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 : β
d : β
instβ : Fact (2 β€ d)
β’ gl d c 0 β 0 | c : β
d : β
instβ : Fact (2 β€ d)
β’ 1 β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ gl d c 0 β 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 : β
d : β
instβ : Fact (2 β€ d)
β’ 1 β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ 1 β 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 : β
d : β
instβ : Fact (2 β€ d)
β’ orderAt (fl (f d) β c) 0 = d | c : β
d : β
instβ : Fact (2 β€ d)
β’ gl d c 0 β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ orderAt (fl (f d) β 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 : β
d : β
instβ : Fact (2 β€ d)
β’ gl d c 0 β 0 | c : β
d : β
instβ : Fact (2 β€ d)
β’ 1 β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ gl d c 0 β 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 : β
d : β
instβ : Fact (2 β€ d)
β’ 1 β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ 1 β 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 : β
d : β
instβ : Fact (2 β€ d)
β’ f d c β = β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ f d c β = β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | set s := superF d | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
β’ SuperNear (fl (f d) β c) d {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ SuperNear (fl (f d) β c) d {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
β’ SuperNear (fl (f d) β c) d {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | generalize ht : {z : β | abs z < (max 16 (abs c / 2))β»ΒΉ} = t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ SuperNear (fl (f d) β c) d {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
β’ SuperNear (fl (f d) β c) d t | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ SuperNear (fl (f d) β c) d {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | have cz : β {z}, z β t β abs (c * z ^ d) β€ 1 / 8 := by
intro z m; simp only [β ht, mem_setOf] at m
simp only [Complex.abs.map_mul, Complex.abs.map_pow]
trans abs c * (max 16 (abs c / 2))β»ΒΉ ^ 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 le_trans ?_ (pow_le_pow_right (le_max_of_le_left (by norm_num)) (two_le_d d))
by_cases cb : abs c / 2 β€ 16
rw [max_eq_left cb, pow_two]; linarith
rw [max_eq_right (not_le.mp cb).le, pow_two]; nlinarith | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
β’ SuperNear (fl (f d) β c) d t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
β’ SuperNear (fl (f d) β c) d t | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
β’ SuperNear (fl (f d) β c) d t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | have cz1 : β {z}, z β t β 7 / 8 β€ abs (1 + c * z ^ d) := by
intro z m
calc abs (1 + c * z ^ d)
_ β₯ Complex.abs 1 - abs (c * z ^ d) := by bound
_ β₯ Complex.abs 1 - 1 / 8 := by linarith [cz m]
_ = 7 / 8 := by norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
β’ SuperNear (fl (f d) β c) d t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ SuperNear (fl (f d) β c) d t | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
β’ SuperNear (fl (f d) β c) d t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | have zb : β {z}, z β t β abs z β€ 1 / 8 := by
intro z m; rw [β 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 _ _)) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ SuperNear (fl (f d) β c) d t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ SuperNear (fl (f d) β c) d t | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ SuperNear (fl (f d) β c) d t
TACTIC:
|
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 [β ht]; exact isOpen_lt Complex.continuous_abs continuous_const
t0 := by simp only [β ht, mem_setOf, Complex.abs.map_zero]; bound
t2 := fun {z} m β¦ le_trans (zb m) (by norm_num)
fa := by
intro z m; rw [fl_f]
refine ((analyticAt_id _ _).pow _).div (analyticAt_const.add
(analyticAt_const.mul ((analyticAt_id _ _).pow _))) ?_
rw [β Complex.abs.ne_zero_iff]; exact (lt_of_lt_of_le (by norm_num) (cz1 m)).ne'
ft := by
intro z m; specialize cz1 m; specialize zb m
simp only [fl_f, mem_setOf, map_divβ, Complex.abs.map_pow, β ht] at m β’
refine lt_of_le_of_lt ?_ m; rw [div_le_iff (lt_of_lt_of_le (by norm_num) cz1)]
refine le_trans (pow_le_pow_of_le_one (Complex.abs.nonneg _)
(le_trans zb (by norm_num)) (two_le_d d)) ?_
rw [pow_two]; refine mul_le_mul_of_nonneg_left ?_ (Complex.abs.nonneg _)
exact le_trans zb (le_trans (by norm_num) cz1)
gs' := by
intro z z0 m; simp only [fl_f, div_div_cancel_left' (pow_ne_zero d z0)]
specialize cz1 m
have czp : 0 < abs (1 + c * z ^ d) := lt_of_lt_of_le (by norm_num) cz1
refine le_of_mul_le_mul_right ?_ czp
rw [β Complex.abs.map_mul, mul_sub_right_distrib, one_mul,
inv_mul_cancel (Complex.abs.ne_zero_iff.mp czp.ne'), β sub_sub, sub_self, zero_sub,
Complex.abs.map_neg]
exact le_trans (cz m) (le_trans (by norm_num)
(mul_le_mul_of_nonneg_left cz1 (by norm_num))) } | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ SuperNear (fl (f d) β c) d t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ SuperNear (fl (f d) β c) d t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
β’ β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : z β t
β’ Complex.abs (c * z ^ d) β€ 1 / 8 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
β’ β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | simp only [β ht, mem_setOf] at m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : z β t
β’ Complex.abs (c * z ^ d) β€ 1 / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs (c * z ^ d) β€ 1 / 8 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : z β t
β’ Complex.abs (c * z ^ d) β€ 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] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs (c * z ^ d) β€ 1 / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * Complex.abs z ^ d β€ 1 / 8 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs (c * z ^ d) β€ 1 / 8
TACTIC:
|
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))β»ΒΉ ^ d | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * Complex.abs z ^ d β€ 1 / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * Complex.abs z ^ d β€ Complex.abs c * (max 16 (Complex.abs c / 2))β»ΒΉ ^ d
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * (max 16 (Complex.abs c / 2))β»ΒΉ ^ d β€ 1 / 8 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * Complex.abs z ^ d β€ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * Complex.abs z ^ d β€ Complex.abs c * (max 16 (Complex.abs c / 2))β»ΒΉ ^ d
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * (max 16 (Complex.abs c / 2))β»ΒΉ ^ d β€ 1 / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * (max 16 (Complex.abs c / 2))β»ΒΉ ^ d β€ 1 / 8 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * Complex.abs z ^ d β€ Complex.abs c * (max 16 (Complex.abs c / 2))β»ΒΉ ^ d
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * (max 16 (Complex.abs c / 2))β»ΒΉ ^ d β€ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [inv_pow, mul_inv_le_iff] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * (max 16 (Complex.abs c / 2))β»ΒΉ ^ d β€ 1 / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d * (1 / 8)
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < max 16 (Complex.abs c / 2) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c * (max 16 (Complex.abs c / 2))β»ΒΉ ^ d β€ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | swap | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d * (1 / 8)
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < max 16 (Complex.abs c / 2) ^ d | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < max 16 (Complex.abs c / 2) ^ d
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d * (1 / 8) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d * (1 / 8)
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < max 16 (Complex.abs c / 2) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < max 16 (Complex.abs c / 2) ^ d
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d * (1 / 8) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d * (1 / 8) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < max 16 (Complex.abs c / 2) ^ d
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d * (1 / 8)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [mul_one_div] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d * (1 / 8) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d / 8 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d * (1 / 8)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [le_div_iff, mul_comm] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < 8 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | swap | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < 8
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < 8
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < 8
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d
TACTIC:
|
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)) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | by_cases cb : abs c / 2 β€ 16 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2 | case pos
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2
case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2
TACTIC:
|
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
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2
case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2 | case pos
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ 16 * 16
case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2
case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | linarith | case pos
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ 16 * 16
case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2 | case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ 16 * 16
case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2
TACTIC:
|
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
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2 | case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ Complex.abs c / 2 * (Complex.abs c / 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ max 16 (Complex.abs c / 2) ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | nlinarith | case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ Complex.abs c / 2 * (Complex.abs c / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
cb : Β¬Complex.abs c / 2 β€ 16
β’ 8 * Complex.abs c β€ Complex.abs c / 2 * (Complex.abs c / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 1 β€ 16 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
z : β
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 1 β€ 16
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
β’ β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
β’ 7 / 8 β€ Complex.abs (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
β’ β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | calc abs (1 + c * z ^ d)
_ β₯ Complex.abs 1 - abs (c * z ^ d) := by bound
_ β₯ Complex.abs 1 - 1 / 8 := by linarith [cz m]
_ = 7 / 8 := by norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
β’ 7 / 8 β€ Complex.abs (1 + c * z ^ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
β’ 7 / 8 β€ Complex.abs (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
β’ Complex.abs (1 + c * z ^ d) β₯ Complex.abs 1 - Complex.abs (c * z ^ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
β’ Complex.abs (1 + c * z ^ d) β₯ Complex.abs 1 - Complex.abs (c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | linarith [cz m] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
β’ Complex.abs 1 - Complex.abs (c * z ^ d) β₯ Complex.abs 1 - 1 / 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
β’ Complex.abs 1 - Complex.abs (c * z ^ d) β₯ Complex.abs 1 - 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
β’ Complex.abs 1 - 1 / 8 = 7 / 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
β’ Complex.abs 1 - 1 / 8 = 7 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ β {z : β}, z β t β Complex.abs z β€ 1 / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β t
β’ Complex.abs z β€ 1 / 8 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ β {z : β}, z β t β Complex.abs z β€ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [β ht] at m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β t
β’ Complex.abs z β€ 1 / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ Complex.abs z β€ 1 / 8 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β t
β’ Complex.abs z β€ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | refine le_trans (le_of_lt m) ?_ | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ Complex.abs z β€ 1 / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ (max 16 (Complex.abs c / 2))β»ΒΉ β€ 1 / 8 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ Complex.abs z β€ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [one_div] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ (max 16 (Complex.abs c / 2))β»ΒΉ β€ 1 / 8 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ (max 16 (Complex.abs c / 2))β»ΒΉ β€ 8β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ (max 16 (Complex.abs c / 2))β»ΒΉ β€ 1 / 8
TACTIC:
|
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 _ _)) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ (max 16 (Complex.abs c / 2))β»ΒΉ β€ 8β»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ (max 16 (Complex.abs c / 2))β»ΒΉ β€ 8β»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ 0 < 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ 0 < 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ 8 β€ 16 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
z : β
m : z β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ 8 β€ 16
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [β ht] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ IsOpen t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ IsOpen {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ IsOpen t
TACTIC:
|
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 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ IsOpen {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ IsOpen {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | simp only [β ht, mem_setOf, Complex.abs.map_zero] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ 0 β t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ 0 < (max 16 (Complex.abs c / 2))β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ 0 β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | bound | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ 0 < (max 16 (Complex.abs c / 2))β»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ 0 < (max 16 (Complex.abs c / 2))β»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ 1 / 8 β€ 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ 1 / 8 β€ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ AnalyticOn β (fl (f d) β c) t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ AnalyticAt β (fl (f d) β c) z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ AnalyticOn β (fl (f d) β c) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [fl_f] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ AnalyticAt β (fl (f d) β c) z | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ AnalyticAt β ((fun c z => z ^ d / (1 + c * z ^ d)) c) z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ AnalyticAt β (fl (f d) β c) z
TACTIC:
|
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 _))) ?_ | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ AnalyticAt β ((fun c z => z ^ d / (1 + c * z ^ d)) c) z | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ 1 + c * z ^ d β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ AnalyticAt β ((fun c z => z ^ d / (1 + c * z ^ d)) c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [β Complex.abs.ne_zero_iff] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ 1 + c * z ^ d β 0 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ Complex.abs (1 + c * z ^ d) β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ 1 + c * z ^ d β 0
TACTIC:
|
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' | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ Complex.abs (1 + c * z ^ d) β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ Complex.abs (1 + c * z ^ d) β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ 0 < 7 / 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ 0 < 7 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ MapsTo (fl (f d) β c) t t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ fl (f d) β c z β t | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ MapsTo (fl (f d) β c) t t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | specialize cz1 m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ fl (f d) β c z β t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ fl (f d) β c z β t | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
β’ fl (f d) β c z β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | specialize zb m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ fl (f d) β c z β t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
β’ fl (f d) β c z β t | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ fl (f d) β c z β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | simp only [fl_f, mem_setOf, map_divβ, Complex.abs.map_pow, β ht] at m β’ | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
β’ fl (f d) β c z β t | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ d / Complex.abs (1 + c * z ^ d) < (max 16 (Complex.abs c / 2))β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
β’ fl (f d) β c z β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | refine lt_of_le_of_lt ?_ m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ d / Complex.abs (1 + c * z ^ d) < (max 16 (Complex.abs c / 2))β»ΒΉ | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ d / Complex.abs (1 + c * z ^ d) β€ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ d / Complex.abs (1 + c * z ^ d) < (max 16 (Complex.abs c / 2))β»ΒΉ
TACTIC:
|
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)] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ d / Complex.abs (1 + c * z ^ d) β€ Complex.abs z | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ d β€ Complex.abs z * Complex.abs (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ d / Complex.abs (1 + c * z ^ d) β€ Complex.abs z
TACTIC:
|
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)) ?_ | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ d β€ Complex.abs z * Complex.abs (1 + c * z ^ d) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ 2 β€ Complex.abs z * Complex.abs (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ d β€ Complex.abs z * Complex.abs (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [pow_two] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ 2 β€ Complex.abs z * Complex.abs (1 + c * z ^ d) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z * Complex.abs z β€ Complex.abs z * Complex.abs (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z ^ 2 β€ Complex.abs z * Complex.abs (1 + c * z ^ d)
TACTIC:
|
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 _) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z * Complex.abs z β€ Complex.abs z * Complex.abs (1 + c * z ^ d) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z β€ Complex.abs (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z * Complex.abs z β€ Complex.abs z * Complex.abs (1 + c * z ^ d)
TACTIC:
|
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) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z β€ Complex.abs (1 + c * z ^ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ Complex.abs z β€ Complex.abs (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < 7 / 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 0 < 7 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 1 / 8 β€ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 1 / 8 β€ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 1 / 8 β€ 7 / 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
z : β
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : Complex.abs z β€ 1 / 8
m : Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ
β’ 1 / 8 β€ 7 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | intro z z0 m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ β {z : β}, z β 0 β z β t β Complex.abs (fl (f d) β c z / z ^ d - 1) β€ 1 / 4 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
β’ Complex.abs (fl (f d) β c z / z ^ d - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
β’ β {z : β}, z β 0 β z β t β Complex.abs (fl (f d) β c z / z ^ d - 1) β€ 1 / 4
TACTIC:
|
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)] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
β’ Complex.abs (fl (f d) β c z / z ^ d - 1) β€ 1 / 4 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
β’ Complex.abs (fl (f d) β c z / z ^ d - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | specialize cz1 m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) β€ 1 / 4 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
cz1 : β {z : β}, z β t β 7 / 8 β€ Complex.abs (1 + c * z ^ d)
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) β€ 1 / 4
TACTIC:
|
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 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) β€ 1 / 4 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) β€ 1 / 4
TACTIC:
|
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 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) β€ 1 / 4 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) * Complex.abs (1 + c * z ^ d) β€ 1 / 4 * Complex.abs (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | rw [β Complex.abs.map_mul, mul_sub_right_distrib, one_mul,
inv_mul_cancel (Complex.abs.ne_zero_iff.mp czp.ne'), β sub_sub, sub_self, zero_sub,
Complex.abs.map_neg] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) * Complex.abs (1 + c * z ^ d) β€ 1 / 4 * Complex.abs (1 + c * z ^ d) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ Complex.abs (c * z ^ d) β€ 1 / 4 * Complex.abs (1 + c * z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ Complex.abs ((1 + c * z ^ d)β»ΒΉ - 1) * Complex.abs (1 + c * z ^ d) β€ 1 / 4 * Complex.abs (1 + c * z ^ d)
TACTIC:
|
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))) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ Complex.abs (c * z ^ d) β€ 1 / 4 * Complex.abs (1 + c * z ^ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ Complex.abs (c * z ^ d) β€ 1 / 4 * Complex.abs (1 + c * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ 0 < 7 / 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
β’ 0 < 7 / 8
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ 1 / 8 β€ 1 / 4 * (7 / 8) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ 1 / 8 β€ 1 / 4 * (7 / 8)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | superNearF | [203, 1] | [256, 59] | norm_num | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ 0 β€ 1 / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
t : Set β
ht : {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ} = t
cz : β {z : β}, z β t β Complex.abs (c * z ^ d) β€ 1 / 8
zb : β {z : β}, z β t β Complex.abs z β€ 1 / 8
z : β
z0 : z β 0
m : z β t
cz1 : 7 / 8 β€ Complex.abs (1 + c * z ^ d)
czp : 0 < Complex.abs (1 + c * z ^ d)
β’ 0 β€ 1 / 4
TACTIC:
|
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 : β
d : β
instβ : Fact (2 β€ d)
z : π
β’ Critical (f d c) z β z = 0 β¨ z = β | case hβ
c : β
d : β
instβ : Fact (2 β€ d)
β’ Critical (f d c) β β β = 0 β¨ β = β
case hβ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
β’ Critical (f d c) βz β βz = 0 β¨ βz = β | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : π
β’ Critical (f d c) z β z = 0 β¨ z = β
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 hβ
c : β
d : β
instβ : Fact (2 β€ d)
β’ Critical (f d c) β β β = 0 β¨ β = β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
c : β
d : β
instβ : Fact (2 β€ d)
β’ Critical (f d c) β β β = 0 β¨ β = β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | have zx : β x : β, (0 : β βL[β] β) x = 0 := fun x β¦ ContinuousLinearMap.zero_apply _ | case hβ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
β’ Critical (f d c) βz β βz = 0 β¨ βz = β | case hβ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ Critical (f d c) βz β βz = 0 β¨ βz = β | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
β’ Critical (f d c) βz β βz = 0 β¨ βz = β
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, β deriv_fderiv, deriv_f', ContinuousLinearMap.ext_iff,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, Algebra.id.smul_eq_mul,
one_mul, mul_eq_zero, Nat.cast_eq_zero, d_ne_zero, false_or_iff,
pow_eq_zero_iff (d_minus_one_pos d).ne', zx] | case hβ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ Critical (f d c) βz β βz = 0 β¨ βz = β | case hβ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ (β (x : β), x = 0 β¨ z = 0) β z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ Critical (f d c) βz β βz = 0 β¨ βz = β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | constructor | case hβ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ (β (x : β), x = 0 β¨ z = 0) β z = 0 | case hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ (β (x : β), x = 0 β¨ z = 0) β z = 0
case hβ.mpr
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ z = 0 β β (x : β), x = 0 β¨ z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ (β (x : β), x = 0 β¨ z = 0) β z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | intro h | case hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ (β (x : β), x = 0 β¨ z = 0) β z = 0 | case hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
h : β (x : β), x = 0 β¨ z = 0
β’ z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ (β (x : β), x = 0 β¨ z = 0) β 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 hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
h : β (x : β), x = 0 β¨ z = 0
β’ z = 0 | case hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
h : 1 = 0 β¨ z = 0
β’ z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
h : β (x : β), x = 0 β¨ z = 0
β’ 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 hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
h : 1 = 0 β¨ z = 0
β’ z = 0 | case hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
h : z = 0
β’ z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
h : 1 = 0 β¨ z = 0
β’ z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | critical_f | [266, 1] | [281, 31] | exact h | case hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
h : z = 0
β’ z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.mp
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
h : z = 0
β’ 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 β¦ Or.inr h | case hβ.mpr
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ z = 0 β β (x : β), x = 0 β¨ z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ.mpr
c : β
d : β
instβ : Fact (2 β€ d)
z : β
zx : β (x : β), 0 x = 0
β’ z = 0 β β (x : β), x = 0 β¨ 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 : β
d : β
instβ : Fact (2 β€ d)
β’ c β multibrot d β (c, βc) β β―.basin | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ c β multibrot d β (c, βc) β β―.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 : β
d : β
instβ : Fact (2 β€ d)
β’ c β multibrot d β (c, 0) β β―.basin | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ c β multibrot d β (c, 0) β s.basin | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ c β multibrot d β (c, 0) β β―.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 : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ c β multibrot d β (c, 0) β s.basin | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ (β n, (c, (f d c)^[n] βc) β β―.near) β β n, (c, (f d c)^[n] 0) β s.near | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ c β multibrot d β (c, 0) β 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 : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n] βc) β β―.near) β β n, (c, (f d c)^[n] 0) β s.near | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n + 1] 0) β β―.near) β β n, (c, (f d c)^[n] 0) β s.near | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n] βc) β β―.near) β β n, (c, (f d c)^[n] 0) β s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | constructor | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n + 1] 0) β β―.near) β β n, (c, (f d c)^[n] 0) β s.near | case mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n + 1] 0) β β―.near) β β n, (c, (f d c)^[n] 0) β s.near
case mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n] 0) β s.near) β β n, (c, (f d c)^[n + 1] 0) β β―.near | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n + 1] 0) β β―.near) β β n, (c, (f d c)^[n] 0) β s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | intro n | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0 | c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
n : β
β’ (f d c)^[n] βc = (f d c)^[n + 1] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β (n : β), (f d c)^[n] βc = (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 : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
n : β
β’ (f d c)^[n] βc = (f d c)^[n + 1] 0 | case zero
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ (f d c)^[0] βc = (f d c)^[0 + 1] 0
case succ
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
n : β
h : (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (f d c)^[n + 1] βc = (f d c)^[n + 1 + 1] 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
n : β
β’ (f d c)^[n] βc = (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 : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ (f d c)^[0] βc = (f d c)^[0 + 1] 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ (f d c)^[0] βc = (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 : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
n : β
h : (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (f d c)^[n + 1] βc = (f d c)^[n + 1 + 1] 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
n : β
h : (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (f d c)^[n + 1] βc = (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 β¨n, hβ© | case mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n + 1] 0) β β―.near) β β n, (c, (f d c)^[n] 0) β s.near | case mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n + 1] 0) β β―.near
β’ β n, (c, (f d c)^[n] 0) β s.near | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n + 1] 0) β β―.near) β β n, (c, (f d c)^[n] 0) β s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | exact β¨n + 1, hβ© | case mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n + 1] 0) β β―.near
β’ β n, (c, (f d c)^[n] 0) β s.near | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n + 1] 0) β β―.near
β’ β n, (c, (f d c)^[n] 0) β s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | intro β¨n, hβ© | case mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n] 0) β s.near) β β n, (c, (f d c)^[n + 1] 0) β β―.near | case mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n] 0) β s.near
β’ β n, (c, (f d c)^[n + 1] 0) β β―.near | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
β’ (β n, (c, (f d c)^[n] 0) β s.near) β β n, (c, (f d c)^[n + 1] 0) β β―.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 : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n] 0) β s.near
β’ β n, (c, (f d c)^[n + 1] 0) β β―.near | case h
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n] 0) β s.near
β’ (c, (f d c)^[n + 1] 0) β β―.near | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
c : β
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
e : β (n : β), (f d c)^[n] βc = (f d c)^[n + 1] 0
n : β
h : (c, (f d c)^[n] 0) β s.near
β’ β n, (c, (f d c)^[n + 1] 0) β β―.near
TACTIC:
|
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