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stringclasses 147
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stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
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stringlengths 1
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_prod | [52, 1] | [53, 23] | simp | a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = (insert 0 (Finset.image (fun n => n + 1) N)).prod (cons a f) | a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = cons a f 0 * N.prod fun x => cons a f (x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = (insert 0 (Finset.image (fun n => n + 1) N)).prod (cons a f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_prod | [52, 1] | [53, 23] | rfl | a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = cons a f 0 * N.prod fun x => cons a f (x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = cons a f 0 * N.prod fun x => cons a f (x + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | rw [Set.range] | ⊢ Set.range push = {N | 0 ∈ N} | ⊢ {x | ∃ y, push y = x} = {N | 0 ∈ N} | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Set.range push = {N | 0 ∈ N}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | apply Set.ext | ⊢ {x | ∃ y, push y = x} = {N | 0 ∈ N} | case h
⊢ ∀ (x : Finset ℕ), x ∈ {x | ∃ y, push y = x} ↔ x ∈ {N | 0 ∈ N} | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ {x | ∃ y, push y = x} = {N | 0 ∈ N}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | simp | case h
⊢ ∀ (x : Finset ℕ), x ∈ {x | ∃ y, push y = x} ↔ x ∈ {N | 0 ∈ N} | case h
⊢ ∀ (x : Finset ℕ), (∃ y, push y = x) ↔ 0 ∈ x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
⊢ ∀ (x : Finset ℕ), x ∈ {x | ∃ y, push y = x} ↔ x ∈ {N | 0 ∈ N}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | intro N | case h
⊢ ∀ (x : Finset ℕ), (∃ y, push y = x) ↔ 0 ∈ x | case h
N : Finset ℕ
⊢ (∃ y, push y = N) ↔ 0 ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h
⊢ ∀ (x : Finset ℕ), (∃ y, push y = x) ↔ 0 ∈ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | constructor | case h
N : Finset ℕ
⊢ (∃ y, push y = N) ↔ 0 ∈ N | case h.mp
N : Finset ℕ
⊢ (∃ y, push y = N) → 0 ∈ N
case h.mpr
N : Finset ℕ
⊢ 0 ∈ N → ∃ y, push y = N | Please generate a tactic in lean4 to solve the state.
STATE:
case h
N : Finset ℕ
⊢ (∃ y, push y = N) ↔ 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | intro h | case h.mp
N : Finset ℕ
⊢ (∃ y, push y = N) → 0 ∈ N | case h.mp
N : Finset ℕ
h : ∃ y, push y = N
⊢ 0 ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
N : Finset ℕ
⊢ (∃ y, push y = N) → 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | rcases h with ⟨M, H⟩ | case h.mp
N : Finset ℕ
h : ∃ y, push y = N
⊢ 0 ∈ N | case h.mp.intro
N M : Finset ℕ
H : push M = N
⊢ 0 ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
N : Finset ℕ
h : ∃ y, push y = N
⊢ 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | rw [push] at H | case h.mp.intro
N M : Finset ℕ
H : push M = N
⊢ 0 ∈ N | case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
N M : Finset ℕ
H : push M = N
⊢ 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | rw [← H] | case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ N | case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ insert 0 (Finset.image (fun n => n + 1) M) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | exact Finset.mem_insert_self 0 _ | case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ insert 0 (Finset.image (fun n => n + 1) M) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
N M : Finset ℕ
H : insert 0 (Finset.image (fun n => n + 1) M) = N
⊢ 0 ∈ insert 0 (Finset.image (fun n => n + 1) M)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | intro N0 | case h.mpr
N : Finset ℕ
⊢ 0 ∈ N → ∃ y, push y = N | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ ∃ y, push y = N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : Finset ℕ
⊢ 0 ∈ N → ∃ y, push y = N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | exists pop N | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ ∃ y, push y = N | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ push (pop N) = N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ ∃ y, push y = N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | rw [push_pop] | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ push (pop N) = N | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ insert 0 N = N | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ push (pop N) = N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_range | [56, 1] | [59, 76] | exact Finset.insert_eq_of_mem N0 | case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ insert 0 N = N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
N : Finset ℕ
N0 : 0 ∈ N
⊢ insert 0 N = N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | apply Filter.comap_embedding_atTop | ⊢ Filter.comap push atTop = atTop | case hm
⊢ ∀ (b₁ b₂ : Finset ℕ), push b₁ ≤ push b₂ ↔ b₁ ≤ b₂
case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Filter.comap push atTop = atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | exact @push_le_push | case hm
⊢ ∀ (b₁ b₂ : Finset ℕ), push b₁ ≤ push b₂ ↔ b₁ ≤ b₂
case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b | case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b | Please generate a tactic in lean4 to solve the state.
STATE:
case hm
⊢ ∀ (b₁ b₂ : Finset ℕ), push b₁ ≤ push b₂ ↔ b₁ ≤ b₂
case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | intro N | case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b | case hu
N : Finset ℕ
⊢ ∃ b, N ≤ push b | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
⊢ ∀ (c : Finset ℕ), ∃ b, c ≤ push b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | exists pop N | case hu
N : Finset ℕ
⊢ ∃ b, N ≤ push b | case hu
N : Finset ℕ
⊢ N ≤ push (pop N) | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
N : Finset ℕ
⊢ ∃ b, N ≤ push b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | rw [push_pop] | case hu
N : Finset ℕ
⊢ N ≤ push (pop N) | case hu
N : Finset ℕ
⊢ N ≤ insert 0 N | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
N : Finset ℕ
⊢ N ≤ push (pop N)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_comap_atTop | [61, 1] | [64, 45] | simp | case hu
N : Finset ℕ
⊢ N ≤ insert 0 N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hu
N : Finset ℕ
⊢ N ≤ insert 0 N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | nth_rw 1 [← push_comap_atTop] | A : Type
f : Finset ℕ → A
l : Filter A
⊢ Filter.Tendsto (f ∘ push) atTop l ↔ Filter.Tendsto f atTop l | A : Type
f : Finset ℕ → A
l : Filter A
⊢ Filter.Tendsto (f ∘ push) (Filter.comap push atTop) l ↔ Filter.Tendsto f atTop l | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : Finset ℕ → A
l : Filter A
⊢ Filter.Tendsto (f ∘ push) atTop l ↔ Filter.Tendsto f atTop l
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | apply Filter.tendsto_comap'_iff | A : Type
f : Finset ℕ → A
l : Filter A
⊢ Filter.Tendsto (f ∘ push) (Filter.comap push atTop) l ↔ Filter.Tendsto f atTop l | case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ Set.range push ∈ atTop | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : Finset ℕ → A
l : Filter A
⊢ Filter.Tendsto (f ∘ push) (Filter.comap push atTop) l ↔ Filter.Tendsto f atTop l
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | rw [push_range] | case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ Set.range push ∈ atTop | case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ {N | 0 ∈ N} ∈ atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ Set.range push ∈ atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | have h : {N : Finset ℕ | 0 ∈ N} = {N : Finset ℕ | {0} ≤ N} := by simp | case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ {N | 0 ∈ N} ∈ atTop | case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | 0 ∈ N} ∈ atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type
f : Finset ℕ → A
l : Filter A
⊢ {N | 0 ∈ N} ∈ atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | rw [h] | case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | 0 ∈ N} ∈ atTop | case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | {0} ≤ N} ∈ atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | 0 ∈ N} ∈ atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | exact Filter.mem_atTop _ | case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | {0} ≤ N} ∈ atTop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A : Type
f : Finset ℕ → A
l : Filter A
h : {N | 0 ∈ N} = {N | {0} ≤ N}
⊢ {N | {0} ≤ N} ∈ atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | tendsto_comp_push | [67, 1] | [72, 35] | simp | A : Type
f : Finset ℕ → A
l : Filter A
⊢ {N | 0 ∈ N} = {N | {0} ≤ N} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type
f : Finset ℕ → A
l : Filter A
⊢ {N | 0 ∈ N} = {N | {0} ≤ N}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | induction' N using Finset.induction with n N Nn h | N : Finset ℕ
f : ℕ → ℂ
⊢ Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n) | case empty
f : ℕ → ℂ
⊢ Complex.abs (∅.sum fun n => f n) ≤ ∅.sum fun n => Complex.abs (f n)
case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs ((insert n N).sum fun n => f n) ≤ (insert n N).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
N : Finset ℕ
f : ℕ → ℂ
⊢ Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | simp | case empty
f : ℕ → ℂ
⊢ Complex.abs (∅.sum fun n => f n) ≤ ∅.sum fun n => Complex.abs (f n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
f : ℕ → ℂ
⊢ Complex.abs (∅.sum fun n => f n) ≤ ∅.sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | rw [Finset.sum_insert Nn] | case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs ((insert n N).sum fun n => f n) ≤ (insert n N).sum fun n => Complex.abs (f n) | case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ (insert n N).sum fun n => Complex.abs (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs ((insert n N).sum fun n => f n) ≤ (insert n N).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | rw [Finset.sum_insert Nn] | case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ (insert n N).sum fun n => Complex.abs (f n) | case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ (insert n N).sum fun n => Complex.abs (f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | trans abs (f n) + abs (N.sum fun n ↦ f n) | case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x) | f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + Complex.abs (N.sum fun n => f n)
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n) + Complex.abs (N.sum fun n => f n) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | exact Complex.abs.add_le _ _ | f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + Complex.abs (N.sum fun n => f n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n + N.sum fun x => f x) ≤ Complex.abs (f n) + Complex.abs (N.sum fun n => f n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | apply add_le_add_left | f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n) + Complex.abs (N.sum fun n => f n) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x) | case bc
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (N.sum fun n => f n) ≤ N.sum fun x => Complex.abs (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (f n) + Complex.abs (N.sum fun n => f n) ≤ Complex.abs (f n) + N.sum fun x => Complex.abs (f x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | finset_complex_abs_sum_le | [75, 1] | [81, 40] | assumption | case bc
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (N.sum fun n => f n) ≤ N.sum fun x => Complex.abs (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc
f : ℕ → ℂ
n : ℕ
N : Finset ℕ
Nn : n ∉ N
h : Complex.abs (N.sum fun n => f n) ≤ N.sum fun n => Complex.abs (f n)
⊢ Complex.abs (N.sum fun n => f n) ≤ N.sum fun x => Complex.abs (f x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | rw [Finset.subset_iff] | A B : Finset ℕ
⊢ B ⊆ A ∪ B \ A | A B : Finset ℕ
⊢ ∀ ⦃x : ℕ⦄, x ∈ B → x ∈ A ∪ B \ A | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ B ⊆ A ∪ B \ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | intro x Bx | A B : Finset ℕ
⊢ ∀ ⦃x : ℕ⦄, x ∈ B → x ∈ A ∪ B \ A | A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∪ B \ A | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ ∀ ⦃x : ℕ⦄, x ∈ B → x ∈ A ∪ B \ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | rw [Finset.mem_union, Finset.mem_sdiff] | A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∪ B \ A | A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∪ B \ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | by_cases Ax : x ∈ A | A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A | case pos
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A
case neg
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | left | case pos
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A | case pos.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | exact Ax | case pos.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∈ A
⊢ x ∈ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | right | case neg
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A | case neg.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ B ∧ x ∉ A | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ A ∨ x ∈ B ∧ x ∉ A
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | subset_union_sdiff | [83, 1] | [88, 26] | exact ⟨Bx, Ax⟩ | case neg.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ B ∧ x ∉ A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
A B : Finset ℕ
x : ℕ
Bx : x ∈ B
Ax : x ∉ A
⊢ x ∈ B ∧ x ∉ A
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:
|
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