Datasets:
pkuAI4M
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lean_github

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https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
generalize hMp : M.sum (fun k : β„• ↦ p k fun _ ↦ y) = Mp
case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d dppr : dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 ⊒ dist (M.sum fun b => (cauchyPowerSeries g c (↑r) b) fun x => y) (g (c + y)) < e
case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d dppr : dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp ⊒ dist Mp (g (c + y)) < e
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d dppr : dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 ⊒ dist (M.sum fun b => (cauchyPowerSeries g c (↑r) b) fun x => y) (g (c + y)) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [hMp] at dppr
case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d dppr : dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp ⊒ dist Mp (g (c + y)) < e
case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d Mp : β„‚ dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 hMp : (M.sum fun k => (p k) fun x => y) = Mp ⊒ dist Mp (g (c + y)) < e
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d dppr : dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp ⊒ dist Mp (g (c + y)) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
generalize hMpr : M.sum (fun k ↦ pr n k fun _ ↦ y) = Mpr
case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d Mp : β„‚ dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 hMp : (M.sum fun k => (p k) fun x => y) = Mp ⊒ dist Mp (g (c + y)) < e
case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d Mp : β„‚ dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ dist Mp (g (c + y)) < e
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d Mp : β„‚ dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 hMp : (M.sum fun k => (p k) fun x => y) = Mp ⊒ dist Mp (g (c + y)) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [hMpr] at dpf dppr
case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d Mp : β„‚ dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ dist Mp (g (c + y)) < e
case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ dist Mp (g (c + y)) < e
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d Mp : β„‚ dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ dist Mp (g (c + y)) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
calc dist Mp (g (c + y)) _ ≀ dist Mp (f n (c + y)) + dist (f n (c + y)) (g (c + y)) := dist_triangle _ _ _ _ ≀ dist Mp Mpr + dist Mpr (f n (c + y)) + d := by bound _ ≀ e / 4 + d + d := by bound _ = e / 4 + 2 * (1 - a) * (e / 4) := by rw [← d4]; ring _ ≀ e / 4 + 2 * (1 - 0) * (e / 4) := by bound _ = 3 / 4 * e := by ring _ < 1 * e := (mul_lt_mul_of_pos_right (by norm_num) ep) _ = e := by simp
case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ dist Mp (g (c + y)) < e
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro.intro.intro I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ dist Mp (g (c + y)) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
intro n
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r ⊒ βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r ⊒ βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
have cs := cauchy_on_cball_radius rp (hb n)
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c ↑r) c ↑r ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
have pn : pr n = cauchyPowerSeries (f n) c r := rfl
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c ↑r) c ↑r ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c ↑r) c ↑r pn : pr n = cauchyPowerSeries (f n) c ↑r ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c ↑r) c ↑r ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [← pn] at cs
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c ↑r) c ↑r pn : pr n = cauchyPowerSeries (f n) c ↑r ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I cs : HasFPowerSeriesOnBall (f n) (pr n) c ↑r pn : pr n = cauchyPowerSeries (f n) c ↑r ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c ↑r) c ↑r pn : pr n = cauchyPowerSeries (f n) c ↑r ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
exact cs
