simpa only [h] using tendstoUniformlyOn_empty | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : K = ∅
⊢ TendstoUniformlyOn (F * G) (f * g) p K | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : K = ∅
⊢ TendstoUniformlyOn (F * G) (f * g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_compact | [142, 1] | [154, 35] | case neg =>
replace h : K.Nonempty := Set.nonempty_iff_ne_empty.2 h
have h2 : ContinuousOn (norm ∘ f) K := continuous_norm.comp_continuousOn hf
have h3 : ContinuousOn (norm ∘ g) K := continuous_norm.comp_continuousOn hg
obtain ⟨xf, _, h4⟩ : ∃ x ∈ K, ∀ y ∈ K, ‖f y‖ ≤ ‖f x‖ := hK.exists_forall_ge h h2
obtain ⟨xg, _, h5⟩ : ∃ x ∈ K, ∀ y ∈ K, ‖g y‖ ≤ ‖g x‖ := hK.exists_forall_ge h h3
exact hF.mul_of_bound hG h4 h5 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : ¬K = ∅
⊢ TendstoUniformlyOn (F * G) (f * g) p K | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : ¬K = ∅
⊢ TendstoUniformlyOn (F * G) (f * g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_compact | [142, 1] | [154, 35] | simpa only [h] using tendstoUniformlyOn_empty | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : K = ∅
⊢ TendstoUniformlyOn (F * G) (f * g) p K | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : K = ∅
⊢ TendstoUniformlyOn (F * G) (f * g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_compact | [142, 1] | [154, 35] | replace h : K.Nonempty := Set.nonempty_iff_ne_empty.2 h | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : ¬K = ∅
⊢ TendstoUniformlyOn (F * G) (f * g) p K | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
⊢ TendstoUniformlyOn (F * G) (f * g) p K | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : ¬K = ∅
⊢ TendstoUniformlyOn (F * G) (f * g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_compact | [142, 1] | [154, 35] | have h2 : ContinuousOn (norm ∘ f) K := continuous_norm.comp_continuousOn hf | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
⊢ TendstoUniformlyOn (F * G) (f * g) p K | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
⊢ TendstoUniformlyOn (F * G) (f * g) p K | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
⊢ TendstoUniformlyOn (F * G) (f * g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_compact | [142, 1] | [154, 35] | have h3 : ContinuousOn (norm ∘ g) K := continuous_norm.comp_continuousOn hg | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
⊢ TendstoUniformlyOn (F * G) (f * g) p K | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
h3 : ContinuousOn (norm ∘ g) K
⊢ TendstoUniformlyOn (F * G) (f * g) p K | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
⊢ TendstoUniformlyOn (F * G) (f * g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_compact | [142, 1] | [154, 35] | obtain ⟨xf, _, h4⟩ : ∃ x ∈ K, ∀ y ∈ K, ‖f y‖ ≤ ‖f x‖ := hK.exists_forall_ge h h2 | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
h3 : ContinuousOn (norm ∘ g) K
⊢ TendstoUniformlyOn (F * G) (f * g) p K | case intro.intro
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
h3 : ContinuousOn (norm ∘ g) K
xf : α
left✝ : xf ∈ K
h4 : ∀ y ∈ K, ‖f y‖ ≤ ‖f xf‖
⊢ TendstoUniformlyOn (F * G) (f * g) p K | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
h3 : ContinuousOn (norm ∘ g) K
⊢ TendstoUniformlyOn (F * G) (f * g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_compact | [142, 1] | [154, 35] | obtain ⟨xg, _, h5⟩ : ∃ x ∈ K, ∀ y ∈ K, ‖g y‖ ≤ ‖g x‖ := hK.exists_forall_ge h h3 | case intro.intro
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
h3 : ContinuousOn (norm ∘ g) K
xf : α
left✝ : xf ∈ K
h4 : ∀ y ∈ K, ‖f y‖ ≤ ‖f xf‖
⊢ TendstoUniformlyOn (F * G) (f * g) p K | case intro.intro.intro.intro
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
h3 : ContinuousOn (norm ∘ g) K
xf : α
left✝¹ : xf ∈ K
h4 : ∀ y ∈ K, ‖f y‖ ≤ ‖f xf‖
xg : α
left✝ : xg ∈ K
h5 : ∀ y ∈ K, ‖g y‖ ≤ ‖g xg‖
⊢ TendstoUniformlyOn (F * G) (f * g) p K | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
h3 : ContinuousOn (norm ∘ g) K
xf : α
left✝ : xf ∈ K
h4 : ∀ y ∈ K, ‖f y‖ ≤ ‖f xf‖
⊢ TendstoUniformlyOn (F * G) (f * g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.mul_of_compact | [142, 1] | [154, 35] | exact hF.mul_of_bound hG h4 h5 | case intro.intro.intro.intro
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
h3 : ContinuousOn (norm ∘ g) K
xf : α
left✝¹ : xf ∈ K
h4 : ∀ y ∈ K, ‖f y‖ ≤ ‖f xf‖
xg : α
left✝ : xg ∈ K
h5 : ∀ y ∈ K, ‖g y‖ ≤ ‖g xg‖
⊢ TendstoUniformlyOn (F * G) (f * g) p K | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hK : IsCompact K
h : Set.Nonempty K
h2 : ContinuousOn (norm ∘ f) K
h3 : ContinuousOn (norm ∘ g) K
xf : α
left✝¹ : xf ∈ K
h4 : ∀ y ∈ K, ‖f y‖ ≤ ‖f xf‖
xg : α
left✝ : xg ∈ K
h5 : ∀ y ∈ K, ‖g y‖ ≤ ‖g xg‖
⊢ TendstoUniformlyOn (F * G) (f * g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.div_of_compact | [156, 1] | [160, 99] | simpa [div_eq_mul_inv] using hF.mul_of_compact (hG.inv_of_compact hg hK hgK) hf (hg.inv₀ hgK) hK | 𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hgK : ∀ z ∈ K, g z ≠ 0
hK : IsCompact K
⊢ TendstoUniformlyOn (F / G) (f / g) p K | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type u_1
ι : Type u_2
α : Type u_3
s K : Set α
inst✝¹ : NormedField 𝕜
F G : ι → α → 𝕜
f g : α → 𝕜
x y : 𝕜
η η' : ℝ
p : Filter ι
mf mg : ℝ
inst✝ : TopologicalSpace α
hF : TendstoUniformlyOn F f p K
hG : TendstoUniformlyOn G g p K
hf : ContinuousOn f K
hg : ContinuousOn g K
hgK : ∀ z ∈ K, g z ≠ 0
hK : IsCompact K
⊢ TendstoUniformlyOn (F / G) (f / g) p K
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | Filter.Eventually.exists' | [167, 1] | [169, 89] | simpa [and_comm, exists_prop] using (frequently_nhdsWithin_iff.mp h.frequently).exists | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
P : ℝ → Prop
t₀ : ℝ
h : ∀ᶠ (t : ℝ) in 𝓝[>] t₀, P t
⊢ ∃ t > t₀, P t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
P : ℝ → Prop
t₀ : ℝ
h : ∀ᶠ (t : ℝ) in 𝓝[>] t₀, P t
⊢ ∃ t > t₀, P t
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | order_eq_zero_iff | [171, 1] | [180, 33] | rw [hp.locally_zero_iff] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ ∀ᶠ (z : ℂ) in 𝓝 z₀, f z = 0 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ ∀ᶠ (z : ℂ) in 𝓝 z₀, f z = 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | order_eq_zero_iff | [171, 1] | [180, 33] | by_cases h : p = 0 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0 | case pos
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : p = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0
case neg
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | order_eq_zero_iff | [171, 1] | [180, 33] | case pos => simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : p = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : p = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | order_eq_zero_iff | [171, 1] | [180, 33] | case neg =>
simp [FormalMultilinearSeries.order_eq_zero_iff h, h]
ext1
rw [hp.coeff_zero, hz₀]; rfl | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | order_eq_zero_iff | [171, 1] | [180, 33] | simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : p = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : p = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | order_eq_zero_iff | [171, 1] | [180, 33] | simp [FormalMultilinearSeries.order_eq_zero_iff h, h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
⊢ p 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
⊢ FormalMultilinearSeries.order p = 0 ↔ p = 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | order_eq_zero_iff | [171, 1] | [180, 33] | ext1 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
⊢ p 0 = 0 | case H
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
x✝ : Fin 0 → ℂ
⊢ (p 0) x✝ = 0 x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
⊢ p 0 = 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | order_eq_zero_iff | [171, 1] | [180, 33] | rw [hp.coeff_zero, hz₀] | case H
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
x✝ : Fin 0 → ℂ
⊢ (p 0) x✝ = 0 x✝ | case H
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
x✝ : Fin 0 → ℂ
⊢ 0 = 0 x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case H
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
x✝ : Fin 0 → ℂ
⊢ (p 0) x✝ = 0 x✝
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | order_eq_zero_iff | [171, 1] | [180, 33] | rfl | case H
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
x✝ : Fin 0 → ℂ
⊢ 0 = 0 x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
h : ¬p = 0
x✝ : Fin 0 → ℂ
⊢ 0 = 0 x✝
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | order_pos_iff | [182, 1] | [185, 57] | simp [pos_iff_ne_zero, (order_eq_zero_iff hp hz₀).not] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
⊢ 0 < FormalMultilinearSeries.order p ↔ ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
hz₀ : f z₀ = 0
⊢ 0 < FormalMultilinearSeries.order p ↔ ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | cindex_pos | [187, 1] | [191, 89] | obtain ⟨p, hp⟩ := h1 | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
h1 : AnalyticAt ℂ f z₀
h2 : f z₀ = 0
h3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r f ≠ 0 | case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
h2 : f z₀ = 0
h3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r f ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
h1 : AnalyticAt ℂ f z₀
h2 : f z₀ = 0
h3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r f ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | cindex_pos | [187, 1] | [191, 89] | filter_upwards [cindex_eventually_eq_order hp] with r h4 | case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
h2 : f z₀ = 0
h3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r f ≠ 0 | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r✝ : ℝ
U : Set ℂ
h2 : f z₀ = 0
h3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
r : ℝ
h4 : cindex z₀ r f = ↑(FormalMultilinearSeries.order p)
⊢ cindex z₀ r f ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r : ℝ
U : Set ℂ
h2 : f z₀ = 0
h3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
⊢ ∀ᶠ (r : ℝ) in 𝓝[>] 0, cindex z₀ r f ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | cindex_pos | [187, 1] | [191, 89] | simpa [h4, order_eq_zero_iff hp h2] using h3.frequently.filter_mono nhdsWithin_le_nhds | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r✝ : ℝ
U : Set ℂ
h2 : f z₀ = 0
h3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
r : ℝ
h4 : cindex z₀ r f = ↑(FormalMultilinearSeries.order p)
⊢ cindex z₀ r f ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p✝ : Filter ι
r✝ : ℝ
U : Set ℂ
h2 : f z₀ = 0
h3 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ 0
p : FormalMultilinearSeries ℂ ℂ ℂ
hp : HasFPowerSeriesAt f p z₀
r : ℝ
h4 : cindex z₀ r f = ↑(FormalMultilinearSeries.order p)
⊢ cindex z₀ r f ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | by_cases h : K = ∅ | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0 | case pos
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : K = ∅
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0
case neg
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | case pos => simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : K = ∅
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : K = ∅
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | case neg =>
obtain ⟨z₀, h1, h2⟩ : ∃ z₀ ∈ K, ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖ :=
hK.exists_forall_le (nonempty_iff_ne_empty.2 h) (continuous_norm.comp_continuousOn hf1)
have h3 := tendstoUniformlyOn_iff.1 F_conv (‖f z₀‖) (norm_pos_iff.2 (hf2 _ h1))
filter_upwards [h3] with n hn z hz h
specialize hn z hz
specialize h2 z hz
simp [h] at hn h2
linarith | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : K = ∅
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : K = ∅
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | obtain ⟨z₀, h1, h2⟩ : ∃ z₀ ∈ K, ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖ :=
hK.exists_forall_le (nonempty_iff_ne_empty.2 h) (continuous_norm.comp_continuousOn hf1) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0 | case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | have h3 := tendstoUniformlyOn_iff.1 F_conv (‖f z₀‖) (norm_pos_iff.2 (hf2 _ h1)) | case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0 | case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | filter_upwards [h3] with n hn z hz h | case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0 | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
hn : ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
z : ℂ
hz : z ∈ K
h : F n z = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
⊢ ∀ᶠ (n : ι) in p, ∀ z ∈ K, F n z ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | specialize hn z hz | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
hn : ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
z : ℂ
hz : z ∈ K
h : F n z = 0
⊢ False | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
z : ℂ
hz : z ∈ K
h : F n z = 0
hn : dist (f z) (F n z) < ‖f z₀‖
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
hn : ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
z : ℂ
hz : z ∈ K
h : F n z = 0
⊢ False
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | specialize h2 z hz | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
z : ℂ
hz : z ∈ K
h : F n z = 0
hn : dist (f z) (F n z) < ‖f z₀‖
⊢ False | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
z : ℂ
hz : z ∈ K
h : F n z = 0
hn : dist (f z) (F n z) < ‖f z₀‖
h2 : ‖f z₀‖ ≤ ‖f z‖
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h2 : ∀ z ∈ K, ‖f z₀‖ ≤ ‖f z‖
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
z : ℂ
hz : z ∈ K
h : F n z = 0
hn : dist (f z) (F n z) < ‖f z₀‖
⊢ False
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | simp [h] at hn h2 | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
z : ℂ
hz : z ∈ K
h : F n z = 0
hn : dist (f z) (F n z) < ‖f z₀‖
h2 : ‖f z₀‖ ≤ ‖f z‖
⊢ False | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
z : ℂ
hz : z ∈ K
h : F n z = 0
h2 : Complex.abs (f z₀) ≤ Complex.abs (f z)
hn : Complex.abs (f z) < Complex.abs (f z₀)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
z : ℂ
hz : z ∈ K
h : F n z = 0
hn : dist (f z) (F n z) < ‖f z₀‖
h2 : ‖f z₀‖ ≤ ‖f z‖
⊢ False
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | hurwitz2_1 | [194, 1] | [207, 13] | linarith | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
z : ℂ
hz : z ∈ K
h : F n z = 0
h2 : Complex.abs (f z₀) ≤ Complex.abs (f z)
hn : Complex.abs (f z) < Complex.abs (f z₀)
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀✝ : ℂ
p : Filter ι
r : ℝ
U K : Set ℂ
hK : IsCompact K
F_conv : TendstoUniformlyOn F f p K
hf1 : ContinuousOn f K
hf2 : ∀ z ∈ K, f z ≠ 0
h✝ : ¬K = ∅
z₀ : ℂ
h1 : z₀ ∈ K
h3 : ∀ᶠ (n : ι) in p, ∀ x ∈ K, dist (f x) (F n x) < ‖f z₀‖
n : ι
z : ℂ
hz : z ∈ K
h : F n z = 0
h2 : Complex.abs (f z₀) ≤ Complex.abs (f z)
hn : Complex.abs (f z) < Complex.abs (f z₀)
⊢ False
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | by_cases h : NeBot p | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z)) | case pos
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))
case neg
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : ¬NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | case neg => simp at h; simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : ¬NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : ¬NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | case pos =>
have f_cont : ContinuousOn f (sphere z₀ r) := F_conv.continuousOn F_cont
rw [Metric.tendsto_nhds]
intro ε hε
have twopir_ne_zero : 2 * Real.pi * r ≠ 0 := by simp [Real.pi_ne_zero, hr.ne.symm]
have : (2 * Real.pi * r)⁻¹ * ε > 0 :=
mul_pos (inv_pos.mpr (mul_pos (mul_pos two_pos Real.pi_pos) hr)) hε.lt
filter_upwards [tendstoUniformlyOn_iff.mp F_conv ((2 * Real.pi * r)⁻¹ * ε) this, F_cont] with n h h'
simp_rw [dist_comm (f _) _, Complex.dist_eq, ← Complex.norm_eq_abs] at h
rw [Complex.dist_eq, ← circleIntegral.integral_sub (h'.circleIntegrable hr.le) (f_cont.circleIntegrable hr.le),
← Complex.norm_eq_abs]
have : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε := by
have : z₀ + r ∈ sphere z₀ r := by simp [hr.le, Real.norm_eq_abs]
exact ⟨z₀ + r, this, h _ this⟩
convert circleIntegral.norm_integral_lt_of_norm_le_const_of_lt hr (h'.sub f_cont) (λ z hz => (h z hz).le) this
field_simp [hr.ne, Real.pi_ne_zero, two_ne_zero]; ring | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | simp at h | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : ¬NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z)) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : p = ⊥
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : ¬NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | simp [h] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : p = ⊥
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : p = ⊥
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | have f_cont : ContinuousOn f (sphere z₀ r) := F_conv.continuousOn F_cont | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z)) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z)) | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | rw [Metric.tendsto_nhds] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z)) | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
⊢ ∀ ε > 0, ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | intro ε hε | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
⊢ ∀ ε > 0, ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
⊢ ∀ ε > 0, ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | have twopir_ne_zero : 2 * Real.pi * r ≠ 0 := by simp [Real.pi_ne_zero, hr.ne.symm] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
⊢ ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | have : (2 * Real.pi * r)⁻¹ * ε > 0 :=
mul_pos (inv_pos.mpr (mul_pos (mul_pos two_pos Real.pi_pos) hr)) hε.lt | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
⊢ ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
⊢ ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
⊢ ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | filter_upwards [tendstoUniformlyOn_iff.mp F_conv ((2 * Real.pi * r)⁻¹ * ε) this, F_cont] with n h h' | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
⊢ ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h : ∀ x ∈ sphere z₀ r, dist (f x) (F n x) < (2 * Real.pi * r)⁻¹ * ε
h' : ContinuousOn (F n) (sphere z₀ r)
⊢ dist (∮ (z : ℂ) in C(z₀, r), F n z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
⊢ ∀ᶠ (x : ι) in p, dist (∮ (z : ℂ) in C(z₀, r), F x z) (∮ (z : ℂ) in C(z₀, r), f z) < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | simp_rw [dist_comm (f _) _, Complex.dist_eq, ← Complex.norm_eq_abs] at h | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h : ∀ x ∈ sphere z₀ r, dist (f x) (F n x) < (2 * Real.pi * r)⁻¹ * ε
h' : ContinuousOn (F n) (sphere z₀ r)
⊢ dist (∮ (z : ℂ) in C(z₀, r), F n z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ dist (∮ (z : ℂ) in C(z₀, r), F n z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h : ∀ x ∈ sphere z₀ r, dist (f x) (F n x) < (2 * Real.pi * r)⁻¹ * ε
h' : ContinuousOn (F n) (sphere z₀ r)
⊢ dist (∮ (z : ℂ) in C(z₀, r), F n z) (∮ (z : ℂ) in C(z₀, r), f z) < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | rw [Complex.dist_eq, ← circleIntegral.integral_sub (h'.circleIntegrable hr.le) (f_cont.circleIntegrable hr.le),
← Complex.norm_eq_abs] | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ dist (∮ (z : ℂ) in C(z₀, r), F n z) (∮ (z : ℂ) in C(z₀, r), f z) < ε | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ‖∮ (z : ℂ) in C(z₀, r), F n z - f z‖ < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ dist (∮ (z : ℂ) in C(z₀, r), F n z) (∮ (z : ℂ) in C(z₀, r), f z) < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | have : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε := by
have : z₀ + r ∈ sphere z₀ r := by simp [hr.le, Real.norm_eq_abs]
exact ⟨z₀ + r, this, h _ this⟩ | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ‖∮ (z : ℂ) in C(z₀, r), F n z - f z‖ < ε | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
this : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ‖∮ (z : ℂ) in C(z₀, r), F n z - f z‖ < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ‖∮ (z : ℂ) in C(z₀, r), F n z - f z‖ < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | convert circleIntegral.norm_integral_lt_of_norm_le_const_of_lt hr (h'.sub f_cont) (λ z hz => (h z hz).le) this | case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
this : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ‖∮ (z : ℂ) in C(z₀, r), F n z - f z‖ < ε | case h.e'_4
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
this : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ε = 2 * Real.pi * r * ((2 * Real.pi * r)⁻¹ * ε) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
this : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ‖∮ (z : ℂ) in C(z₀, r), F n z - f z‖ < ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | field_simp [hr.ne, Real.pi_ne_zero, two_ne_zero] | case h.e'_4
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
this : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ε = 2 * Real.pi * r * ((2 * Real.pi * r)⁻¹ * ε) | case h.e'_4
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
this : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ε * (2 * Real.pi * r) = 2 * Real.pi * r * ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
this : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ε = 2 * Real.pi * r * ((2 * Real.pi * r)⁻¹ * ε)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | ring | case h.e'_4
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
this : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ε * (2 * Real.pi * r) = 2 * Real.pi * r * ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
this : ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ε * (2 * Real.pi * r) = 2 * Real.pi * r * ε
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | simp [Real.pi_ne_zero, hr.ne.symm] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
⊢ 2 * Real.pi * r ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
⊢ 2 * Real.pi * r ≠ 0
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/hurwitz.lean | TendstoUniformlyOn.tendsto_circle_integral | [209, 1] | [231, 59] | have : z₀ + r ∈ sphere z₀ r := by simp [hr.le, Real.norm_eq_abs] | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε | ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this✝ : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
this : z₀ + ↑r ∈ sphere z₀ r
⊢ ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h✝ : NeBot p
f_cont : ContinuousOn f (sphere z₀ r)
ε : ℝ
hε : ε > 0
twopir_ne_zero : 2 * Real.pi * r ≠ 0
this : (2 * Real.pi * r)⁻¹ * ε > 0
n : ι
h' : ContinuousOn (F n) (sphere z₀ r)
h : ∀ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
⊢ ∃ x ∈ sphere z₀ r, ‖F n x - f x‖ < (2 * Real.pi * r)⁻¹ * ε
TACTIC:
|
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