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stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
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| start
stringlengths 6
10
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stringlengths 6
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stringlengths 1
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2.09M
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | use(h _ (qt (m n (le_refl _)))).1, fun k nk ↦ (h _ (qt (m k nk))).2 | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : ∀ (k : ℕ), n ≤ k → s.potential c ((f c)^[k] z) < q
⊢ (c, (f c)^[n] z) ∈ s.near ∧ ∀ (k : ℕ), n ≤ k → mfderiv I I (s.bottcherNear c) ((f c)^[k] z) ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : ∀ (k : ℕ), n ≤ k → s.potential c ((f c)^[k] z) < q
⊢ (c, (f c)^[n] z) ∈ s.near ∧ ∀ (k : ℕ), n ≤ k → mfderiv I I (s.bottcherNear c) ((f c)^[k] z) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | apply
(mfderiv_ne_zero_eventually (s.bottcherNear_holomorphic _ (s.mem_near c)).along_snd
(s.bottcherNear_mfderiv_ne_zero c)).mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
⊢ ∀ᶠ (z : S) in 𝓝 a, (c, z) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) z ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
⊢ ∀ᶠ (x : S) in 𝓝 a,
mfderiv I I (fun y => s.bottcherNear c y) x ≠ 0 → (c, x) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
⊢ ∀ᶠ (z : S) in 𝓝 a, (c, z) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) z ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | apply ((s.isOpen_near.snd_preimage c).eventually_mem (s.mem_near c)).mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
⊢ ∀ᶠ (x : S) in 𝓝 a,
mfderiv I I (fun y => s.bottcherNear c y) x ≠ 0 → (c, x) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) x ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
⊢ ∀ᶠ (x : S) in 𝓝 a,
x ∈ {b | (c, b) ∈ s.near} →
mfderiv I I (fun y => s.bottcherNear c y) x ≠ 0 → (c, x) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
⊢ ∀ᶠ (x : S) in 𝓝 a,
mfderiv I I (fun y => s.bottcherNear c y) x ≠ 0 → (c, x) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | refine eventually_of_forall fun z m nc ↦ ?_ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
⊢ ∀ᶠ (x : S) in 𝓝 a,
x ∈ {b | (c, b) ∈ s.near} →
mfderiv I I (fun y => s.bottcherNear c y) x ≠ 0 → (c, x) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) x ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
z : S
m : z ∈ {b | (c, b) ∈ s.near}
nc : mfderiv I I (fun y => s.bottcherNear c y) z ≠ 0
⊢ (c, z) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) z ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
⊢ ∀ᶠ (x : S) in 𝓝 a,
x ∈ {b | (c, b) ∈ s.near} →
mfderiv I I (fun y => s.bottcherNear c y) x ≠ 0 → (c, x) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | use m, nc | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
z : S
m : z ∈ {b | (c, b) ∈ s.near}
nc : mfderiv I I (fun y => s.bottcherNear c y) z ≠ 0
⊢ (c, z) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) z ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
z : S
m : z ∈ {b | (c, b) ∈ s.near}
nc : mfderiv I I (fun y => s.bottcherNear c y) z ≠ 0
⊢ (c, z) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) z ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | intro k nk | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
⊢ ∀ (k : ℕ), n ≤ k → s.potential c ((f c)^[k] z) < q | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
⊢ s.potential c ((f c)^[k] z) < q | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
⊢ ∀ (k : ℕ), n ≤ k → s.potential c ((f c)^[k] z) < q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | refine lt_of_le_of_lt ?_ pq | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
⊢ s.potential c ((f c)^[k] z) < q | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
⊢ s.potential c ((f c)^[k] z) ≤ p ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
⊢ s.potential c ((f c)^[k] z) < q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | simp only [s.potential_eqn_iter] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
⊢ s.potential c ((f c)^[k] z) ≤ p ^ n | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
⊢ s.potential c z ^ d ^ k ≤ p ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
⊢ s.potential c ((f c)^[k] z) ≤ p ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | have dn := (Nat.lt_pow_self s.d1 k).le | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
⊢ s.potential c z ^ d ^ k ≤ p ^ n | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
dn : k ≤ d ^ k
⊢ s.potential c z ^ d ^ k ≤ p ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
⊢ s.potential c z ^ d ^ k ≤ p ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | apply _root_.trans (pow_le_pow_of_le_one s.potential_nonneg s.potential_le_one dn) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
dn : k ≤ d ^ k
⊢ s.potential c z ^ d ^ k ≤ p ^ n | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
dn : k ≤ d ^ k
⊢ s.potential c z ^ k ≤ p ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
dn : k ≤ d ^ k
⊢ s.potential c z ^ d ^ k ≤ p ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | refine _root_.trans (pow_le_pow_left s.potential_nonneg m _) ?_ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
dn : k ≤ d ^ k
⊢ s.potential c z ^ k ≤ p ^ n | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
dn : k ≤ d ^ k
⊢ p ^ k ≤ p ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
dn : k ≤ d ^ k
⊢ s.potential c z ^ k ≤ p ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.has_nice_n | [477, 1] | [495, 70] | exact pow_le_pow_of_le_one (_root_.trans s.potential_nonneg m) p1.le nk | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
dn : k ≤ d ^ k
⊢ p ^ k ≤ p ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n✝ : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
t : Set S
h : ∀ y ∈ t, (c, y) ∈ s.near ∧ mfderiv I I (s.bottcherNear c) y ≠ 0
q : ℝ
qp : 0 < q
qt : {z | s.potential c z < q} ⊆ t
n : ℕ
pq : p ^ n < q
z : S
m : s.potential c z ≤ p
k : ℕ
nk : n ≤ k
dn : k ≤ d ^ k
⊢ p ^ k ≤ p ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.nice_np | [501, 1] | [505, 42] | have q : p < 1 ∧ OnePreimage s := ⟨p1, op⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
⊢ s.IsNiceN c p (s.np c p) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
q : p < 1 ∧ OnePreimage s
⊢ s.IsNiceN c p (s.np c p) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
⊢ s.IsNiceN c p (s.np c p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.nice_np | [501, 1] | [505, 42] | simp only [Super.np, q, true_and_iff, dif_pos] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
q : p < 1 ∧ OnePreimage s
⊢ s.IsNiceN c p (s.np c p) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
q : p < 1 ∧ OnePreimage s
⊢ s.IsNiceN c p (Nat.find ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
q : p < 1 ∧ OnePreimage s
⊢ s.IsNiceN c p (s.np c p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.nice_np | [501, 1] | [505, 42] | exact Nat.find_spec (s.has_nice_n c p1) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
q : p < 1 ∧ OnePreimage s
⊢ s.IsNiceN c p (Nat.find ⋯) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : ℝ
p1 : p < 1
op : OnePreimage s
q : p < 1 ∧ OnePreimage s
⊢ s.IsNiceN c p (Nat.find ⋯)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.np_zero | [507, 1] | [508, 100] | simp only [Super.np, zero_lt_one, op, true_and_iff, dif_pos, Nat.find_eq_zero, Super.isNice_zero] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
op : OnePreimage s
⊢ s.np c 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
op : OnePreimage s
⊢ s.np c 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.np_mono | [510, 1] | [515, 87] | have p01 : p0 < 1 := lt_of_le_of_lt le p11 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
⊢ s.np c p0 ≤ s.np c p1 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
⊢ s.np c p0 ≤ s.np c p1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
⊢ s.np c p0 ≤ s.np c p1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.np_mono | [510, 1] | [515, 87] | have e : s.np c p0 = Nat.find (s.has_nice_n c p01) := by
simp only [Super.np, p01, op, true_and_iff, dif_pos] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
⊢ s.np c p0 ≤ s.np c p1 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
e : s.np c p0 = Nat.find ⋯
⊢ s.np c p0 ≤ s.np c p1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
⊢ s.np c p0 ≤ s.np c p1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.np_mono | [510, 1] | [515, 87] | rw [e] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
e : s.np c p0 = Nat.find ⋯
⊢ s.np c p0 ≤ s.np c p1 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
e : s.np c p0 = Nat.find ⋯
⊢ Nat.find ⋯ ≤ s.np c p1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
e : s.np c p0 = Nat.find ⋯
⊢ s.np c p0 ≤ s.np c p1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.np_mono | [510, 1] | [515, 87] | apply Nat.find_min' | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
e : s.np c p0 = Nat.find ⋯
⊢ Nat.find ⋯ ≤ s.np c p1 | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
e : s.np c p0 = Nat.find ⋯
⊢ s.IsNiceN c p0 (s.np c p1) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
e : s.np c p0 = Nat.find ⋯
⊢ Nat.find ⋯ ≤ s.np c p1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.np_mono | [510, 1] | [515, 87] | exact fun z zp ↦ s.nice_np c p11 _ (_root_.trans zp le) | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
e : s.np c p0 = Nat.find ⋯
⊢ s.IsNiceN c p0 (s.np c p1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
e : s.np c p0 = Nat.find ⋯
⊢ s.IsNiceN c p0 (s.np c p1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.np_mono | [510, 1] | [515, 87] | simp only [Super.np, p01, op, true_and_iff, dif_pos] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
⊢ s.np c p0 = Nat.find ⋯ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p0 p1 : ℝ
le : p0 ≤ p1
p11 : p1 < 1
op : OnePreimage s
p01 : p0 < 1
⊢ s.np c p0 = Nat.find ⋯
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | have c0 : 0 < abs c := lt_trans (by norm_num) c16 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ ⋯.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
⊢ ⋯.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ ⋯.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | have z0 : 0 < abs z := lt_of_lt_of_le c0 cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
⊢ ⋯.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
⊢ ⋯.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
⊢ ⋯.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | set s := superF d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
⊢ ⋯.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
⊢ ⋯.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | set t := closedBall (0 : ℂ) (abs c)⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | suffices e : EqOn (fun z : ℂ ↦ s.bottcher c (z : 𝕊)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t by
have z0' : z ≠ 0 := Complex.abs.ne_zero_iff.mp z0.ne'
convert @e z⁻¹ _; rw [inv_coe (inv_ne_zero z0'), inv_inv]
simp only [mem_closedBall, Complex.dist_eq, sub_zero, map_inv₀, inv_le_inv z0 c0, cz, t] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
⊢ EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | have a0 : HolomorphicOn I I (fun z : ℂ ↦ s.bottcher c (z : 𝕊)⁻¹) t := by
intro z m
refine (s.bottcher_holomorphicOn _ ?_).along_snd.comp (holomorphic_inv.comp holomorphic_coe _)
simp only [mem_closedBall, Complex.dist_eq, sub_zero, t] at m
by_cases z0 : z = 0; simp only [z0, coe_zero, inv_zero']; exact s.post_a c
rw [inv_coe z0]; refine postcritical_large (by linarith) ?_
rwa [map_inv₀, le_inv c0]; exact Complex.abs.pos z0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
⊢ EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
⊢ EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
⊢ EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | have a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t := by
intro z m; apply AnalyticAt.holomorphicAt
apply bottcherNear_analytic_z (superNearF d c)
simp only [mem_setOf, mem_closedBall, Complex.dist_eq, sub_zero, t] at m ⊢
refine lt_of_le_of_lt m ?_
refine inv_lt_inv_of_lt (lt_of_lt_of_le (by norm_num) (le_max_left _ _)) ?_
exact max_lt c16 (half_lt_self (lt_trans (by norm_num) c16)) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
⊢ EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
⊢ EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | refine (a0.eq_of_locally_eq a1 (convex_closedBall _ _).isPreconnected ?_).self_of_nhdsSet | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ ∃ x ∈ t, (𝓝 x).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | use 0, mem_closedBall_self (by bound) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ ∃ x ∈ t, (𝓝 x).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ ∃ x ∈ t, (𝓝 x).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | have e : ∀ᶠ z in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (z : 𝕊)⁻¹ := by
simp only [Super.bottcherNear, extChartAt_inf_apply, inv_inv, toComplex_coe,
RiemannSphere.inv_inf, toComplex_zero, sub_zero, Super.fl, eq_self_iff_true,
Filter.eventually_true] | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | refine Filter.EventuallyEq.trans ?_ (Filter.EventuallyEq.symm e) | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) fun x => s.bottcherNear c (↑x)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | have i : Tendsto (fun z : ℂ ↦ (z : 𝕊)⁻¹) (𝓝 0) (𝓝 ∞) := by
have h : ContinuousAt (fun z : ℂ ↦ (z : 𝕊)⁻¹) 0 :=
(RiemannSphere.continuous_inv.comp continuous_coe).continuousAt
simp only [ContinuousAt, coe_zero, inv_zero'] at h; exact h | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) fun x => s.bottcherNear c (↑x)⁻¹ | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
i : Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞)
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) fun x => s.bottcherNear c (↑x)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) fun x => s.bottcherNear c (↑x)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | exact i.eventually (s.bottcher_eq_bottcherNear c) | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
i : Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞)
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) fun x => s.bottcherNear c (↑x)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
i : Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞)
⊢ (𝓝 0).EventuallyEq (fun z => s.bottcher c (↑z)⁻¹) fun x => s.bottcherNear c (↑x)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 16 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 16
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | have z0' : z ≠ 0 := Complex.abs.ne_zero_iff.mp z0.ne' | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | convert @e z⁻¹ _ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹ | case h.e'_2.h.e'_13
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ ↑z = (↑z⁻¹)⁻¹
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ z⁻¹ ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ s.bottcher c ↑z = bottcherNear (fl (f d) ∞ c) d z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | rw [inv_coe (inv_ne_zero z0'), inv_inv] | case h.e'_2.h.e'_13
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ ↑z = (↑z⁻¹)⁻¹
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ z⁻¹ ∈ t | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ z⁻¹ ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_13
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ ↑z = (↑z⁻¹)⁻¹
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ z⁻¹ ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | simp only [mem_closedBall, Complex.dist_eq, sub_zero, map_inv₀, inv_le_inv z0 c0, cz, t] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ z⁻¹ ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
e : EqOn (fun z => s.bottcher c (↑z)⁻¹) (bottcherNear (fl (f d) ∞ c) d) t
z0' : z ≠ 0
⊢ z⁻¹ ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | intro z m | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
⊢ HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : z ∈ t
⊢ HolomorphicAt I I (fun z => s.bottcher c (↑z)⁻¹) z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
⊢ HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | refine (s.bottcher_holomorphicOn _ ?_).along_snd.comp (holomorphic_inv.comp holomorphic_coe _) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : z ∈ t
⊢ HolomorphicAt I I (fun z => s.bottcher c (↑z)⁻¹) z | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : z ∈ t
⊢ (c, (↑z)⁻¹) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : z ∈ t
⊢ HolomorphicAt I I (fun z => s.bottcher c (↑z)⁻¹) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | simp only [mem_closedBall, Complex.dist_eq, sub_zero, t] at m | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : z ∈ t
⊢ (c, (↑z)⁻¹) ∈ s.post | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ (c, (↑z)⁻¹) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : z ∈ t
⊢ (c, (↑z)⁻¹) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | by_cases z0 : z = 0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ (c, (↑z)⁻¹) ∈ s.post | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ (c, (↑z)⁻¹) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | simp only [z0, coe_zero, inv_zero'] | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : z = 0
⊢ (c, ∞) ∈ s.post
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | exact s.post_a c | case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : z = 0
⊢ (c, ∞) ∈ s.post
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : z = 0
⊢ (c, ∞) ∈ s.post
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | rw [inv_coe z0] | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, ↑z⁻¹) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, (↑z)⁻¹) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | refine postcritical_large (by linarith) ?_ | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, ↑z⁻¹) ∈ s.post | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ Complex.abs (c, ↑z⁻¹).1 ≤ Complex.abs z⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ (c, ↑z⁻¹) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | rwa [map_inv₀, le_inv c0] | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ Complex.abs (c, ↑z⁻¹).1 ≤ Complex.abs z⁻¹ | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ 0 < Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ Complex.abs (c, ↑z⁻¹).1 ≤ Complex.abs z⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | exact Complex.abs.pos z0 | case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ 0 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ 0 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | linarith | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ 4 ≤ Complex.abs (c, ↑z⁻¹).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0✝ : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
z0 : ¬z = 0
⊢ 4 ≤ Complex.abs (c, ↑z⁻¹).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | intro z m | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
⊢ HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : z ∈ t
⊢ HolomorphicAt I I (bottcherNear (fl (f d) ∞ c) d) z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
⊢ HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | apply AnalyticAt.holomorphicAt | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : z ∈ t
⊢ HolomorphicAt I I (bottcherNear (fl (f d) ∞ c) d) z | case fa
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : z ∈ t
⊢ AnalyticAt ℂ (bottcherNear (fl (f d) ∞ c) d) z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : z ∈ t
⊢ HolomorphicAt I I (bottcherNear (fl (f d) ∞ c) d) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | apply bottcherNear_analytic_z (superNearF d c) | case fa
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : z ∈ t
⊢ AnalyticAt ℂ (bottcherNear (fl (f d) ∞ c) d) z | case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : z ∈ t
⊢ z ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : z ∈ t
⊢ AnalyticAt ℂ (bottcherNear (fl (f d) ∞ c) d) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | simp only [mem_setOf, mem_closedBall, Complex.dist_eq, sub_zero, t] at m ⊢ | case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : z ∈ t
⊢ z ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} | case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : z ∈ t
⊢ z ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | refine lt_of_le_of_lt m ?_ | case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹ | case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ (Complex.abs c)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | refine inv_lt_inv_of_lt (lt_of_lt_of_le (by norm_num) (le_max_left _ _)) ?_ | case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ (Complex.abs c)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹ | case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ max 16 (Complex.abs c / 2) < Complex.abs c | Please generate a tactic in lean4 to solve the state.
STATE:
case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ (Complex.abs c)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | exact max_lt c16 (half_lt_self (lt_trans (by norm_num) c16)) | case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ max 16 (Complex.abs c / 2) < Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case fa.a
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ max 16 (Complex.abs c / 2) < Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ 0 < 16 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z✝ : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z✝
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z✝
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
z : ℂ
m : Complex.abs z ≤ (Complex.abs c)⁻¹
⊢ 0 < 16
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ 0 ≤ (Complex.abs c)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ 0 ≤ (Complex.abs c)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | simp only [Super.bottcherNear, extChartAt_inf_apply, inv_inv, toComplex_coe,
RiemannSphere.inv_inf, toComplex_zero, sub_zero, Super.fl, eq_self_iff_true,
Filter.eventually_true] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
⊢ ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | have h : ContinuousAt (fun z : ℂ ↦ (z : 𝕊)⁻¹) 0 :=
(RiemannSphere.continuous_inv.comp continuous_coe).continuousAt | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
⊢ Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
h : ContinuousAt (fun z => (↑z)⁻¹) 0
⊢ Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
⊢ Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | simp only [ContinuousAt, coe_zero, inv_zero'] at h | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
h : ContinuousAt (fun z => (↑z)⁻¹) 0
⊢ Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
h : Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞)
⊢ Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
h : ContinuousAt (fun z => (↑z)⁻¹) 0
⊢ Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | bottcher_eq_bottcherNear_z | [39, 1] | [74, 52] | exact h | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
h : Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞)
⊢ Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
c0 : 0 < Complex.abs c
z0 : 0 < Complex.abs z
s : Super (f d) d ∞ := superF d
t : Set ℂ := closedBall 0 (Complex.abs c)⁻¹
a0 : HolomorphicOn I I (fun z => s.bottcher c (↑z)⁻¹) t
a1 : HolomorphicOn I I (bottcherNear (fl (f d) ∞ c) d) t
e : ∀ᶠ (z : ℂ) in 𝓝 0, bottcherNear (fl (f d) ∞ c) d z = s.bottcherNear c (↑z)⁻¹
h : Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞)
⊢ Tendsto (fun z => (↑z)⁻¹) (𝓝 0) (𝓝 ∞)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | inv_mem_t | [82, 1] | [86, 83] | simp only [mem_setOf, map_inv₀] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ z⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ (Complex.abs z)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ z⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | inv_mem_t | [82, 1] | [86, 83] | refine inv_lt_inv_of_lt (lt_of_lt_of_le (by norm_num) (le_max_left _ _)) ?_ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ (Complex.abs z)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ max 16 (Complex.abs c / 2) < Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ (Complex.abs z)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | inv_mem_t | [82, 1] | [86, 83] | exact lt_of_lt_of_le (max_lt c16 (half_lt_self (lt_trans (by norm_num) c16))) cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ max 16 (Complex.abs c / 2) < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ max 16 (Complex.abs c / 2) < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | inv_mem_t | [82, 1] | [86, 83] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 16 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 16
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | set s := superF d | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | have z0 : abs z ≠ 0 := (lt_of_lt_of_le (lt_trans (by norm_num) c16) cz).ne' | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | have i8 : (abs z)⁻¹ ≤ 1 / 8 := by
rw [one_div]; apply inv_le_inv_of_le; norm_num
exact le_trans (by norm_num) (le_trans c16.le cz) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | have i1 : (abs z)⁻¹ ≤ 1 := le_trans i8 (by norm_num) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | simp only [term] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
⊢ Complex.abs (g (fl (f d) ∞ c) d ((fl (f d) ∞ c)^[n] z⁻¹) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤
2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
⊢ Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | have wc := iterates_converge (superNearF d c) n (inv_mem_t c16 cz) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
⊢ Complex.abs (g (fl (f d) ∞ c) d ((fl (f d) ∞ c)^[n] z⁻¹) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤
2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
wc : Complex.abs ((fl (f d) ∞ c)^[n] z⁻¹) ≤ (5 / 8) ^ n * Complex.abs z⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d ((fl (f d) ∞ c)^[n] z⁻¹) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤
2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
⊢ Complex.abs (g (fl (f d) ∞ c) d ((fl (f d) ∞ c)^[n] z⁻¹) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤
2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | generalize hw : (fl (f d) ∞ c)^[n] z⁻¹ = w | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
wc : Complex.abs ((fl (f d) ∞ c)^[n] z⁻¹) ≤ (5 / 8) ^ n * Complex.abs z⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d ((fl (f d) ∞ c)^[n] z⁻¹) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤
2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
wc : Complex.abs ((fl (f d) ∞ c)^[n] z⁻¹) ≤ (5 / 8) ^ n * Complex.abs z⁻¹
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
wc : Complex.abs ((fl (f d) ∞ c)^[n] z⁻¹) ≤ (5 / 8) ^ n * Complex.abs z⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d ((fl (f d) ∞ c)^[n] z⁻¹) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤
2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | rw [hw] at wc | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
wc : Complex.abs ((fl (f d) ∞ c)^[n] z⁻¹) ≤ (5 / 8) ^ n * Complex.abs z⁻¹
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * Complex.abs z⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
wc : Complex.abs ((fl (f d) ∞ c)^[n] z⁻¹) ≤ (5 / 8) ^ n * Complex.abs z⁻¹
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | replace wc : abs w ≤ (abs z)⁻¹ := by
rw [map_inv₀] at wc
exact le_trans wc (mul_le_of_le_one_left (inv_nonneg.mpr (Complex.abs.nonneg _)) (by bound)) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * Complex.abs z⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * Complex.abs z⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | have cw : abs (c * w ^ d) ≤ (abs z)⁻¹ := by
simp only [Complex.abs.map_mul, Complex.abs.map_pow]
calc abs c * abs w ^ d
_ ≤ abs z * (abs z)⁻¹ ^ d := by bound
_ ≤ abs z * (abs z)⁻¹ ^ 2 := by bound
_ = (abs z)⁻¹ := by rw [pow_two]; field_simp [z0] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | have cw2 : abs (c * w ^ d) ≤ 1 / 2 := le_trans cw (le_trans i8 (by norm_num)) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | simp only [gl_f, gl] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d)⁻¹ ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | rw [Complex.inv_cpow, ← Complex.cpow_neg] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d)⁻¹ ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d)⁻¹ ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | swap | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi | case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
⊢ 0 < 16 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
⊢ 0 < 16
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | rw [one_div] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ (Complex.abs z)⁻¹ ≤ 1 / 8 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ (Complex.abs z)⁻¹ ≤ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | apply inv_le_inv_of_le | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹ | case ha
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 0 < 8
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | norm_num | case ha
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 0 < 8
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z | case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 0 < 8
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | exact le_trans (by norm_num) (le_trans c16.le cz) | case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ 16 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ 16
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ 1 / 8 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ 1 / 8 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | rw [map_inv₀] at wc | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * Complex.abs z⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs w ≤ (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs w ≤ (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * Complex.abs z⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs w ≤ (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | exact le_trans wc (mul_le_of_le_one_left (inv_nonneg.mpr (Complex.abs.nonneg _)) (by bound)) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs w ≤ (Complex.abs z)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs w ≤ (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ (5 / 8) ^ n ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ (5 / 8) ^ n ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | simp only [Complex.abs.map_mul, Complex.abs.map_pow] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs c * Complex.abs w ^ d ≤ (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | calc abs c * abs w ^ d
_ ≤ abs z * (abs z)⁻¹ ^ d := by bound
_ ≤ abs z * (abs z)⁻¹ ^ 2 := by bound
_ = (abs z)⁻¹ := by rw [pow_two]; field_simp [z0] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs c * Complex.abs w ^ d ≤ (Complex.abs z)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs c * Complex.abs w ^ d ≤ (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs c * Complex.abs w ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs c * Complex.abs w ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | rw [pow_two] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ 2 = (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * ((Complex.abs z)⁻¹ * (Complex.abs z)⁻¹) = (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ 2 = (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | field_simp [z0] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * ((Complex.abs z)⁻¹ * (Complex.abs z)⁻¹) = (Complex.abs z)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * ((Complex.abs z)⁻¹ * (Complex.abs z)⁻¹) = (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
⊢ 1 / 8 ≤ 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
⊢ 1 / 8 ≤ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Bottcher.lean | term_approx | [89, 1] | [131, 66] | refine (lt_of_le_of_lt (le_abs_self _) (lt_of_le_of_lt ?_ (half_lt_self Real.pi_pos))).ne | case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi | case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ |(1 + c * w ^ d).arg| ≤ Real.pi / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi
TACTIC:
|
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