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I cs : HasFPowerSeriesOnBall (f n) (pr n) c ↑r pn : pr n = cauchyPowerSeries (f n) c ↑r ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ o : IsOpen s h : βˆ€ (n : I), AnalyticOn β„‚ (f n) s u : TendstoUniformlyOn f g atTop s c : β„‚ hc : c ∈ s r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r hb : βˆ€ (n : I), AnalyticOn β„‚ (f n) (closedBall c ↑r) pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r n : I cs : HasFPowerSeriesOnBall (f n) (pr n) c ↑r pn : pr n = cauchyPowerSeries (f n) c ↑r ⊒ HasFPowerSeriesOnBall (f n) (pr n) c ↑r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
bound
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ u : TendstoUniformlyOn f g atTop s c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r ⊒ a β‰₯ 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ u : TendstoUniformlyOn f g atTop s c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r ⊒ a β‰₯ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
bound
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ u : TendstoUniformlyOn f g atTop s c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 ⊒ 1 - a > 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ u : TendstoUniformlyOn f g atTop s c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 ⊒ 1 - a > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [← d4]
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ u : TendstoUniformlyOn f g atTop s c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d ⊒ d > 0
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ u : TendstoUniformlyOn f g atTop s c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d ⊒ (1 - a) * (e / 4) > 0
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ u : TendstoUniformlyOn f g atTop s c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d ⊒ d > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
bound
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ u : TendstoUniformlyOn f g atTop s c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d ⊒ (1 - a) * (e / 4) > 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ u : TendstoUniformlyOn f g atTop s c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d ⊒ (1 - a) * (e / 4) > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
apply le_of_lt
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ dist (f n (c + y)) (g (c + y)) ≀ d
case a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ dist (f n (c + y)) (g (c + y)) < d
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ dist (f n (c + y)) (g (c + y)) ≀ d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [dist_comm]
case a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ dist (f n (c + y)) (g (c + y)) < d
case a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ dist (g (c + y)) (f n (c + y)) < d
Please generate a tactic in lean4 to solve the state. STATE: case a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ dist (f n (c + y)) (g (c + y)) < d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
refine hn (c + y) ?_
case a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ dist (g (c + y)) (f n (c + y)) < d
case a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ c + y ∈ s
Please generate a tactic in lean4 to solve the state. STATE: case a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ dist (g (c + y)) (f n (c + y)) < d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
apply cb
case a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ c + y ∈ s
case a.a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ c + y ∈ closedBall c ↑r
Please generate a tactic in lean4 to solve the state. STATE: case a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ c + y ∈ s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
simp
case a.a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ c + y ∈ closedBall c ↑r
case a.a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ Complex.abs y ≀ ↑r
Please generate a tactic in lean4 to solve the state. STATE: case a.a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ c + y ∈ closedBall c ↑r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
exact yr.le
case a.a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ Complex.abs y ≀ ↑r
no goals
Please generate a tactic in lean4 to solve the state. STATE: case a.a I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yb : y ∈ EMetric.ball 0 ↑r yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d ⊒ Complex.abs y ≀ ↑r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
trans M.sum fun k : β„• ↦ dist (p k fun _ ↦ y) (pr n k fun _ ↦ y)
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) ≀ e / 4
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) ≀ M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y) I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) ≀ e / 4
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) ≀ e / 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
apply dist_sum_sum_le M (fun k : β„• ↦ p k fun _ ↦ y) fun k : β„• ↦ pr n k fun _ ↦ y
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) ≀ M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y) I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) ≀ e / 4
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) ≀ e / 4
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) ≀ M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y) I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) ≀ e / 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
trans M.sum fun k ↦ a ^ k * d
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) ≀ e / 4
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) ≀ M.sum fun k => a ^ k * d I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => a ^ k * d) ≀ e / 4
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) ≀ e / 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
apply Finset.sum_le_sum
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) ≀ M.sum fun k => a ^ k * d
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ βˆ€ i ∈ M, dist ((p i) fun x => y) ((pr n i) fun x => y) ≀ a ^ i * d
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) ≀ M.sum fun k => a ^ k * d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
intro k _
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ βˆ€ i ∈ M, dist ((p i) fun x => y) ((pr n i) fun x => y) ≀ a ^ i * d
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ a ^ k * d
Please generate a tactic in lean4 to solve the state. STATE: case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ βˆ€ i ∈ M, dist ((p i) fun x => y) ((pr n i) fun x => y) ≀ a ^ i * d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
have hak : a ^ k = abs y ^ k * r⁻¹ ^ k := by calc (abs y / r) ^ k _ = (abs y * r⁻¹) ^ k := by rw [div_eq_mul_inv, NNReal.coe_inv] _ = abs y ^ k * r⁻¹ ^ k := mul_pow _ _ _
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ a ^ k * d
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ a ^ k * d
Please generate a tactic in lean4 to solve the state. STATE: case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ a ^ k * d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [hak]
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ a ^ k * d
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d
Please generate a tactic in lean4 to solve the state. STATE: case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ a ^ k * d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
generalize hd' : d.toNNReal = d'
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d
Please generate a tactic in lean4 to solve the state. STATE: case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
have dd : (d' : ℝ) = d := by rw [← hd']; exact Real.coe_toNNReal d dp.le
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d
Please generate a tactic in lean4 to solve the state. STATE: case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
have hcb : βˆ€ z, z ∈ closedBall c r β†’ abs (g z - f n z) ≀ d' := by intro z zb; exact _root_.trans (hn z (cb zb)).le (le_of_eq dd.symm)
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d hcb : βˆ€ z ∈ closedBall c ↑r, Complex.abs (g z - f n z) ≀ ↑d' ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d
Please generate a tactic in lean4 to solve the state. STATE: case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
exact _root_.trans (cauchy_dist k y rp cg (cf n) hcb) (mul_le_mul_of_nonneg_left (le_of_eq dd) (by bound))
case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d hcb : βˆ€ z ∈ closedBall c ↑r, Complex.abs (g z - f n z) ≀ ↑d' ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d hcb : βˆ€ z ∈ closedBall c ↑r, Complex.abs (g z - f n z) ≀ ↑d' ⊒ dist ((p k) fun x => y) ((pr n k) fun x => y) ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k * d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
calc (abs y / r) ^ k _ = (abs y * r⁻¹) ^ k := by rw [div_eq_mul_inv, NNReal.coe_inv] _ = abs y ^ k * r⁻¹ ^ k := mul_pow _ _ _
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M ⊒ a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M ⊒ a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [div_eq_mul_inv, NNReal.coe_inv]
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M ⊒ (Complex.abs y / ↑r) ^ k = (Complex.abs y * ↑r⁻¹) ^ k
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M ⊒ (Complex.abs y / ↑r) ^ k = (Complex.abs y * ↑r⁻¹) ^ k TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [← hd']
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' ⊒ ↑d' = d
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' ⊒ ↑d.toNNReal = d
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' ⊒ ↑d' = d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
exact Real.coe_toNNReal d dp.le
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' ⊒ ↑d.toNNReal = d
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' ⊒ ↑d.toNNReal = d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
intro z zb
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d ⊒ βˆ€ z ∈ closedBall c ↑r, Complex.abs (g z - f n z) ≀ ↑d'
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d z : β„‚ zb : z ∈ closedBall c ↑r ⊒ Complex.abs (g z - f n z) ≀ ↑d'
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d ⊒ βˆ€ z ∈ closedBall c ↑r, Complex.abs (g z - f n z) ≀ ↑d' TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
exact _root_.trans (hn z (cb zb)).le (le_of_eq dd.symm)
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d z : β„‚ zb : z ∈ closedBall c ↑r ⊒ Complex.abs (g z - f n z) ≀ ↑d'
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d z : β„‚ zb : z ∈ closedBall c ↑r ⊒ Complex.abs (g z - f n z) ≀ ↑d' TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
bound
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d hcb : βˆ€ z ∈ closedBall c ↑r, Complex.abs (g z - f n z) ≀ ↑d' ⊒ 0 ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d k : β„• a✝ : k ∈ M hak : a ^ k = Complex.abs y ^ k * ↑r⁻¹ ^ k d' : ℝβ‰₯0 hd' : d.toNNReal = d' dd : ↑d' = d hcb : βˆ€ z ∈ closedBall c ↑r, Complex.abs (g z - f n z) ≀ ↑d' ⊒ 0 ≀ Complex.abs y ^ k * ↑r⁻¹ ^ k TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
have pgb : (M.sum fun k ↦ a ^ k) ≀ (1 - a)⁻¹ := partial_geometric_bound M a0 a1
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => a ^ k * d) ≀ e / 4
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (M.sum fun k => a ^ k * d) ≀ e / 4
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d ⊒ (M.sum fun k => a ^ k * d) ≀ e / 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
calc (M.sum fun k ↦ a ^ k * d) = (M.sum fun k ↦ a ^ k) * d := by rw [← Finset.sum_mul] _ ≀ (1 - a)⁻¹ * d := by bound _ = (1 - a)⁻¹ * ((1 - a) * (e / 4)) := by rw [← d4] _ = (1 - a) * (1 - a)⁻¹ * (e / 4) := by ring _ = 1 * (e / 4) := by rw [mul_inv_cancel a1p.ne'] _ = e / 4 := by ring
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (M.sum fun k => a ^ k * d) ≀ e / 4
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (M.sum fun k => a ^ k * d) ≀ e / 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [← Finset.sum_mul]
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (M.sum fun k => a ^ k * d) = (M.sum fun k => a ^ k) * d
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (M.sum fun k => a ^ k * d) = (M.sum fun k => a ^ k) * d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
bound
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (M.sum fun k => a ^ k) * d ≀ (1 - a)⁻¹ * d
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (M.sum fun k => a ^ k) * d ≀ (1 - a)⁻¹ * d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [← d4]
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (1 - a)⁻¹ * d = (1 - a)⁻¹ * ((1 - a) * (e / 4))
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (1 - a)⁻¹ * d = (1 - a)⁻¹ * ((1 - a) * (e / 4)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
ring
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (1 - a)⁻¹ * ((1 - a) * (e / 4)) = (1 - a) * (1 - a)⁻¹ * (e / 4)
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (1 - a)⁻¹ * ((1 - a) * (e / 4)) = (1 - a) * (1 - a)⁻¹ * (e / 4) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [mul_inv_cancel a1p.ne']
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (1 - a) * (1 - a)⁻¹ * (e / 4) = 1 * (e / 4)
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ (1 - a) * (1 - a)⁻¹ * (e / 4) = 1 * (e / 4) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
ring
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ 1 * (e / 4) = e / 4
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) ≀ d pgb : (M.sum fun k => a ^ k) ≀ (1 - a)⁻¹ ⊒ 1 * (e / 4) = e / 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
bound
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ dist Mp (f n (c + y)) + dist (f n (c + y)) (g (c + y)) ≀ dist Mp Mpr + dist Mpr (f n (c + y)) + d
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ dist Mp (f n (c + y)) + dist (f n (c + y)) (g (c + y)) ≀ dist Mp Mpr + dist Mpr (f n (c + y)) + d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
bound
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ dist Mp Mpr + dist Mpr (f n (c + y)) + d ≀ e / 4 + d + d
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ dist Mp Mpr + dist Mpr (f n (c + y)) + d ≀ e / 4 + d + d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
rw [← d4]
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ e / 4 + d + d = e / 4 + 2 * (1 - a) * (e / 4)
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ e / 4 + (1 - a) * (e / 4) + (1 - a) * (e / 4) = e / 4 + 2 * (1 - a) * (e / 4)
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ e / 4 + d + d = e / 4 + 2 * (1 - a) * (e / 4) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
ring
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ e / 4 + (1 - a) * (e / 4) + (1 - a) * (e / 4) = e / 4 + 2 * (1 - a) * (e / 4)
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ e / 4 + (1 - a) * (e / 4) + (1 - a) * (e / 4) = e / 4 + 2 * (1 - a) * (e / 4) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
bound
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ e / 4 + 2 * (1 - a) * (e / 4) ≀ e / 4 + 2 * (1 - 0) * (e / 4)
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ e / 4 + 2 * (1 - a) * (e / 4) ≀ e / 4 + 2 * (1 - 0) * (e / 4) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
ring
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ e / 4 + 2 * (1 - 0) * (e / 4) = 3 / 4 * e
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ e / 4 + 2 * (1 - 0) * (e / 4) = 3 / 4 * e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
norm_num
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ 3 / 4 < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ 3 / 4 < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Analytic/Uniform.lean
uniform_analytic_lim
[189, 1]
[270, 21]
simp
I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ 1 * e = e
no goals
Please generate a tactic in lean4 to solve the state. STATE: I : Type inst✝¹ : Lattice I inst✝ : Nonempty I f : I β†’ β„‚ β†’ β„‚ g : β„‚ β†’ β„‚ s : Set β„‚ c : β„‚ r : ℝβ‰₯0 cb : closedBall c ↑r βŠ† s rp : 0 < r pr : I β†’ FormalMultilinearSeries β„‚ β„‚ β„‚ := fun n => cauchyPowerSeries (f n) c ↑r hpf : βˆ€ (n : I), HasFPowerSeriesOnBall (f n) (pr n) c ↑r cf : βˆ€ (n : I), ContinuousOn (f n) (closedBall c ↑r) cg : ContinuousOn g (closedBall c ↑r) p : FormalMultilinearSeries β„‚ β„‚ β„‚ := cauchyPowerSeries g c ↑r y : β„‚ yr : Complex.abs y < ↑r a : ℝ := Complex.abs y / ↑r a0 : a β‰₯ 0 a1 : a < 1 a1p : 1 - a > 0 e : ℝ ep : e > 0 d : ℝ d4 : (1 - a) * (e / 4) = d dp : d > 0 n : I hn : βˆ€ x ∈ s, dist (g x) (f n x) < d dfg : dist (f n (c + y)) (g (c + y)) ≀ d M : Finset β„• Mp : β„‚ hMp : (M.sum fun k => (p k) fun x => y) = Mp Mpr : β„‚ dppr : dist Mp Mpr ≀ e / 4 dpf : dist Mpr (f n (c + y)) ≀ d hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr ⊒ 1 * e = e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
by_cases z0 : z = 0
z w : β„‚ n : β„• ⊒ (z ^ w) ^ n = z ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n) case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: z w : β„‚ n : β„• ⊒ (z ^ w) ^ n = z ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
simp only [Complex.cpow_def_of_ne_zero z0, ← Complex.exp_nat_mul]
case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n)
case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (↑n * (z.log * w)).exp = (z.log * (w * ↑n)).exp
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
ring_nf
case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (↑n * (z.log * w)).exp = (z.log * (w * ↑n)).exp
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (↑n * (z.log * w)).exp = (z.log * (w * ↑n)).exp TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
rw [z0]
case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
by_cases w0 : w = 0
case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
by_cases n0 : n = 0
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
have wn0 : w * n β‰  0 := mul_ne_zero w0 (Nat.cast_ne_zero.mpr n0)
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
rw [Complex.zero_cpow w0]
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0 ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
rw [Complex.zero_cpow wn0]
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0 ^ (w * ↑n)
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
exact zero_pow n0
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
rw [w0]
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ 0) ^ n = 0 ^ (0 * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
simp
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ 0) ^ n = 0 ^ (0 * ↑n)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ 0) ^ n = 0 ^ (0 * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
rw [n0]
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ 0 = 0 ^ (w * ↑0)
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
simp
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ 0 = 0 ^ (w * ↑0)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ 0 = 0 ^ (w * ↑0) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
rw [potential_image, bad_potential_image]
c : β„š Γ— β„š n : β„• r : Floating ⊒ potential_image ↑c ∈ approx (bad_potential_image n r c)
c : β„š Γ— β„š n : β„• r : Floating ⊒ (let p := potential 2 ↑↑c; let t := p ^ 16; lerp t ↑clear ↑outside) ∈ approx (let c := Box.ofRat c; let p := c.potential c n r; let t := p.1.sqr.sqr.sqr.sqr; let i := lerp t clear' outside'; i.quantize)
Please generate a tactic in lean4 to solve the state. STATE: c : β„š Γ— β„š n : β„• r : Floating ⊒ potential_image ↑c ∈ approx (bad_potential_image n r c) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
have e : βˆ€ p : ℝ, p^16 = (((p^2)^2)^2)^2 := by intro p; rw [←pow_mul, ←pow_mul, ←pow_mul]
c : β„š Γ— β„š n : β„• r : Floating ⊒ (let p := potential 2 ↑↑c; let t := p ^ 16; lerp t ↑clear ↑outside) ∈ approx (let c := Box.ofRat c; let p := c.potential c n r; let t := p.1.sqr.sqr.sqr.sqr; let i := lerp t clear' outside'; i.quantize)
c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ (let p := potential 2 ↑↑c; let t := p ^ 16; lerp t ↑clear ↑outside) ∈ approx (let c := Box.ofRat c; let p := c.potential c n r; let t := p.1.sqr.sqr.sqr.sqr; let i := lerp t clear' outside'; i.quantize)
Please generate a tactic in lean4 to solve the state. STATE: c : β„š Γ— β„š n : β„• r : Floating ⊒ (let p := potential 2 ↑↑c; let t := p ^ 16; lerp t ↑clear ↑outside) ∈ approx (let c := Box.ofRat c; let p := c.potential c n r; let t := p.1.sqr.sqr.sqr.sqr; let i := lerp t clear' outside'; i.quantize) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
simp only [far', outside', clear', green', e]
c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ (let p := potential 2 ↑↑c; let t := p ^ 16; lerp t ↑clear ↑outside) ∈ approx (let c := Box.ofRat c; let p := c.potential c n r; let t := p.1.sqr.sqr.sqr.sqr; let i := lerp t clear' outside'; i.quantize)
c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈ approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat).quantize
Please generate a tactic in lean4 to solve the state. STATE: c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ (let p := potential 2 ↑↑c; let t := p ^ 16; lerp t ↑clear ↑outside) ∈ approx (let c := Box.ofRat c; let p := c.potential c n r; let t := p.1.sqr.sqr.sqr.sqr; let i := lerp t clear' outside'; i.quantize) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
apply Color.mem_approx_quantize
c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈ approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat).quantize
case cm c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈ approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat)
Please generate a tactic in lean4 to solve the state. STATE: c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈ approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat).quantize TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
apply mem_approx_lerp
case cm c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈ approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat)
case cm.tm c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ (((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr case cm.xm c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ ↑clear ∈ approx clear.ofRat case cm.ym c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ ↑outside ∈ approx outside.ofRat
Please generate a tactic in lean4 to solve the state. STATE: case cm c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ lerp ((((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2) ↑clear ↑outside ∈ approx (lerp ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr clear.ofRat outside.ofRat) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
intro p
c : β„š Γ— β„š n : β„• r : Floating ⊒ βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
c : β„š Γ— β„š n : β„• r : Floating p : ℝ ⊒ p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
Please generate a tactic in lean4 to solve the state. STATE: c : β„š Γ— β„š n : β„• r : Floating ⊒ βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
rw [←pow_mul, ←pow_mul, ←pow_mul]
c : β„š Γ— β„š n : β„• r : Floating p : ℝ ⊒ p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : β„š Γ— β„š n : β„• r : Floating p : ℝ ⊒ p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
apply Interval.mem_approx_sqr
case cm.tm c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ (((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr
case cm.tm.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ ((potential 2 ↑↑c ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr
Please generate a tactic in lean4 to solve the state. STATE: case cm.tm c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ (((potential 2 ↑↑c ^ 2) ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr.sqr TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
apply Interval.mem_approx_sqr
case cm.tm.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ ((potential 2 ↑↑c ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr
case cm.tm.ax.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ (potential 2 ↑↑c ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr
Please generate a tactic in lean4 to solve the state. STATE: case cm.tm.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ ((potential 2 ↑↑c ^ 2) ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr.sqr TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
apply Interval.mem_approx_sqr
case cm.tm.ax.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ (potential 2 ↑↑c ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr
case cm.tm.ax.ax.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ potential 2 ↑↑c ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr
Please generate a tactic in lean4 to solve the state. STATE: case cm.tm.ax.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ (potential 2 ↑↑c ^ 2) ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr.sqr TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
apply Interval.mem_approx_sqr
case cm.tm.ax.ax.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ potential 2 ↑↑c ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr
case cm.tm.ax.ax.ax.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ potential 2 ↑↑c ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1
Please generate a tactic in lean4 to solve the state. STATE: case cm.tm.ax.ax.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ potential 2 ↑↑c ^ 2 ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1.sqr TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
mono
case cm.tm.ax.ax.ax.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ potential 2 ↑↑c ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case cm.tm.ax.ax.ax.ax c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ potential 2 ↑↑c ∈ approx ((Box.ofRat c).potential (Box.ofRat c) n r).1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
mono
case cm.xm c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ ↑clear ∈ approx clear.ofRat
no goals
Please generate a tactic in lean4 to solve the state. STATE: case cm.xm c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ ↑clear ∈ approx clear.ofRat TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Render/Mandelbrot.lean
approx_bad_potential_image
[60, 1]
[75, 9]
mono
case cm.ym c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ ↑outside ∈ approx outside.ofRat
no goals
Please generate a tactic in lean4 to solve the state. STATE: case cm.ym c : β„š Γ— β„š n : β„• r : Floating e : βˆ€ (p : ℝ), p ^ 16 = (((p ^ 2) ^ 2) ^ 2) ^ 2 ⊒ ↑outside ∈ approx outside.ofRat TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
by_cases z0 : z = 0
z w : β„‚ n : β„• ⊒ (z ^ w) ^ n = z ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n) case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: z w : β„‚ n : β„• ⊒ (z ^ w) ^ n = z ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
simp only [Complex.cpow_def_of_ne_zero z0, ← Complex.exp_nat_mul]
case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n)
case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (↑n * (z.log * w)).exp = (z.log * (w * ↑n)).exp
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
ring_nf
case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (↑n * (z.log * w)).exp = (z.log * (w * ↑n)).exp
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : Β¬z = 0 ⊒ (↑n * (z.log * w)).exp = (z.log * (w * ↑n)).exp TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
rw [z0]
case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (z ^ w) ^ n = z ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
by_cases w0 : w = 0
case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
by_cases n0 : n = 0
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
have wn0 : w * n β‰  0 := mul_ne_zero w0 (Nat.cast_ne_zero.mpr n0)
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
rw [Complex.zero_cpow w0]
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0 ^ (w * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
rw [Complex.zero_cpow wn0]
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0 ^ (w * ↑n)
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
exact zero_pow n0
case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : Β¬n = 0 wn0 : w * ↑n β‰  0 ⊒ 0 ^ n = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
rw [w0]
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ 0) ^ n = 0 ^ (0 * ↑n)
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
simp
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ 0) ^ n = 0 ^ (0 * ↑n)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 w0 : w = 0 ⊒ (0 ^ 0) ^ n = 0 ^ (0 * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
rw [n0]
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n)
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ 0 = 0 ^ (w * ↑0)
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ n = 0 ^ (w * ↑n) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Pow.lean
pow_mul_nat
[7, 1]
[17, 10]
simp
case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ 0 = 0 ^ (w * ↑0)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos z w : β„‚ n : β„• z0 : z = 0 w0 : Β¬w = 0 n0 : n = 0 ⊒ (0 ^ w) ^ 0 = 0 ^ (w * ↑0) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNearM.lean
attracts_shift
[48, 1]
[50, 60]
simp only [Attracts, ← Function.iterate_add_apply]
S : Type inst✝⁴ : TopologicalSpace S inst✝³ : CompactSpace S inst✝² : T3Space S inst✝¹ : ChartedSpace β„‚ S inst✝ : AnalyticManifold I S f✝ : β„‚ β†’ S β†’ S c : β„‚ a✝ z✝ : S d n : β„• f : S β†’ S z a : S k : β„• ⊒ Attracts f (f^[k] z) a ↔ Attracts f z a
S : Type inst✝⁴ : TopologicalSpace S inst✝³ : CompactSpace S inst✝² : T3Space S inst✝¹ : ChartedSpace β„‚ S inst✝ : AnalyticManifold I S f✝ : β„‚ β†’ S β†’ S c : β„‚ a✝ z✝ : S d n : β„• f : S β†’ S z a : S k : β„• ⊒ Tendsto (fun n => f^[n + k] z) atTop (𝓝 a) ↔ Tendsto (fun n => f^[n] z) atTop (𝓝 a)
Please generate a tactic in lean4 to solve the state. STATE: S : Type inst✝⁴ : TopologicalSpace S inst✝³ : CompactSpace S inst✝² : T3Space S inst✝¹ : ChartedSpace β„‚ S inst✝ : AnalyticManifold I S f✝ : β„‚ β†’ S β†’ S c : β„‚ a✝ z✝ : S d n : β„• f : S β†’ S z a : S k : β„• ⊒ Attracts f (f^[k] z) a ↔ Attracts f z a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNearM.lean
attracts_shift
[48, 1]
[50, 60]
apply @Filter.tendsto_add_atTop_iff_nat _ fun n ↦ f^[n] z
S : Type inst✝⁴ : TopologicalSpace S inst✝³ : CompactSpace S inst✝² : T3Space S inst✝¹ : ChartedSpace β„‚ S inst✝ : AnalyticManifold I S f✝ : β„‚ β†’ S β†’ S c : β„‚ a✝ z✝ : S d n : β„• f : S β†’ S z a : S k : β„• ⊒ Tendsto (fun n => f^[n + k] z) atTop (𝓝 a) ↔ Tendsto (fun n => f^[n] z) atTop (𝓝 a)
no goals
Please generate a tactic in lean4 to solve the state. STATE: S : Type inst✝⁴ : TopologicalSpace S inst✝³ : CompactSpace S inst✝² : T3Space S inst✝¹ : ChartedSpace β„‚ S inst✝ : AnalyticManifold I S f✝ : β„‚ β†’ S β†’ S c : β„‚ a✝ z✝ : S d n : β„• f : S β†’ S z a : S k : β„• ⊒ Tendsto (fun n => f^[n + k] z) atTop (𝓝 a) ↔ Tendsto (fun n => f^[n] z) atTop (𝓝 a) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNearM.lean
Super.dp
[69, 1]
[69, 74]
norm_num
S : Type inst✝⁴ : TopologicalSpace S inst✝³ : CompactSpace S inst✝² : T3Space S inst✝¹ : ChartedSpace β„‚ S inst✝ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c : β„‚ a z : S d n : β„• s : Super f d a ⊒ 0 < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: S : Type inst✝⁴ : TopologicalSpace S inst✝³ : CompactSpace S inst✝² : T3Space S inst✝¹ : ChartedSpace β„‚ S inst✝ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c : β„‚ a z : S d n : β„• s : Super f d a ⊒ 0 < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNearM.lean
Super.d1
[71, 1]
[71, 80]
norm_num
S : Type inst✝⁴ : TopologicalSpace S inst✝³ : CompactSpace S inst✝² : T3Space S inst✝¹ : ChartedSpace β„‚ S inst✝ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c : β„‚ a z : S d n : β„• s : Super f d a ⊒ 1 < 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: S : Type inst✝⁴ : TopologicalSpace S inst✝³ : CompactSpace S inst✝² : T3Space S inst✝¹ : ChartedSpace β„‚ S inst✝ : AnalyticManifold I S f : β„‚ β†’ S β†’ S c : β„‚ a z : S d n : β„• s : Super f d a ⊒ 1 < 2 TACTIC: