Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files
xet
Community
1
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
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2.09M
input
stringlengths
73
2.09M
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
near_one_avoids_zero
[165, 1]
[166, 73]
exact Complex.slitPlane_ne_zero (mem_slitPlane_of_near_one h)
z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ z β‰  0
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ z β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
derivWithin.clog
[172, 1]
[178, 40]
have hz := DifferentiableWithinAt.hasDerivWithinAt hf
f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z
f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 hz : HasDerivWithinAt f (derivWithin f s z) s z ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
derivWithin.clog
[172, 1]
[178, 40]
have h := HasDerivWithinAt.clog hz hx
f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 hz : HasDerivWithinAt f (derivWithin f s z) s z ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z
f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 hz : HasDerivWithinAt f (derivWithin f s z) s z h : HasDerivWithinAt (fun t => (f t).log) (derivWithin f s z / f z) s z ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 hz : HasDerivWithinAt f (derivWithin f s z) s z ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
derivWithin.clog
[172, 1]
[178, 40]
have u := o.uniqueDiffWithinAt (π•œ := β„‚) zs
f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 hz : HasDerivWithinAt f (derivWithin f s z) s z h : HasDerivWithinAt (fun t => (f t).log) (derivWithin f s z / f z) s z ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z
f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 hz : HasDerivWithinAt f (derivWithin f s z) s z h : HasDerivWithinAt (fun t => (f t).log) (derivWithin f s z / f z) s z u : UniqueDiffWithinAt β„‚ s z ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 hz : HasDerivWithinAt f (derivWithin f s z) s z h : HasDerivWithinAt (fun t => (f t).log) (derivWithin f s z / f z) s z ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
derivWithin.clog
[172, 1]
[178, 40]
rw [HasDerivWithinAt.derivWithin h u]
f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 hz : HasDerivWithinAt f (derivWithin f s z) s z h : HasDerivWithinAt (fun t => (f t).log) (derivWithin f s z / f z) s z u : UniqueDiffWithinAt β„‚ s z ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ s : Set β„‚ o : IsOpen s zs : z ∈ s hf : DifferentiableWithinAt β„‚ f s z hx : (f z).re > 0 ∨ (f z).im β‰  0 hz : HasDerivWithinAt f (derivWithin f s z) s z h : HasDerivWithinAt (fun t => (f t).log) (derivWithin f s z / f z) s z u : UniqueDiffWithinAt β„‚ s z ⊒ derivWithin (fun z => (f z).log) s z = derivWithin f s z / f z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
by_cases rp : r ≀ 0
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z case neg z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : Β¬r ≀ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
have a0 : abs z < 0 := lt_of_lt_of_le h rp
case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
have a0' : abs z β‰₯ 0 := by bound
case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 a0' : Complex.abs z β‰₯ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
exfalso
case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 a0' : Complex.abs z β‰₯ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 a0' : Complex.abs z β‰₯ 0 ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 a0' : Complex.abs z β‰₯ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
linarith [a0, a0']
case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 a0' : Complex.abs z β‰₯ 0 ⊒ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 a0' : Complex.abs z β‰₯ 0 ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
bound
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 ⊒ Complex.abs z β‰₯ 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : r ≀ 0 a0 : Complex.abs z < 0 ⊒ Complex.abs z β‰₯ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
simp only [not_le] at rp
case neg z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : Β¬r ≀ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case neg z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : Β¬r ≀ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
simp only [Complex.log_one, sub_zero, Complex.norm_eq_abs, one_div, add_sub_cancel_left] at L
case neg z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r L : β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€– ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case neg z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r L : Complex.abs (1 + z).log ≀ (1 - r)⁻¹ * Complex.abs z ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r L : β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€– ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
simpa only [one_div, ge_iff_le]
case neg z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r L : Complex.abs (1 + z).log ≀ (1 - r)⁻¹ * Complex.abs z ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r L : Complex.abs (1 + z).log ≀ (1 - r)⁻¹ * Complex.abs z ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
generalize hs : Metric.ball (1:β„‚) r = s
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€– TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
have o : IsOpen s := by rw [← hs]; exact Metric.isOpen_ball
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€– TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
have s1z : 1 + z ∈ s := by simp [← hs]; assumption
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€– TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
have s1 : (1:β„‚) ∈ s := by simp [← hs]; assumption
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€– TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
have sp : βˆ€ w : β„‚, w ∈ s β†’ w.re > 0 ∨ w.im β‰  0 := by intro w ws apply mem_slitPlane_of_near_one simp only [Metric.mem_ball, Complex.dist_eq, ← hs] at ws calc abs (w - 1) < r := by assumption _ < 1 := r1
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€– TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
have sa : βˆ€ w : β„‚, w ∈ s β†’ abs w β‰₯ 1 - r := by intro w ws simp only [Metric.mem_ball, Complex.dist_eq, ← hs] at ws calc abs w = abs (1 + (w - 1)) := by ring_nf _ β‰₯ abs (1 : β„‚) - abs (w - 1) := by bound _ β‰₯ abs (1 : β„‚) - r := by bound _ = 1 - r := by rw [Complex.abs.map_one]
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€– TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
refine Convex.norm_image_sub_le_of_norm_derivWithin_le ?_ ?_ ?_ s1 s1z
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€–
case refine_1 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ DifferentiableOn β„‚ log s case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ βˆ€ x ∈ s, β€–derivWithin log s xβ€– ≀ 1 / (1 - r) case refine_3 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ Convex ℝ s
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ β€–(1 + z).log - log 1β€– ≀ 1 / (1 - r) * β€–1 + z - 1β€– TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
rw [← hs]
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s ⊒ IsOpen s
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s ⊒ IsOpen (Metric.ball 1 r)
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s ⊒ IsOpen s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
exact Metric.isOpen_ball
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s ⊒ IsOpen (Metric.ball 1 r)
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s ⊒ IsOpen (Metric.ball 1 r) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
simp [← hs]
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s ⊒ 1 + z ∈ s
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s ⊒ Complex.abs z < r
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s ⊒ 1 + z ∈ s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
assumption
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s ⊒ Complex.abs z < r
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s ⊒ Complex.abs z < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
simp [← hs]
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s ⊒ 1 ∈ s
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s ⊒ 0 < r
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s ⊒ 1 ∈ s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
assumption
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s ⊒ 0 < r
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s ⊒ 0 < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
intro w ws
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s ⊒ βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s ⊒ βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
apply mem_slitPlane_of_near_one
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case a z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : w ∈ s ⊒ Complex.abs (w - 1) < 1
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
simp only [Metric.mem_ball, Complex.dist_eq, ← hs] at ws
case a z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : w ∈ s ⊒ Complex.abs (w - 1) < 1
case a z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs (w - 1) < 1
Please generate a tactic in lean4 to solve the state. STATE: case a z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : w ∈ s ⊒ Complex.abs (w - 1) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
calc abs (w - 1) < r := by assumption _ < 1 := r1
case a z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs (w - 1) < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs (w - 1) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
assumption
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs (w - 1) < r
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs (w - 1) < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
intro w ws
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 ⊒ βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : w ∈ s ⊒ Complex.abs w β‰₯ 1 - r
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 ⊒ βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
simp only [Metric.mem_ball, Complex.dist_eq, ← hs] at ws
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : w ∈ s ⊒ Complex.abs w β‰₯ 1 - r
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs w β‰₯ 1 - r
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : w ∈ s ⊒ Complex.abs w β‰₯ 1 - r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
calc abs w = abs (1 + (w - 1)) := by ring_nf _ β‰₯ abs (1 : β„‚) - abs (w - 1) := by bound _ β‰₯ abs (1 : β„‚) - r := by bound _ = 1 - r := by rw [Complex.abs.map_one]
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs w β‰₯ 1 - r
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs w β‰₯ 1 - r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
ring_nf
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs w = Complex.abs (1 + (w - 1))
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs w = Complex.abs (1 + (w - 1)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
bound
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs (1 + (w - 1)) β‰₯ Complex.abs 1 - Complex.abs (w - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs (1 + (w - 1)) β‰₯ Complex.abs 1 - Complex.abs (w - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
bound
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs 1 - Complex.abs (w - 1) β‰₯ Complex.abs 1 - r
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs 1 - Complex.abs (w - 1) β‰₯ Complex.abs 1 - r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
rw [Complex.abs.map_one]
z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs 1 - r = 1 - r
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 w : β„‚ ws : Complex.abs (w - 1) < r ⊒ Complex.abs 1 - r = 1 - r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
exact DifferentiableOn.clog differentiableOn_id sp
case refine_1 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ DifferentiableOn β„‚ log s
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ DifferentiableOn β„‚ log s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
intro w ws
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ βˆ€ x ∈ s, β€–derivWithin log s xβ€– ≀ 1 / (1 - r)
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ β€–derivWithin log s wβ€– ≀ 1 / (1 - r)
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ βˆ€ x ∈ s, β€–derivWithin log s xβ€– ≀ 1 / (1 - r) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
rw [derivWithin.clog o ws, derivWithin.cid o ws]
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ β€–derivWithin log s wβ€– ≀ 1 / (1 - r)
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ β€–1 / wβ€– ≀ 1 / (1 - r) case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ β€–derivWithin log s wβ€– ≀ 1 / (1 - r) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
simp only [one_div, norm_inv, Complex.norm_eq_abs]
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ β€–1 / wβ€– ≀ 1 / (1 - r) case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ (Complex.abs w)⁻¹ ≀ (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ β€–1 / wβ€– ≀ 1 / (1 - r) case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
rw [inv_le]
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ (Complex.abs w)⁻¹ ≀ (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ (1 - r)⁻¹⁻¹ ≀ Complex.abs w case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ (Complex.abs w)⁻¹ ≀ (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
have aw := sa w ws
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ (1 - r)⁻¹⁻¹ ≀ Complex.abs w case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : Complex.abs w β‰₯ 1 - r ⊒ (1 - r)⁻¹⁻¹ ≀ Complex.abs w case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ (1 - r)⁻¹⁻¹ ≀ Complex.abs w case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
simp at aw
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : Complex.abs w β‰₯ 1 - r ⊒ (1 - r)⁻¹⁻¹ ≀ Complex.abs w case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : 1 ≀ Complex.abs w + r ⊒ (1 - r)⁻¹⁻¹ ≀ Complex.abs w case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : Complex.abs w β‰₯ 1 - r ⊒ (1 - r)⁻¹⁻¹ ≀ Complex.abs w case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
field_simp
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : 1 ≀ Complex.abs w + r ⊒ (1 - r)⁻¹⁻¹ ≀ Complex.abs w case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : 1 ≀ Complex.abs w + r ⊒ 1 ≀ Complex.abs w + r case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : 1 ≀ Complex.abs w + r ⊒ (1 - r)⁻¹⁻¹ ≀ Complex.abs w case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
assumption
case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : 1 ≀ Complex.abs w + r ⊒ 1 ≀ Complex.abs w + r case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : 1 ≀ Complex.abs w + r ⊒ 1 ≀ Complex.abs w + r case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
have aw := sa w ws
case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : Complex.abs w β‰₯ 1 - r ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
linarith
case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : Complex.abs w β‰₯ 1 - r ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.ha z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s aw : Complex.abs w β‰₯ 1 - r ⊒ 0 < Complex.abs w case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
norm_num
case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ r < 1 case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ 0 < (1 - r)⁻¹ case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
assumption
case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ r < 1 case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.hb z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ r < 1 case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
exact differentiableWithinAt_id
case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.hf z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ DifferentiableWithinAt β„‚ (fun z => z) s w case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
exact sp w ws
case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.hx z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r w : β„‚ ws : w ∈ s ⊒ w.re > 0 ∨ w.im β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
rw [← hs]
case refine_3 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ Convex ℝ s
case refine_3 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ Convex ℝ (Metric.ball 1 r)
Please generate a tactic in lean4 to solve the state. STATE: case refine_3 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ Convex ℝ s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_log1p_small
[180, 1]
[218, 36]
exact convex_ball _ _
case refine_3 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ Convex ℝ (Metric.ball 1 r)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_3 z : β„‚ r : ℝ r1 : r < 1 h : Complex.abs z < r rp : 0 < r s : Set β„‚ hs : Metric.ball 1 r = s o : IsOpen s s1z : 1 + z ∈ s s1 : 1 ∈ s sp : βˆ€ w ∈ s, w.re > 0 ∨ w.im β‰  0 sa : βˆ€ w ∈ s, Complex.abs w β‰₯ 1 - r ⊒ Convex ℝ (Metric.ball 1 r) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
apply le_of_forall_pos_lt_add
a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e ⊒ a ≀ b
case h a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e ⊒ βˆ€ (Ξ΅ : ℝ), 0 < Ξ΅ β†’ a < b + Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e ⊒ a ≀ b TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
intro e ep
case h a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e ⊒ βˆ€ (Ξ΅ : ℝ), 0 < Ξ΅ β†’ a < b + Ξ΅
case h a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e ⊒ a < b + e
Please generate a tactic in lean4 to solve the state. STATE: case h a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e ⊒ βˆ€ (Ξ΅ : ℝ), 0 < Ξ΅ β†’ a < b + Ξ΅ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
by_cases et : e β‰₯ t
case h a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e ⊒ a < b + e
case pos a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e β‰₯ t ⊒ a < b + e case neg a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : Β¬e β‰₯ t ⊒ a < b + e
Please generate a tactic in lean4 to solve the state. STATE: case h a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e ⊒ a < b + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
specialize h (t/2) (by bound) (by bound)
case pos a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e β‰₯ t ⊒ a < b + e
case pos a b t : ℝ tp : 0 < t e : ℝ ep : 0 < e et : e β‰₯ t h : a ≀ b + t / 2 ⊒ a < b + e
Please generate a tactic in lean4 to solve the state. STATE: case pos a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e β‰₯ t ⊒ a < b + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
calc a ≀ b + t/2 := h _ ≀ b + e/2 := by bound _ < b + e := by bound
case pos a b t : ℝ tp : 0 < t e : ℝ ep : 0 < e et : e β‰₯ t h : a ≀ b + t / 2 ⊒ a < b + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos a b t : ℝ tp : 0 < t e : ℝ ep : 0 < e et : e β‰₯ t h : a ≀ b + t / 2 ⊒ a < b + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
bound
a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e β‰₯ t ⊒ 0 < t / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e β‰₯ t ⊒ 0 < t / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
bound
a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e β‰₯ t ⊒ t / 2 < t
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e β‰₯ t ⊒ t / 2 < t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
bound
a b t : ℝ tp : 0 < t e : ℝ ep : 0 < e et : e β‰₯ t h : a ≀ b + t / 2 ⊒ b + t / 2 ≀ b + e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b t : ℝ tp : 0 < t e : ℝ ep : 0 < e et : e β‰₯ t h : a ≀ b + t / 2 ⊒ b + t / 2 ≀ b + e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
bound
a b t : ℝ tp : 0 < t e : ℝ ep : 0 < e et : e β‰₯ t h : a ≀ b + t / 2 ⊒ b + e / 2 < b + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b t : ℝ tp : 0 < t e : ℝ ep : 0 < e et : e β‰₯ t h : a ≀ b + t / 2 ⊒ b + e / 2 < b + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
simp only [ge_iff_le, not_le] at et
case neg a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : Β¬e β‰₯ t ⊒ a < b + e
case neg a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e < t ⊒ a < b + e
Please generate a tactic in lean4 to solve the state. STATE: case neg a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : Β¬e β‰₯ t ⊒ a < b + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
calc a ≀ b + e/2 := h (e/2) (by bound) (by linarith) _ < b + e := by bound
case neg a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e < t ⊒ a < b + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e < t ⊒ a < b + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
bound
a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e < t ⊒ 0 < e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e < t ⊒ 0 < e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
linarith
a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e < t ⊒ e / 2 < t
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e < t ⊒ e / 2 < t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
le_of_forall_small_le_add
[220, 1]
[231, 28]
bound
a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e < t ⊒ b + e / 2 < b + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b t : ℝ tp : 0 < t h : βˆ€ (e : ℝ), 0 < e β†’ e < t β†’ a ≀ b + e e : ℝ ep : 0 < e et : e < t ⊒ b + e / 2 < b + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
one_over_one_sub_le
[233, 1]
[239, 21]
intro xp xh
x : ℝ ⊒ 0 ≀ x β†’ x ≀ 1 / 2 β†’ 1 / (1 - x) ≀ 1 + 2 * x
x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 ⊒ 1 / (1 - x) ≀ 1 + 2 * x
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ ⊒ 0 ≀ x β†’ x ≀ 1 / 2 β†’ 1 / (1 - x) ≀ 1 + 2 * x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
one_over_one_sub_le
[233, 1]
[239, 21]
have x1 : 1 - x > 0 := by linarith
x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 ⊒ 1 / (1 - x) ≀ 1 + 2 * x
x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ 1 / (1 - x) ≀ 1 + 2 * x
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 ⊒ 1 / (1 - x) ≀ 1 + 2 * x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
one_over_one_sub_le
[233, 1]
[239, 21]
rw [div_le_iff x1]
x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ 1 / (1 - x) ≀ 1 + 2 * x
x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ 1 ≀ (1 + 2 * x) * (1 - x)
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ 1 / (1 - x) ≀ 1 + 2 * x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
one_over_one_sub_le
[233, 1]
[239, 21]
calc (1 + 2*x) * (1 - x) = 1 + x * (1 - 2*x) := by ring _ β‰₯ 1 + x * (1 - 2 * (1/2)) := by bound _ = 1 := by ring
x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ 1 ≀ (1 + 2 * x) * (1 - x)
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ 1 ≀ (1 + 2 * x) * (1 - x) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
one_over_one_sub_le
[233, 1]
[239, 21]
linarith
x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 ⊒ 1 - x > 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 ⊒ 1 - x > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
one_over_one_sub_le
[233, 1]
[239, 21]
ring
x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ (1 + 2 * x) * (1 - x) = 1 + x * (1 - 2 * x)
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ (1 + 2 * x) * (1 - x) = 1 + x * (1 - 2 * x) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
one_over_one_sub_le
[233, 1]
[239, 21]
bound
x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ 1 + x * (1 - 2 * x) β‰₯ 1 + x * (1 - 2 * (1 / 2))
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ 1 + x * (1 - 2 * x) β‰₯ 1 + x * (1 - 2 * (1 / 2)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
one_over_one_sub_le
[233, 1]
[239, 21]
ring
x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ 1 + x * (1 - 2 * (1 / 2)) = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xp : 0 ≀ x xh : x ≀ 1 / 2 x1 : 1 - x > 0 ⊒ 1 + x * (1 - 2 * (1 / 2)) = 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
intro e ep
f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r ⊒ βˆ€ (e : ℝ), 0 < e β†’ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r ⊒ βˆ€ (e : ℝ), 0 < e β†’ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
rcases Metric.continuousAt_iff.mp fc e ep with ⟨s,sp,sc⟩
f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
case intro.intro f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, dist x z < s β†’ dist (f x) (f z) < e ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
simp_rw [ Complex.dist_eq ] at sc
case intro.intro f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, dist x z < s β†’ dist (f x) (f z) < e ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
case intro.intro f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, dist x z < s β†’ dist (f x) (f z) < e ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
by_cases sr : s ≀ r
case intro.intro f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
case pos f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : s ≀ r ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : Β¬s ≀ r ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
exists s
case pos f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : s ≀ r ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : Β¬s ≀ r ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : Β¬s ≀ r ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
Please generate a tactic in lean4 to solve the state. STATE: case pos f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : s ≀ r ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : Β¬s ≀ r ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
simp only [not_le] at sr
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : Β¬s ≀ r ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : Β¬s ≀ r ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
exists r
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ 0 < r ∧ r ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < r β†’ Complex.abs (f w - f z) < e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ βˆƒ s, 0 < s ∧ s ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
refine ⟨rp, by bound, ?_⟩
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ 0 < r ∧ r ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < r β†’ Complex.abs (f w - f z) < e
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ βˆ€ {w : β„‚}, Complex.abs (w - z) < r β†’ Complex.abs (f w - f z) < e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ 0 < r ∧ r ≀ r ∧ βˆ€ {w : β„‚}, Complex.abs (w - z) < r β†’ Complex.abs (f w - f z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
intro w wr
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ βˆ€ {w : β„‚}, Complex.abs (w - z) < r β†’ Complex.abs (f w - f z) < e
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ Complex.abs (f w - f z) < e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ βˆ€ {w : β„‚}, Complex.abs (w - z) < r β†’ Complex.abs (f w - f z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
refine @sc w ?_
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ Complex.abs (f w - f z) < e
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ Complex.abs (w - z) < s
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ Complex.abs (f w - f z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
trans r
case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ Complex.abs (w - z) < s
f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ Complex.abs (w - z) < r f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ r < s
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ Complex.abs (w - z) < s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
assumption
f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ Complex.abs (w - z) < r f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ r < s
f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ r < s
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ Complex.abs (w - z) < r f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ r < s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
assumption
f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ r < s
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s w : β„‚ wr : Complex.abs (w - z) < r ⊒ r < s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Metric.continuous_near
[241, 1]
[252, 34]
bound
f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ r ≀ r
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ r : ℝ fc : ContinuousAt f z rp : 0 < r e : ℝ ep : 0 < e s : ℝ sp : s > 0 sc : βˆ€ {x : β„‚}, Complex.abs (x - z) < s β†’ Complex.abs (f x - f z) < e sr : r < s ⊒ r ≀ r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
by_cases rz : abs z < r
z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
case pos z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : Complex.abs z < r ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : Β¬Complex.abs z < r ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
simp only [not_lt] at rz
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : Β¬Complex.abs z < r ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : Β¬Complex.abs z < r ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
have azp : 0 < abs z := Complex.abs.pos nz
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
generalize ha : 1 - r/2/abs z = a
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
have a0 : 0 ≀ a := by rw [←ha, sub_nonneg, div_le_one azp]; exact _root_.trans (by bound) rz
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
have a1 : a < 1 := by rw [←ha, sub_lt_self_iff]; positivity
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a a1 : a < 1 ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
generalize hw : ↑a * z = w
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a a1 : a < 1 ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a a1 : a < 1 w : β„‚ hw : ↑a * z = w ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a a1 : a < 1 ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
use w
case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a a1 : a < 1 w : β„‚ hw : ↑a * z = w ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
case h z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a a1 : a < 1 w : β„‚ hw : ↑a * z = w ⊒ Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a a1 : a < 1 w : β„‚ hw : ↑a * z = w ⊒ βˆƒ w, Complex.abs (w - z) < r ∧ Complex.abs w < Complex.abs z TACTIC: