Datasets:
pkuAI4M
/
lean_github

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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:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
constructor
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
case h.left 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 case h.right 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 < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
use 0
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 h z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : Complex.abs z < r ⊒ Complex.abs (0 - z) < r ∧ Complex.abs 0 < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: 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 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
simp only [zero_sub, map_neg_eq_map, rz, map_zero, AbsoluteValue.pos_iff, ne_eq, nz, not_false_eq_true, and_self]
case h z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : Complex.abs z < r ⊒ Complex.abs (0 - z) < r ∧ Complex.abs 0 < Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h z : β„‚ nz : z β‰  0 r : ℝ rp : 0 < r rz : Complex.abs z < r ⊒ Complex.abs (0 - z) < r ∧ Complex.abs 0 < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
rw [←ha, sub_nonneg, div_le_one azp]
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 ⊒ 0 ≀ a
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 ⊒ r / 2 ≀ 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 rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a ⊒ 0 ≀ a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
exact _root_.trans (by bound) rz
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 ⊒ r / 2 ≀ Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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 ⊒ r / 2 ≀ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
bound
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 ⊒ r / 2 ≀ r
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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 ⊒ r / 2 ≀ r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
rw [←ha, sub_lt_self_iff]
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 ⊒ a < 1
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 ⊒ 0 < r / 2 / 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 rz : r ≀ Complex.abs z azp : 0 < Complex.abs z a : ℝ ha : 1 - r / 2 / Complex.abs z = a a0 : 0 ≀ a ⊒ a < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
positivity
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 ⊒ 0 < r / 2 / Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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 ⊒ 0 < r / 2 / Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
rw [←hw,←ha]
case h.left 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
case h.left 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 (↑(1 - r / 2 / Complex.abs z) * z - z) < r
Please generate a tactic in lean4 to solve the state. STATE: case h.left 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 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
simp only [Complex.ofReal_sub, Complex.ofReal_one, Complex.ofReal_div]
case h.left 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 (↑(1 - r / 2 / Complex.abs z) * z - z) < r
case h.left 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 ((1 - ↑r / ↑2 / ↑(Complex.abs z)) * z - z) < r
Please generate a tactic in lean4 to solve the state. STATE: case h.left 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 (↑(1 - r / 2 / Complex.abs z) * z - z) < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
rw [mul_sub_right_distrib]
case h.left 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 ((1 - ↑r / ↑2 / ↑(Complex.abs z)) * z - z) < r
case h.left 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 (1 * z - ↑r / ↑2 / ↑(Complex.abs z) * z - z) < r
Please generate a tactic in lean4 to solve the state. STATE: case h.left 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 ((1 - ↑r / ↑2 / ↑(Complex.abs z)) * z - z) < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
simp only [one_mul, sub_sub_cancel_left, AbsoluteValue.map_neg, AbsoluteValue.map_mul, map_divβ‚€, Complex.abs_ofReal, Complex.abs_two, Complex.abs_abs, abs_of_pos rp, div_mul (r/2), div_mul_cancelβ‚€ _ azp.ne', div_one, abs_two, div_self azp.ne']
case h.left 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 (1 * z - ↑r / ↑2 / ↑(Complex.abs z) * z - z) < r
case h.left 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 ⊒ r / 2 < r
Please generate a tactic in lean4 to solve the state. STATE: case h.left 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 (1 * z - ↑r / ↑2 / ↑(Complex.abs z) * z - z) < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
bound
case h.left 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 ⊒ r / 2 < r
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.left 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 ⊒ r / 2 < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
simp only [←hw, AbsoluteValue.map_mul, Complex.abs_ofReal, abs_of_nonneg a0]
case h.right 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 < Complex.abs z
case h.right 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 ⊒ a * Complex.abs z < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case h.right 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 < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
calc a * abs z < 1 * abs z := mul_lt_mul_of_pos_right a1 azp _ = abs z := by ring
case h.right 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 ⊒ a * Complex.abs z < Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.right 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 ⊒ a * Complex.abs z < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
slightly_smaller
[254, 1]
[276, 27]
ring
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 ⊒ 1 * Complex.abs z = Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: 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 ⊒ 1 * Complex.abs z = Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
frequently_smaller
[279, 1]
[282, 74]
simp only [Filter.Frequently, Metric.eventually_nhds_iff, not_exists, not_forall, not_not, Complex.dist_eq, not_and]
z : β„‚ z0 : z β‰  0 ⊒ βˆƒαΆ  (w : β„‚) in 𝓝 z, Complex.abs w < Complex.abs z
z : β„‚ z0 : z β‰  0 ⊒ βˆ€ x > 0, βˆƒ x_1, βˆƒ (_ : Complex.abs (x_1 - z) < x), Complex.abs x_1 < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ z0 : z β‰  0 ⊒ βˆƒαΆ  (w : β„‚) in 𝓝 z, Complex.abs w < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
frequently_smaller
[279, 1]
[282, 74]
intro r rp
z : β„‚ z0 : z β‰  0 ⊒ βˆ€ x > 0, βˆƒ x_1, βˆƒ (_ : Complex.abs (x_1 - z) < x), Complex.abs x_1 < Complex.abs z
z : β„‚ z0 : z β‰  0 r : ℝ rp : r > 0 ⊒ βˆƒ x, βˆƒ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ z0 : z β‰  0 ⊒ βˆ€ x > 0, βˆƒ x_1, βˆƒ (_ : Complex.abs (x_1 - z) < x), Complex.abs x_1 < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
frequently_smaller
[279, 1]
[282, 74]
rcases slightly_smaller z0 rp with ⟨w, b, lt⟩
z : β„‚ z0 : z β‰  0 r : ℝ rp : r > 0 ⊒ βˆƒ x, βˆƒ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z
case intro.intro z : β„‚ z0 : z β‰  0 r : ℝ rp : r > 0 w : β„‚ b : Complex.abs (w - z) < r lt : Complex.abs w < Complex.abs z ⊒ βˆƒ x, βˆƒ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ z0 : z β‰  0 r : ℝ rp : r > 0 ⊒ βˆƒ x, βˆƒ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
frequently_smaller
[279, 1]
[282, 74]
use w, b, lt
case intro.intro z : β„‚ z0 : z β‰  0 r : ℝ rp : r > 0 w : β„‚ b : Complex.abs (w - z) < r lt : Complex.abs w < Complex.abs z ⊒ βˆƒ x, βˆƒ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro z : β„‚ z0 : z β‰  0 r : ℝ rp : r > 0 w : β„‚ b : Complex.abs (w - z) < r lt : Complex.abs w < Complex.abs z ⊒ βˆƒ x, βˆƒ (_ : Complex.abs (x - z) < r), Complex.abs x < Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
by_cases nz : z = 0
f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z ⊒ Complex.abs (f z) ≀ c * Complex.abs z
case pos f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : z = 0 ⊒ Complex.abs (f z) ≀ c * Complex.abs z case neg f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 ⊒ Complex.abs (f z) ≀ c * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z ⊒ Complex.abs (f z) ≀ c * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
apply le_of_forall_small_le_add zero_lt_one
case neg f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 ⊒ Complex.abs (f z) ≀ c * Complex.abs z
case neg f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 ⊒ βˆ€ (e : ℝ), 0 < e β†’ e < 1 β†’ Complex.abs (f z) ≀ c * Complex.abs z + e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 ⊒ Complex.abs (f z) ≀ c * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
intro e ep _
case neg f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 ⊒ βˆ€ (e : ℝ), 0 < e β†’ e < 1 β†’ Complex.abs (f z) ≀ c * Complex.abs z + e
case neg f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 ⊒ βˆ€ (e : ℝ), 0 < e β†’ e < 1 β†’ Complex.abs (f z) ≀ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
rcases Metric.continuous_near fc rp e ep with ⟨s,sp,_,sc⟩
case neg f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e
case neg.intro.intro.intro f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
rcases slightly_smaller nz sp with ⟨w,wz,awz⟩
case neg.intro.intro.intro f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e
case neg.intro.intro.intro.intro.intro f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.intro.intro.intro f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
have wr : abs w < r := lt_of_lt_of_le awz zr
case neg.intro.intro.intro.intro.intro f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e
case neg.intro.intro.intro.intro.intro f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.intro.intro.intro.intro.intro f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
calc abs (f z) = abs (f w - (f w - f z)) := by ring_nf _ ≀ abs (f w) + abs (f w - f z) := by bound _ ≀ c * abs w + e := by bound [h w wr, sc wz] _ ≀ c * abs z + e := by bound
case neg.intro.intro.intro.intro.intro f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg.intro.intro.intro.intro.intro f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ Complex.abs (f z) ≀ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
refine h z ?_
case pos f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : z = 0 ⊒ Complex.abs (f z) ≀ c * Complex.abs z
case pos f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : z = 0 ⊒ Complex.abs z < r
Please generate a tactic in lean4 to solve the state. STATE: case pos f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : z = 0 ⊒ Complex.abs (f z) ≀ c * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
rw [nz]
case pos f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : z = 0 ⊒ Complex.abs z < r
case pos f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : z = 0 ⊒ Complex.abs 0 < r
Please generate a tactic in lean4 to solve the state. STATE: case pos f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : z = 0 ⊒ Complex.abs z < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
simpa
case pos f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : z = 0 ⊒ Complex.abs 0 < r
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : z = 0 ⊒ Complex.abs 0 < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
ring_nf
f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ Complex.abs (f z) = Complex.abs (f w - (f w - f z))
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ Complex.abs (f z) = Complex.abs (f w - (f w - f z)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
bound
f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ Complex.abs (f w - (f w - f z)) ≀ Complex.abs (f w) + Complex.abs (f w - f z)
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ Complex.abs (f w - (f w - f z)) ≀ Complex.abs (f w) + Complex.abs (f w - f z) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
bound [h w wr, sc wz]
f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ Complex.abs (f w) + Complex.abs (f w - f z) ≀ c * Complex.abs w + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ Complex.abs (f w) + Complex.abs (f w - f z) ≀ c * Complex.abs w + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_to_strong_small
[284, 1]
[296, 34]
bound
f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ c * Complex.abs w + e ≀ c * Complex.abs z + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ r c : ℝ rp : r > 0 cp : c > 0 zr : Complex.abs z ≀ r fc : ContinuousAt f z h : βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (f z) ≀ c * Complex.abs z nz : Β¬z = 0 e : ℝ ep : 0 < e a✝ : e < 1 s : ℝ sp : 0 < s left✝ : s ≀ r sc : βˆ€ {w : β„‚}, Complex.abs (w - z) < s β†’ Complex.abs (f w - f z) < e w : β„‚ wz : Complex.abs (w - z) < s awz : Complex.abs w < Complex.abs z wr : Complex.abs w < r ⊒ c * Complex.abs w + e ≀ c * Complex.abs z + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
by_cases r0 : r ≀ 0
z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case pos z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : r ≀ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 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 zr : 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
log1p_small'
[298, 1]
[312, 44]
simp only [not_le] at r0
case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : Β¬r ≀ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 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 zr : Complex.abs z ≀ r r0 : 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
log1p_small'
[298, 1]
[312, 44]
have fc : ContinuousAt (fun z ↦ log (1 + z)) z := by apply ContinuousAt.clog; apply ContinuousAt.add; exact continuousAt_const; exact continuousAt_id refine mem_slitPlane_of_near_one ?_ simp only [add_sub_cancel_left, lt_of_le_of_lt zr r1]
case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) 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 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
apply weak_to_strong_small r0 (by bound) zr fc
case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊒ βˆ€ (z : β„‚), Complex.abs z < r β†’ 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 zr : Complex.abs z ≀ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
intro w wr
case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊒ βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z w : β„‚ wr : Complex.abs w < r ⊒ Complex.abs (1 + w).log ≀ 1 / (1 - r) * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊒ βˆ€ (z : β„‚), Complex.abs z < r β†’ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
exact @weak_log1p_small w r (by bound) wr
case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z w : β„‚ wr : Complex.abs w < r ⊒ Complex.abs (1 + w).log ≀ 1 / (1 - r) * Complex.abs w
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z w : β„‚ wr : Complex.abs w < r ⊒ Complex.abs (1 + w).log ≀ 1 / (1 - r) * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
have z0 := le_antisymm (le_trans zr r0) (Complex.abs.nonneg _)
case pos z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : r ≀ 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case pos z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : r ≀ 0 z0 : 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 zr : Complex.abs z ≀ r r0 : 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
log1p_small'
[298, 1]
[312, 44]
simp only [map_eq_zero] at z0
case pos z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : r ≀ 0 z0 : Complex.abs z = 0 ⊒ Complex.abs (1 + z).log ≀ 1 / (1 - r) * Complex.abs z
case pos z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : r ≀ 0 z0 : 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 zr : Complex.abs z ≀ r r0 : r ≀ 0 z0 : 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
log1p_small'
[298, 1]
[312, 44]
simp only [z0, add_zero, Complex.log_one, map_zero, r0, sub_zero, ne_eq, one_ne_zero, not_false_eq_true, div_self, mul_zero, le_refl]
case pos z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : r ≀ 0 z0 : z = 0 ⊒ 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 pos z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : r ≀ 0 z0 : 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
log1p_small'
[298, 1]
[312, 44]
apply ContinuousAt.clog
z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun z => (1 + z).log) z
case h₁ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun t => 1 + t) z case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane
Please generate a tactic in lean4 to solve the state. STATE: z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun z => (1 + z).log) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
apply ContinuousAt.add
case h₁ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun t => 1 + t) z case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane
case h₁.hf z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun x => 1) z case h₁.hg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun x => x) z case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane
Please generate a tactic in lean4 to solve the state. STATE: case h₁ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun t => 1 + t) z case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
exact continuousAt_const
case h₁.hf z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun x => 1) z case h₁.hg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun x => x) z case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane
case h₁.hg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun x => x) z case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane
Please generate a tactic in lean4 to solve the state. STATE: case h₁.hf z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun x => 1) z case h₁.hg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun x => x) z case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
exact continuousAt_id
case h₁.hg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun x => x) z case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane
case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane
Please generate a tactic in lean4 to solve the state. STATE: case h₁.hg z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ ContinuousAt (fun x => x) z case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
refine mem_slitPlane_of_near_one ?_
case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane
case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ Complex.abs (1 + z - 1) < 1
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ 1 + z ∈ slitPlane TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
simp only [add_sub_cancel_left, lt_of_le_of_lt zr r1]
case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ Complex.abs (1 + z - 1) < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ z : β„‚ r : ℝ r1 : r < 1 zr : Complex.abs z ≀ r r0 : 0 < r ⊒ Complex.abs (1 + z - 1) < 1 TACTIC: