url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fc_f | [183, 1] | [185, 49] | rw [gl_zero] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fc_f | [183, 1] | [185, 49] | exact one_ne_zero | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fd_f | [187, 1] | [189, 34] | rw [fl_f', analyticAt_gl.monomial_mul_orderAt gl_frequently_ne_zero, orderAt_eq_zero, add_zero] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ orderAt (fl (f d) ∞ c) 0 = d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ orderAt (fl (f d) ∞ c) 0 = d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fd_f | [187, 1] | [189, 34] | rw [gl_zero] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ gl d c 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | fd_f | [187, 1] | [189, 34] | exact one_ne_zero | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_inf | [191, 1] | [191, 95] | simp only [f, f', lift_inf', eq_self_iff_true, imp_true_iff] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ f d c ∞ = ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ f d c ∞ = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | set s := superF d | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ SuperNear (fl (f d) ∞ c) d {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ SuperNear (fl (f d) ∞ c) d {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻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 : ℂ
⊢ SuperNear (fl (f d) ∞ c) d {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | generalize ht : {z : ℂ | abs z < (max 16 (abs c / 2))⁻¹} = t | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ SuperNear (fl (f d) ∞ c) d {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
⊢ SuperNear (fl (f d) ∞ c) d t | 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 : ℂ
s : Super (f d) d ∞ := superF d
⊢ SuperNear (fl (f d) ∞ c) d {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | have cz : ∀ {z}, z ∈ t → abs (c * z ^ d) ≤ 1 / 8 := by
intro z m; simp only [← ht, mem_setOf] at m
simp only [Complex.abs.map_mul, Complex.abs.map_pow]
trans abs c * (max 16 (abs c / 2))⁻¹ ^ d; bound
rw [inv_pow, mul_inv_le_iff]; swap; bound
rw [mul_one_div]; rw [le_div_iff, mul_comm]; swap; norm_num
refine... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
⊢ SuperNear (fl (f d) ∞ c) d t | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
⊢ SuperNear (fl (f d) ∞ c) d t | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
⊢ SuperNear (fl (f d) ∞ c) d t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | have cz1 : ∀ {z}, z ∈ t → 7 / 8 ≤ abs (1 + c * z ^ d) := by
intro z m
calc abs (1 + c * z ^ d)
_ ≥ Complex.abs 1 - abs (c * z ^ d) := by bound
_ ≥ Complex.abs 1 - 1 / 8 := by linarith [cz m]
_ = 7 / 8 := by norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
⊢ SuperNear (fl (f d) ∞ c) d t | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
⊢ SuperNear (fl (f d) ∞ c) d ... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
⊢ SuperNear (fl (f d) ∞ c) d ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | have zb : ∀ {z}, z ∈ t → abs z ≤ 1 / 8 := by
intro z m; rw [← ht] at m; refine le_trans (le_of_lt m) ?_
rw [one_div]; exact inv_le_inv_of_le (by norm_num) (le_trans (by norm_num) (le_max_left _ _)) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
⊢ SuperNear (fl (f d) ∞ c) d ... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | exact
{ d2 := two_le_d d
fa0 := (s.fla c).along_snd
fd := fd_f
fc := fc_f
o := by rw [← ht]; exact isOpen_lt Complex.continuous_abs continuous_const
t0 := by simp only [← ht, mem_setOf, Complex.abs.map_zero]; bound
t2 := fun {z} m ↦ le_trans (zb m) (by norm_num)
fa := by
intro z m; r... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
⊢ ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : z ∈ t
⊢ Complex.abs (c * z ^ d) ≤ 1 / 8 | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
⊢ ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | simp only [← ht, mem_setOf] at m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : z ∈ t
⊢ Complex.abs (c * z ^ d) ≤ 1 / 8 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs (c * z ^ d) ≤ 1 / 8 | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : z ∈ t
⊢ Complex.abs (c * z ^ d) ≤ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | simp only [Complex.abs.map_mul, Complex.abs.map_pow] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs (c * z ^ d) ≤ 1 / 8 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c * Complex.abs z ^ d ≤ 1 / 8 | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs (c * z ^ d) ≤ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | trans abs c * (max 16 (abs c / 2))⁻¹ ^ d | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c * Complex.abs z ^ d ≤ 1 / 8 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c * Complex.abs z ^ d ≤ Complex.abs c * (max 16 (Complex.abs c / 2))⁻¹ ^ d
... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c * Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c * Complex.abs z ^ d ≤ Complex.abs c * (max 16 (Complex.abs c / 2))⁻¹ ^ d
... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c * (max 16 (Complex.abs c / 2))⁻¹ ^ d ≤ 1 / 8 | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c * Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [inv_pow, mul_inv_le_iff] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c * (max 16 (Complex.abs c / 2))⁻¹ ^ d ≤ 1 / 8 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ d * (1 / 8)
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c * (max 16 (C... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | swap | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ d * (1 / 8)
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 0 < max 16 (Complex.abs c / 2) ^ d
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : ... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c ≤ max 16 (Co... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 0 < max 16 (Complex.abs c / 2) ^ d
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : ... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ d * (1 / 8) | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 0 < max 16 (Complex.abs c ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [mul_one_div] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ d * (1 / 8) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ d / 8 | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c ≤ max 16 (Co... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [le_div_iff, mul_comm] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ d / 8 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 8 * Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ d
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ Complex.abs c ≤ max 16 (Co... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | swap | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 8 * Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ d
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 0 < 8
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super ... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 8 * Complex.abs c ≤ max 16... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 0 < 8
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super ... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 8 * Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ d | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 0 < 8
c✝ : ℂ
d✝ : ℕ
inst✝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | refine le_trans ?_ (pow_le_pow_right (le_max_of_le_left (by norm_num)) (two_le_d d)) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 8 * Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ d | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 8 * Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ 2 | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 8 * Complex.abs c ≤ max 16... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | by_cases cb : abs c / 2 ≤ 16 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 8 * Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ 2 | case pos
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : Complex.abs c / 2 ≤ 16
⊢ 8 * Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ 2... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 8 * Complex.abs c ≤ max 16... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [max_eq_left cb, pow_two] | case pos
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : Complex.abs c / 2 ≤ 16
⊢ 8 * Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ 2... | case pos
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : Complex.abs c / 2 ≤ 16
⊢ 8 * Complex.abs c ≤ 16 * 16
case neg
c✝ : ℂ
d✝ : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : Complex.abs c ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | linarith | case pos
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : Complex.abs c / 2 ≤ 16
⊢ 8 * Complex.abs c ≤ 16 * 16
case neg
c✝ : ℂ
d✝ : ... | case neg
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : ¬Complex.abs c / 2 ≤ 16
⊢ 8 * Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : Complex.abs c ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [max_eq_right (not_le.mp cb).le, pow_two] | case neg
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : ¬Complex.abs c / 2 ≤ 16
⊢ 8 * Complex.abs c ≤ max 16 (Complex.abs c / 2) ^ ... | case neg
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : ¬Complex.abs c / 2 ≤ 16
⊢ 8 * Complex.abs c ≤ Complex.abs c / 2 * (Complex.... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : ¬Complex.abs c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | nlinarith | case neg
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : ¬Complex.abs c / 2 ≤ 16
⊢ 8 * Complex.abs c ≤ Complex.abs c / 2 * (Complex.... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
cb : ¬Complex.abs c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 1 ≤ 16 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
z : ℂ
m : Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹
⊢ 1 ≤ 16
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
⊢ ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
⊢ 7 / 8 ≤ Complex.abs (1 + c * z ^ d) | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
⊢ ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | calc abs (1 + c * z ^ d)
_ ≥ Complex.abs 1 - abs (c * z ^ d) := by bound
_ ≥ Complex.abs 1 - 1 / 8 := by linarith [cz m]
_ = 7 / 8 := by norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
⊢ 7 / 8 ≤ Complex.abs (1 + c * z ^ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
⊢ 7 / 8 ≤ Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
⊢ Complex.abs (1 + c * z ^ d) ≥ Complex.abs 1 - Complex.abs (c * z ^ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
⊢ Complex.abs... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | linarith [cz m] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
⊢ Complex.abs 1 - Complex.abs (c * z ^ d) ≥ Complex.abs 1 - 1 / 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
⊢ Complex.abs... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
⊢ Complex.abs 1 - 1 / 8 = 7 / 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
⊢ Complex.abs... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
⊢ ∀ {z : ℂ}, z ∈ t → Complex.... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
z : ℂ
m : z ∈ t
⊢ Complex.abs... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [← ht] at m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
z : ℂ
m : z ∈ t
⊢ Complex.abs... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
z : ℂ
m : z ∈ {z | Complex.ab... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | refine le_trans (le_of_lt m) ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
z : ℂ
m : z ∈ {z | Complex.ab... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
z : ℂ
m : z ∈ {z | Complex.ab... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [one_div] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
z : ℂ
m : z ∈ {z | Complex.ab... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
z : ℂ
m : z ∈ {z | Complex.ab... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | exact inv_le_inv_of_le (by norm_num) (le_trans (by norm_num) (le_max_left _ _)) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
z : ℂ
m : z ∈ {z | Complex.ab... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
z : ℂ
m : z ∈ {z | Complex.ab... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
z : ℂ
m : z ∈ {z | Complex.ab... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [← ht] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | exact isOpen_lt Complex.continuous_abs continuous_const | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | simp only [← ht, mem_setOf, Complex.abs.map_zero] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [fl_f] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | refine ((analyticAt_id _ _).pow _).div (analyticAt_const.add
(analyticAt_const.mul ((analyticAt_id _ _).pow _))) ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [← Complex.abs.ne_zero_iff] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | exact (lt_of_lt_of_le (by norm_num) (cz1 m)).ne' | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | specialize cz1 m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
m : z ∈ t
cz1 : 7 / 8 ≤ Complex.abs (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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | specialize zb m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
m : z ∈ t
cz1 : 7 / 8 ≤ Complex.abs (1... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
⊢ fl ... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Compl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | simp only [fl_f, mem_setOf, map_div₀, Complex.abs.map_pow, ← ht] at m ⊢ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
⊢ fl ... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
m : z ∈ t
cz1 : 7 / 8 ≤... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | refine lt_of_le_of_lt ?_ m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [div_le_iff (lt_of_lt_of_le (by norm_num) cz1)] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | refine le_trans (pow_le_pow_of_le_one (Complex.abs.nonneg _)
(le_trans zb (by norm_num)) (two_le_d d)) ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [pow_two] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | refine mul_le_mul_of_nonneg_left ?_ (Complex.abs.nonneg _) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | exact le_trans zb (le_trans (by norm_num) cz1) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z ≤ 1 / 8
m : Complex.abs... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
z : ℂ
cz1 : 7 / 8 ≤ Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z z0 m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | simp only [fl_f, div_div_cancel_left' (pow_ne_zero d z0)] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | specialize cz1 m | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / 8 ≤ Complex.abs (1 + c * z ^ d)
zb : ∀ {z : ℂ}, z ∈ t → Compl... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
cz1 : ∀ {z : ℂ}, z ∈ t → 7 / ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | have czp : 0 < abs (1 + c * z ^ d) := lt_of_lt_of_le (by norm_num) cz1 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Compl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | refine le_of_mul_le_mul_right ?_ czp | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Compl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [← Complex.abs.map_mul, mul_sub_right_distrib, one_mul,
inv_mul_cancel (Complex.abs.ne_zero_iff.mp czp.ne'), ← sub_sub, sub_self, zero_sub,
Complex.abs.map_neg] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | 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 : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Compl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | exact le_trans (cz m) (le_trans (by norm_num)
(mul_le_mul_of_nonneg_left cz1 (by norm_num))) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Compl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Compl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Compl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Complex.abs z ≤ 1 / 8
z : ℂ
z0 : z ≠ 0
m : z ∈ t
cz1 : 7 / 8 ≤ Com... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
t : Set ℂ
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} = t
cz : ∀ {z : ℂ}, z ∈ t → Complex.abs (c * z ^ d) ≤ 1 / 8
zb : ∀ {z : ℂ}, z ∈ t → Compl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | induction' z using OnePoint.rec with z | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ Critical (f d c) z ↔ z = 0 ∨ z = ∞ | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Critical (f d c) ∞ ↔ ∞ = 0 ∨ ∞ = ∞
case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ Critical (f d c) ↑z ↔ ↑z = 0 ∨ ↑z = ∞ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ Critical (f d c) z ↔ z = 0 ∨ z = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | simp only [(superF d).critical_a, or_true] | case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Critical (f d c) ∞ ↔ ∞ = 0 ∨ ∞ = ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Critical (f d c) ∞ ↔ ∞ = 0 ∨ ∞ = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | have zx : ∀ x : ℂ, (0 : ℂ →L[ℂ] ℂ) x = 0 := fun x ↦ ContinuousLinearMap.zero_apply _ | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ Critical (f d c) ↑z ↔ ↑z = 0 ∨ ↑z = ∞ | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ Critical (f d c) ↑z ↔ ↑z = 0 ∨ ↑z = ∞ | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ Critical (f d c) ↑z ↔ ↑z = 0 ∨ ↑z = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | simp only [Critical, mfderiv, (holomorphicF (c, z)).along_snd.mdifferentiableAt, if_pos,
ModelWithCorners.Boundaryless.range_eq_univ, fderivWithin_univ, writtenInExtChartAt_coe_f,
RiemannSphere.extChartAt_coe, coePartialEquiv_symm_apply, toComplex_coe, coe_eq_zero,
coe_eq_inf_iff, or_false_iff, ← deriv_fderiv, de... | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ Critical (f d c) ↑z ↔ ↑z = 0 ∨ ↑z = ∞ | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ (∀ (x : ℂ), x = 0 ∨ z = 0) ↔ z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ Critical (f d c) ↑z ↔ ↑z = 0 ∨ ↑z = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | constructor | case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ (∀ (x : ℂ), x = 0 ∨ z = 0) ↔ z = 0 | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ (∀ (x : ℂ), x = 0 ∨ z = 0) → z = 0
case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ z = 0 → ∀ (x : ℂ), x = 0 ∨ z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ (∀ (x : ℂ), x = 0 ∨ z = 0) ↔ z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | intro h | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ (∀ (x : ℂ), x = 0 ∨ z = 0) → z = 0 | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
h : ∀ (x : ℂ), x = 0 ∨ z = 0
⊢ z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ (∀ (x : ℂ), x = 0 ∨ z = 0) → z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | specialize h 1 | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
h : ∀ (x : ℂ), x = 0 ∨ z = 0
⊢ z = 0 | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
h : 1 = 0 ∨ z = 0
⊢ z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
h : ∀ (x : ℂ), x = 0 ∨ z = 0
⊢ z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | simp only [one_mul, mul_eq_zero, one_ne_zero, false_or_iff] at h | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
h : 1 = 0 ∨ z = 0
⊢ z = 0 | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
h : z = 0
⊢ z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
h : 1 = 0 ∨ z = 0
⊢ z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | exact h | case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
h : z = 0
⊢ z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
h : z = 0
⊢ z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | exact fun h x ↦ Or.inr h | case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ z = 0 → ∀ (x : ℂ), x = 0 ∨ z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
zx : ∀ (x : ℂ), 0 x = 0
⊢ z = 0 → ∀ (x : ℂ), x = 0 ∨ z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin' | [284, 1] | [285, 71] | simp only [multibrot, mem_setOf, Super.basin_iff_attracts, Attracts] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ c ∈ multibrot d ↔ (c, ↑c) ∉ ⋯.basin | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ c ∈ multibrot d ↔ (c, ↑c) ∉ ⋯.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | set s := superF d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ c ∈ multibrot d ↔ (c, 0) ∉ ⋯.basin | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ c ∈ multibrot d ↔ (c, 0) ∉ s.basin | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ c ∈ multibrot d ↔ (c, 0) ∉ ⋯.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | simp only [multibrot_basin', not_iff_not, Super.basin, mem_setOf] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ c ∈ multibrot d ↔ (c, 0) ∉ s.basin | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ (∃ n, (c, (f d c)^[n] ↑c) ∈ ⋯.near) ↔ ∃ n, (c, (f d c)^[n] 0) ∈ s.near | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ c ∈ multibrot d ↔ (c, 0) ∉ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | simp only [e] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (∃ n, (c, (f d c)^[n] ↑c) ∈ ⋯.near) ↔ ∃ n, (c, (f d c)^[n] 0) ∈ s.near | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near) ↔ ∃ n, (c, (f d c)^[n] 0) ∈ s.near | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (∃ n, (c, (f d c)^[n] ↑c) ∈ ⋯.near) ↔ ∃ n, (c, (f d c)^[n] 0) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | constructor | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near) ↔ ∃ n, (c, (f d c)^[n] 0) ∈ s.near | case mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near) → ∃ n, (c, (f d c)^[n] 0) ∈ s.near
case mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near) ↔ ∃ n, (c, (f d c)^[n] 0) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | intro n | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
n : ℕ
⊢ (f d c)^[n] ↑c = (f d c)^[n + 1] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | induction' n with n h | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
n : ℕ
⊢ (f d c)^[n] ↑c = (f d c)^[n + 1] 0 | case zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ (f d c)^[0] ↑c = (f d c)^[0 + 1] 0
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
n : ℕ
h : (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (f d c)^[n + 1] ↑c = (f d c)^[n + 1 + 1] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
n : ℕ
⊢ (f d c)^[n] ↑c = (f d c)^[n + 1] 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | simp only [Function.iterate_zero_apply, zero_add, Function.iterate_one, f_0, Nat.zero_eq] | case zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ (f d c)^[0] ↑c = (f d c)^[0 + 1] 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ (f d c)^[0] ↑c = (f d c)^[0 + 1] 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | simp only [Function.iterate_succ_apply', h] | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
n : ℕ
h : (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (f d c)^[n + 1] ↑c = (f d c)^[n + 1 + 1] 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
n : ℕ
h : (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (f d c)^[n + 1] ↑c = (f d c)^[n + 1 + 1] 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | intro ⟨n, h⟩ | case mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near) → ∃ n, (c, (f d c)^[n] 0) ∈ s.near | case mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
n : ℕ
h : (c, (f d c)^[n + 1] 0) ∈ ⋯.near
⊢ ∃ n, (c, (f d c)^[n] 0) ∈ s.near | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near) → ∃ n, (c, (f d c)^[n] 0) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | exact ⟨n + 1, h⟩ | case mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
n : ℕ
h : (c, (f d c)^[n + 1] 0) ∈ ⋯.near
⊢ ∃ n, (c, (f d c)^[n] 0) ∈ s.near | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
n : ℕ
h : (c, (f d c)^[n + 1] 0) ∈ ⋯.near
⊢ ∃ n, (c, (f d c)^[n] 0) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | intro ⟨n, h⟩ | case mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (∃ n, (c, (f d c)^[n] 0) ∈ s.near) → ∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near | case mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
n : ℕ
h : (c, (f d c)^[n] 0) ∈ s.near
⊢ ∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
⊢ (∃ n, (c, (f d c)^[n] 0) ∈ s.near) → ∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | use n | case mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
n : ℕ
h : (c, (f d c)^[n] 0) ∈ s.near
⊢ ∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
n : ℕ
h : (c, (f d c)^[n] 0) ∈ s.near
⊢ (c, (f d c)^[n + 1] 0) ∈ ⋯.near | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
n : ℕ
h : (c, (f d c)^[n] 0) ∈ s.near
⊢ ∃ n, (c, (f d c)^[n + 1] 0) ∈ ⋯.near
TACTIC:
|
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