Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files Community
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https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
cases' h with m m
case a.mpr A B : Finset β„• x : β„• h : x ∈ A ∧ x βˆ‰ B ∨ x ∈ A ∧ x ∈ B ⊒ x ∈ A
case a.mpr.inl A B : Finset β„• x : β„• m : x ∈ A ∧ x βˆ‰ B ⊒ x ∈ A case a.mpr.inr A B : Finset β„• x : β„• m : x ∈ A ∧ x ∈ B ⊒ x ∈ A
Please generate a tactic in lean4 to solve the state. STATE: case a.mpr A B : Finset β„• x : β„• h : x ∈ A ∧ x βˆ‰ B ∨ x ∈ A ∧ x ∈ B ⊒ x ∈ A TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
repeat exact m.1
case a.mpr.inl A B : Finset β„• x : β„• m : x ∈ A ∧ x βˆ‰ B ⊒ x ∈ A case a.mpr.inr A B : Finset β„• x : β„• m : x ∈ A ∧ x ∈ B ⊒ x ∈ A
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.mpr.inl A B : Finset β„• x : β„• m : x ∈ A ∧ x βˆ‰ B ⊒ x ∈ A case a.mpr.inr A B : Finset β„• x : β„• m : x ∈ A ∧ x ∈ B ⊒ x ∈ A TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
exact m.1
case a.mpr.inr A B : Finset β„• x : β„• m : x ∈ A ∧ x ∈ B ⊒ x ∈ A
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.mpr.inr A B : Finset β„• x : β„• m : x ∈ A ∧ x ∈ B ⊒ x ∈ A TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_sum_partition
[97, 1]
[101, 59]
have ha : A = A \ B βˆͺ A ∩ B := finset_partition A B
A B : Finset β„• f : β„• β†’ β„‚ ⊒ A.sum f = (A \ B).sum f + (A ∩ B).sum f
A B : Finset β„• f : β„• β†’ β„‚ ha : A = A \ B βˆͺ A ∩ B ⊒ A.sum f = (A \ B).sum f + (A ∩ B).sum f
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ⊒ A.sum f = (A \ B).sum f + (A ∩ B).sum f TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_sum_partition
[97, 1]
[101, 59]
nth_rw 1 [ha]
A B : Finset β„• f : β„• β†’ β„‚ ha : A = A \ B βˆͺ A ∩ B ⊒ A.sum f = (A \ B).sum f + (A ∩ B).sum f
A B : Finset β„• f : β„• β†’ β„‚ ha : A = A \ B βˆͺ A ∩ B ⊒ (A \ B βˆͺ A ∩ B).sum f = (A \ B).sum f + (A ∩ B).sum f
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ha : A = A \ B βˆͺ A ∩ B ⊒ A.sum f = (A \ B).sum f + (A ∩ B).sum f TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_sum_partition
[97, 1]
[101, 59]
exact Finset.sum_union (Finset.disjoint_sdiff_inter A B)
A B : Finset β„• f : β„• β†’ β„‚ ha : A = A \ B βˆͺ A ∩ B ⊒ (A \ B βˆͺ A ∩ B).sum f = (A \ B).sum f + (A ∩ B).sum f
no goals
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ha : A = A \ B βˆͺ A ∩ B ⊒ (A \ B βˆͺ A ∩ B).sum f = (A \ B).sum f + (A ∩ B).sum f TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_union
[106, 1]
[107, 41]
rw [symmDiff_def, Finset.sup_eq_union]
A B : Finset β„• ⊒ A βˆ† B = A \ B βˆͺ B \ A
no goals
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• ⊒ A βˆ† B = A \ B βˆͺ B \ A TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
rw [finset_sum_partition A B f, finset_sum_partition B A f, Finset.inter_comm B A]
A B : Finset β„• f : β„• β†’ β„‚ ⊒ dist (A.sum f) (B.sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n)
A B : Finset β„• f : β„• β†’ β„‚ ⊒ dist ((A \ B).sum f + (A ∩ B).sum f) ((B \ A).sum f + (A ∩ B).sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n)
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ⊒ dist (A.sum f) (B.sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
rw [dist_add_right ((A \ B).sum f) ((B \ A).sum f) ((A ∩ B).sum f)]
A B : Finset β„• f : β„• β†’ β„‚ ⊒ dist ((A \ B).sum f + (A ∩ B).sum f) ((B \ A).sum f + (A ∩ B).sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n)
A B : Finset β„• f : β„• β†’ β„‚ ⊒ dist ((A \ B).sum f) ((B \ A).sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n)
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ⊒ dist ((A \ B).sum f + (A ∩ B).sum f) ((B \ A).sum f + (A ∩ B).sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
rw [Complex.dist_eq]
A B : Finset β„• f : β„• β†’ β„‚ ⊒ dist ((A \ B).sum f) ((B \ A).sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n)
A B : Finset β„• f : β„• β†’ β„‚ ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n)
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ⊒ dist ((A \ B).sum f) ((B \ A).sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
trans (A \ B).sum (fun n ↦ abs (f n)) + (B \ A).sum (fun n ↦ abs (f n))
A B : Finset β„• f : β„• β†’ β„‚ ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n)
A B : Finset β„• f : β„• β†’ β„‚ ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) A B : Finset β„• f : β„• β†’ β„‚ ⊒ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) ≀ (A βˆ† B).sum fun n => Complex.abs (f n)
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ (A βˆ† B).sum fun n => Complex.abs (f n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
have ha := finset_complex_abs_sum_le (A \ B) f
A B : Finset β„• f : β„• β†’ β„‚ ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
A B : Finset β„• f : β„• β†’ β„‚ ha : Complex.abs ((A \ B).sum fun n => f n) ≀ (A \ B).sum fun n => Complex.abs (f n) ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
have hb := finset_complex_abs_sum_le (B \ A) f
A B : Finset β„• f : β„• β†’ β„‚ ha : Complex.abs ((A \ B).sum fun n => f n) ≀ (A \ B).sum fun n => Complex.abs (f n) ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
A B : Finset β„• f : β„• β†’ β„‚ ha : Complex.abs ((A \ B).sum fun n => f n) ≀ (A \ B).sum fun n => Complex.abs (f n) hb : Complex.abs ((B \ A).sum fun n => f n) ≀ (B \ A).sum fun n => Complex.abs (f n) ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ha : Complex.abs ((A \ B).sum fun n => f n) ≀ (A \ B).sum fun n => Complex.abs (f n) ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
calc abs ((A \ B).sum f - (B \ A).sum f) _ ≀ abs ((A \ B).sum f) + abs ((B \ A).sum f) := by bound _ ≀ (A \ B).sum (fun n ↦ abs (f n)) + (B \ A).sum (fun n ↦ abs (f n)) := by bound
A B : Finset β„• f : β„• β†’ β„‚ ha : Complex.abs ((A \ B).sum fun n => f n) ≀ (A \ B).sum fun n => Complex.abs (f n) hb : Complex.abs ((B \ A).sum fun n => f n) ≀ (B \ A).sum fun n => Complex.abs (f n) ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
no goals
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ha : Complex.abs ((A \ B).sum fun n => f n) ≀ (A \ B).sum fun n => Complex.abs (f n) hb : Complex.abs ((B \ A).sum fun n => f n) ≀ (B \ A).sum fun n => Complex.abs (f n) ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
bound
A B : Finset β„• f : β„• β†’ β„‚ ha : Complex.abs ((A \ B).sum fun n => f n) ≀ (A \ B).sum fun n => Complex.abs (f n) hb : Complex.abs ((B \ A).sum fun n => f n) ≀ (B \ A).sum fun n => Complex.abs (f n) ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ Complex.abs ((A \ B).sum f) + Complex.abs ((B \ A).sum f)
no goals
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ha : Complex.abs ((A \ B).sum fun n => f n) ≀ (A \ B).sum fun n => Complex.abs (f n) hb : Complex.abs ((B \ A).sum fun n => f n) ≀ (B \ A).sum fun n => Complex.abs (f n) ⊒ Complex.abs ((A \ B).sum f - (B \ A).sum f) ≀ Complex.abs ((A \ B).sum f) + Complex.abs ((B \ A).sum f) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
bound
A B : Finset β„• f : β„• β†’ β„‚ ha : Complex.abs ((A \ B).sum fun n => f n) ≀ (A \ B).sum fun n => Complex.abs (f n) hb : Complex.abs ((B \ A).sum fun n => f n) ≀ (B \ A).sum fun n => Complex.abs (f n) ⊒ Complex.abs ((A \ B).sum f) + Complex.abs ((B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)
no goals
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ha : Complex.abs ((A \ B).sum fun n => f n) ≀ (A \ B).sum fun n => Complex.abs (f n) hb : Complex.abs ((B \ A).sum fun n => f n) ≀ (B \ A).sum fun n => Complex.abs (f n) ⊒ Complex.abs ((A \ B).sum f) + Complex.abs ((B \ A).sum f) ≀ ((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
apply le_of_eq
A B : Finset β„• f : β„• β†’ β„‚ ⊒ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) ≀ (A βˆ† B).sum fun n => Complex.abs (f n)
case a A B : Finset β„• f : β„• β†’ β„‚ ⊒ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) = (A βˆ† B).sum fun n => Complex.abs (f n)
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• f : β„• β†’ β„‚ ⊒ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) ≀ (A βˆ† B).sum fun n => Complex.abs (f n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_bound
[109, 1]
[121, 70]
rw [←Finset.sum_union (sdiff_sdiff_disjoint A B), symmDiff_union]
case a A B : Finset β„• f : β„• β†’ β„‚ ⊒ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) = (A βˆ† B).sum fun n => Complex.abs (f n)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a A B : Finset β„• f : β„• β†’ β„‚ ⊒ (((A \ B).sum fun n => Complex.abs (f n)) + (B \ A).sum fun n => Complex.abs (f n)) = (A βˆ† B).sum fun n => Complex.abs (f n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
intro n ab
A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m ⊒ Late (A βˆ† B) m
A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A βˆ† B ⊒ n β‰₯ m
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m ⊒ Late (A βˆ† B) m TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
rw [symmDiff_def, Finset.sup_eq_union, Finset.mem_union] at ab
A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A βˆ† B ⊒ n β‰₯ m
A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A \ B ∨ n ∈ B \ A ⊒ n β‰₯ m
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A βˆ† B ⊒ n β‰₯ m TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
by_contra h
A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A \ B ∨ n ∈ B \ A ⊒ n β‰₯ m
A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A \ B ∨ n ∈ B \ A h : Β¬n β‰₯ m ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A \ B ∨ n ∈ B \ A ⊒ n β‰₯ m TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
simp at h
A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A \ B ∨ n ∈ B \ A h : Β¬n β‰₯ m ⊒ False
A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A \ B ∨ n ∈ B \ A h : n < m ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A \ B ∨ n ∈ B \ A h : Β¬n β‰₯ m ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
cases' ab with a b
A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A \ B ∨ n ∈ B \ A h : n < m ⊒ False
case inl A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m a : n ∈ A \ B ⊒ False case inr A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m b : n ∈ B \ A ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• ab : n ∈ A \ B ∨ n ∈ B \ A h : n < m ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
rw [Finset.mem_sdiff] at a
case inl A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m a : n ∈ A \ B ⊒ False
case inl A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m a : n ∈ A ∧ n βˆ‰ B ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inl A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m a : n ∈ A \ B ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
have h := Finset.mem_of_subset hb (Finset.mem_range.mpr h)
case inl A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m a : n ∈ A ∧ n βˆ‰ B ⊒ False
case inl A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h✝ : n < m a : n ∈ A ∧ n βˆ‰ B h : n ∈ B ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inl A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m a : n ∈ A ∧ n βˆ‰ B ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
exact a.2 h
case inl A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h✝ : n < m a : n ∈ A ∧ n βˆ‰ B h : n ∈ B ⊒ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h✝ : n < m a : n ∈ A ∧ n βˆ‰ B h : n ∈ B ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
rw [Finset.mem_sdiff] at b
case inr A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m b : n ∈ B \ A ⊒ False
case inr A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m b : n ∈ B ∧ n βˆ‰ A ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m b : n ∈ B \ A ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
have h := Finset.mem_of_subset ha (Finset.mem_range.mpr h)
case inr A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m b : n ∈ B ∧ n βˆ‰ A ⊒ False
case inr A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h✝ : n < m b : n ∈ B ∧ n βˆ‰ A h : n ∈ A ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case inr A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h : n < m b : n ∈ B ∧ n βˆ‰ A ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
symmDiff_late
[124, 1]
[135, 16]
exact b.2 h
case inr A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h✝ : n < m b : n ∈ B ∧ n βˆ‰ A h : n ∈ A ⊒ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr A B : Finset β„• m : β„• ha : A β‰₯ Finset.range m hb : B β‰₯ Finset.range m n : β„• h✝ : n < m b : n ∈ B ∧ n βˆ‰ A h : n ∈ A ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sub_near
[138, 1]
[147, 54]
rw [abs_le]
a z : β„‚ ⊒ |Complex.abs (a - z) - Complex.abs a| ≀ Complex.abs z
a z : β„‚ ⊒ -Complex.abs z ≀ Complex.abs (a - z) - Complex.abs a ∧ Complex.abs (a - z) - Complex.abs a ≀ Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: a z : β„‚ ⊒ |Complex.abs (a - z) - Complex.abs a| ≀ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sub_near
[138, 1]
[147, 54]
constructor
a z : β„‚ ⊒ -Complex.abs z ≀ Complex.abs (a - z) - Complex.abs a ∧ Complex.abs (a - z) - Complex.abs a ≀ Complex.abs z
case left a z : β„‚ ⊒ -Complex.abs z ≀ Complex.abs (a - z) - Complex.abs a case right a z : β„‚ ⊒ Complex.abs (a - z) - Complex.abs a ≀ Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: a z : β„‚ ⊒ -Complex.abs z ≀ Complex.abs (a - z) - Complex.abs a ∧ Complex.abs (a - z) - Complex.abs a ≀ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sub_near
[138, 1]
[147, 54]
simp only [neg_le_sub_iff_le_add]
case left a z : β„‚ ⊒ -Complex.abs z ≀ Complex.abs (a - z) - Complex.abs a
case left a z : β„‚ ⊒ Complex.abs a ≀ Complex.abs (a - z) + Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case left a z : β„‚ ⊒ -Complex.abs z ≀ Complex.abs (a - z) - Complex.abs a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sub_near
[138, 1]
[147, 54]
calc abs (a - z) + abs z _ β‰₯ |abs a - abs z| + abs z := by bound _ β‰₯ abs a - abs z + abs z := by bound _ = abs a := by simp only [sub_add_cancel]
case left a z : β„‚ ⊒ Complex.abs a ≀ Complex.abs (a - z) + Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case left a z : β„‚ ⊒ Complex.abs a ≀ Complex.abs (a - z) + Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sub_near
[138, 1]
[147, 54]
bound
a z : β„‚ ⊒ Complex.abs (a - z) + Complex.abs z β‰₯ |Complex.abs a - Complex.abs z| + Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: a z : β„‚ ⊒ Complex.abs (a - z) + Complex.abs z β‰₯ |Complex.abs a - Complex.abs z| + Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sub_near
[138, 1]
[147, 54]
bound
a z : β„‚ ⊒ |Complex.abs a - Complex.abs z| + Complex.abs z β‰₯ Complex.abs a - Complex.abs z + Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: a z : β„‚ ⊒ |Complex.abs a - Complex.abs z| + Complex.abs z β‰₯ Complex.abs a - Complex.abs z + Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sub_near
[138, 1]
[147, 54]
simp only [sub_add_cancel]
a z : β„‚ ⊒ Complex.abs a - Complex.abs z + Complex.abs z = Complex.abs a
no goals
Please generate a tactic in lean4 to solve the state. STATE: a z : β„‚ ⊒ Complex.abs a - Complex.abs z + Complex.abs z = Complex.abs a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sub_near
[138, 1]
[147, 54]
calc abs (a - z) - abs a ≀ abs a + abs z - abs a := by bound _ = abs z := by simp only [add_sub_cancel_left]
case right a z : β„‚ ⊒ Complex.abs (a - z) - Complex.abs a ≀ Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case right a z : β„‚ ⊒ Complex.abs (a - z) - Complex.abs a ≀ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sub_near
[138, 1]
[147, 54]
bound
a z : β„‚ ⊒ Complex.abs (a - z) - Complex.abs a ≀ Complex.abs a + Complex.abs z - Complex.abs a
no goals
Please generate a tactic in lean4 to solve the state. STATE: a z : β„‚ ⊒ Complex.abs (a - z) - Complex.abs a ≀ Complex.abs a + Complex.abs z - Complex.abs a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sub_near
[138, 1]
[147, 54]
simp only [add_sub_cancel_left]
a z : β„‚ ⊒ Complex.abs a + Complex.abs z - Complex.abs a = Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: a z : β„‚ ⊒ Complex.abs a + Complex.abs z - Complex.abs a = Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
add_near
[149, 1]
[152, 13]
have h := sub_near a (-z)
a z : β„‚ ⊒ |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z
a z : β„‚ h : |Complex.abs (a - -z) - Complex.abs a| ≀ Complex.abs (-z) ⊒ |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: a z : β„‚ ⊒ |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
add_near
[149, 1]
[152, 13]
simp only [sub_neg_eq_add, map_neg_eq_map] at h
a z : β„‚ h : |Complex.abs (a - -z) - Complex.abs a| ≀ Complex.abs (-z) ⊒ |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z
a z : β„‚ h : |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z ⊒ |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: a z : β„‚ h : |Complex.abs (a - -z) - Complex.abs a| ≀ Complex.abs (-z) ⊒ |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
add_near
[149, 1]
[152, 13]
assumption
a z : β„‚ h : |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z ⊒ |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: a z : β„‚ h : |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z ⊒ |Complex.abs (a + z) - Complex.abs a| ≀ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
mem_slitPlane_of_near_one
[154, 1]
[163, 13]
intro h
z : β„‚ ⊒ Complex.abs (z - 1) < 1 β†’ z ∈ slitPlane
z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ z ∈ slitPlane
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ ⊒ Complex.abs (z - 1) < 1 β†’ z ∈ slitPlane TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
mem_slitPlane_of_near_one
[154, 1]
[163, 13]
apply Or.inl
z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ z ∈ slitPlane
case h z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ 0 < z.re
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ z ∈ slitPlane TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
mem_slitPlane_of_near_one
[154, 1]
[163, 13]
have hr : (1 - z).re < 1 := by calc (1 - z).re ≀ |(1 - z).re| := le_abs_self (1 - z).re _ ≀ abs (1 - z) := (Complex.abs_re_le_abs _) _ = abs (z - 1) := by rw [←Complex.abs.map_neg (1 - z)]; simp only [neg_sub] _ < 1 := h
case h z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ 0 < z.re
case h z : β„‚ h : Complex.abs (z - 1) < 1 hr : (1 - z).re < 1 ⊒ 0 < z.re
Please generate a tactic in lean4 to solve the state. STATE: case h z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ 0 < z.re TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
mem_slitPlane_of_near_one
[154, 1]
[163, 13]
simp only [Complex.sub_re, Complex.one_re, sub_lt_self_iff] at hr
case h z : β„‚ h : Complex.abs (z - 1) < 1 hr : (1 - z).re < 1 ⊒ 0 < z.re
case h z : β„‚ h : Complex.abs (z - 1) < 1 hr : 0 < z.re ⊒ 0 < z.re
Please generate a tactic in lean4 to solve the state. STATE: case h z : β„‚ h : Complex.abs (z - 1) < 1 hr : (1 - z).re < 1 ⊒ 0 < z.re TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
mem_slitPlane_of_near_one
[154, 1]
[163, 13]
assumption
case h z : β„‚ h : Complex.abs (z - 1) < 1 hr : 0 < z.re ⊒ 0 < z.re
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h z : β„‚ h : Complex.abs (z - 1) < 1 hr : 0 < z.re ⊒ 0 < z.re TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
mem_slitPlane_of_near_one
[154, 1]
[163, 13]
calc (1 - z).re ≀ |(1 - z).re| := le_abs_self (1 - z).re _ ≀ abs (1 - z) := (Complex.abs_re_le_abs _) _ = abs (z - 1) := by rw [←Complex.abs.map_neg (1 - z)]; simp only [neg_sub] _ < 1 := h
z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ (1 - z).re < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ (1 - z).re < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
mem_slitPlane_of_near_one
[154, 1]
[163, 13]
rw [←Complex.abs.map_neg (1 - z)]
z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ Complex.abs (1 - z) = Complex.abs (z - 1)
z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ Complex.abs (-(1 - z)) = Complex.abs (z - 1)
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ Complex.abs (1 - z) = Complex.abs (z - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
mem_slitPlane_of_near_one
[154, 1]
[163, 13]
simp only [neg_sub]
z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ Complex.abs (-(1 - z)) = Complex.abs (z - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ Complex.abs (-(1 - z)) = Complex.abs (z - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
near_one_avoids_zero
[165, 1]
[166, 73]
intro h
z : β„‚ ⊒ Complex.abs (z - 1) < 1 β†’ z β‰  0
z : β„‚ h : Complex.abs (z - 1) < 1 ⊒ z β‰  0
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ ⊒ Complex.abs (z - 1) < 1 β†’ z β‰  0 TACTIC:
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: