Datasets:
pkuAI4M
/
lean_github

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https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_at_zero
[393, 1]
[395, 67]
induction' n with n h
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t n : β„• ⊒ f^[n] 0 = 0
case zero f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t ⊒ f^[0] 0 = 0 case succ f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t n : β„• h : f^[n] 0 = 0 ⊒ f^[n + 1] 0 = 0
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t n : β„• ⊒ f^[n] 0 = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_at_zero
[393, 1]
[395, 67]
simp only [Function.iterate_zero, id]
case zero f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t ⊒ f^[0] 0 = 0 case succ f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t n : β„• h : f^[n] 0 = 0 ⊒ f^[n + 1] 0 = 0
case succ f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t n : β„• h : f^[n] 0 = 0 ⊒ f^[n + 1] 0 = 0
Please generate a tactic in lean4 to solve the state. STATE: case zero f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t ⊒ f^[0] 0 = 0 case succ f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t n : β„• h : f^[n] 0 = 0 ⊒ f^[n + 1] 0 = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_at_zero
[393, 1]
[395, 67]
simp only [Function.iterate_succ', Function.comp_apply, h, s.f0]
case succ f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t n : β„• h : f^[n] 0 = 0 ⊒ f^[n + 1] 0 = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case succ f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t n : β„• h : f^[n] 0 = 0 ⊒ f^[n + 1] 0 = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_at_zero
[398, 1]
[399, 61]
simp only [term, iterates_at_zero s, g0, Complex.one_cpow]
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t n : β„• ⊒ term f d n 0 = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t n : β„• ⊒ term f d n 0 = 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_prod_at_zero
[402, 1]
[403, 47]
simp_rw [tprodOn, term_at_zero s, tprod_one]
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t ⊒ tprodOn (term f d) 0 = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t ⊒ tprodOn (term f d) 0 = 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_monic
[406, 1]
[410, 43]
have dz : HasDerivAt (fun z : β„‚ ↦ z) 1 0 := hasDerivAt_id 0
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t ⊒ HasDerivAt (bottcherNear f d) 1 0
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 ⊒ HasDerivAt (bottcherNear f d) 1 0
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t ⊒ HasDerivAt (bottcherNear f d) 1 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_monic
[406, 1]
[410, 43]
have db := HasDerivAt.mul dz (term_prod_analytic_z s 0 s.t0).differentiableAt.hasDerivAt
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 ⊒ HasDerivAt (bottcherNear f d) 1 0
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 db : HasDerivAt (fun y => y * tprodOn (term f d) y) (1 * tprodOn (term f d) 0 + 0 * deriv (tprodOn (term f d)) 0) 0 ⊒ HasDerivAt (bottcherNear f d) 1 0
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 ⊒ HasDerivAt (bottcherNear f d) 1 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_monic
[406, 1]
[410, 43]
simp only [one_mul, MulZeroClass.zero_mul, add_zero] at db
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 db : HasDerivAt (fun y => y * tprodOn (term f d) y) (1 * tprodOn (term f d) 0 + 0 * deriv (tprodOn (term f d)) 0) 0 ⊒ HasDerivAt (bottcherNear f d) 1 0
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 db : HasDerivAt (fun y => y * tprodOn (term f d) y) (tprodOn (term f d) 0) 0 ⊒ HasDerivAt (bottcherNear f d) 1 0
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 db : HasDerivAt (fun y => y * tprodOn (term f d) y) (1 * tprodOn (term f d) 0 + 0 * deriv (tprodOn (term f d)) 0) 0 ⊒ HasDerivAt (bottcherNear f d) 1 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_monic
[406, 1]
[410, 43]
rw [term_prod_at_zero s] at db
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 db : HasDerivAt (fun y => y * tprodOn (term f d) y) (tprodOn (term f d) 0) 0 ⊒ HasDerivAt (bottcherNear f d) 1 0
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 db : HasDerivAt (fun y => y * tprodOn (term f d) y) 1 0 ⊒ HasDerivAt (bottcherNear f d) 1 0
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 db : HasDerivAt (fun y => y * tprodOn (term f d) y) (tprodOn (term f d) 0) 0 ⊒ HasDerivAt (bottcherNear f d) 1 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_monic
[406, 1]
[410, 43]
exact db
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 db : HasDerivAt (fun y => y * tprodOn (term f d) y) 1 0 ⊒ HasDerivAt (bottcherNear f d) 1 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t dz : HasDerivAt (fun z => z) 1 0 db : HasDerivAt (fun y => y * tprodOn (term f d) y) 1 0 ⊒ HasDerivAt (bottcherNear f d) 1 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_zero
[413, 1]
[414, 50]
simp only [bottcherNear, MulZeroClass.zero_mul]
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ ⊒ bottcherNear f d 0 = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ ⊒ bottcherNear f d 0 = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
by_cases z0 : z = 0
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0)
case pos f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : z = 0 ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0) case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0)
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
simp only [z0, iterates_at_zero s, tendsto_const_nhds]
case pos f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : z = 0 ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0) case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0)
case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0)
Please generate a tactic in lean4 to solve the state. STATE: case pos f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : z = 0 ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0) case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
rw [Metric.tendsto_atTop]
case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0)
case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 ⊒ βˆ€ Ξ΅ > 0, βˆƒ N, βˆ€ n β‰₯ N, dist (f^[n] z) 0 < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 ⊒ Tendsto (fun n => f^[n] z) atTop (𝓝 0) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
intro e ep
case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 ⊒ βˆ€ Ξ΅ > 0, βˆƒ N, βˆ€ n β‰₯ N, dist (f^[n] z) 0 < Ξ΅
case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, dist (f^[n] z) 0 < e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 ⊒ βˆ€ Ξ΅ > 0, βˆƒ N, βˆ€ n β‰₯ N, dist (f^[n] z) 0 < Ξ΅ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
simp only [Complex.dist_eq, sub_zero]
case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, dist (f^[n] z) 0 < e
case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, dist (f^[n] z) 0 < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
have xp : e / abs z > 0 := div_pos ep (Complex.abs.pos z0)
case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e
case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
rcases exists_pow_lt_of_lt_one xp (by norm_num : (5 / 8 : ℝ) < 1) with ⟨N, Nb⟩
case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e
case neg.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N < e / Complex.abs z ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
simp only [lt_div_iff (Complex.abs.pos z0)] at Nb
case neg.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N < e / Complex.abs z ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e
case neg.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e
Please generate a tactic in lean4 to solve the state. STATE: case neg.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N < e / Complex.abs z ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
use N
case neg.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e
case h f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e ⊒ βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e
Please generate a tactic in lean4 to solve the state. STATE: case neg.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e ⊒ βˆƒ N, βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
intro n nN
case h f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e ⊒ βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e
case h f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e n : β„• nN : n β‰₯ N ⊒ Complex.abs (f^[n] z) < e
Please generate a tactic in lean4 to solve the state. STATE: case h f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e ⊒ βˆ€ n β‰₯ N, Complex.abs (f^[n] z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
refine lt_of_le_of_lt (iterates_converge s n zt) (lt_of_le_of_lt ?_ Nb)
case h f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e n : β„• nN : n β‰₯ N ⊒ Complex.abs (f^[n] z) < e
case h f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e n : β„• nN : n β‰₯ N ⊒ (5 / 8) ^ n * Complex.abs z ≀ (5 / 8) ^ N * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: case h f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e n : β„• nN : n β‰₯ N ⊒ Complex.abs (f^[n] z) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
bound
case h f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e n : β„• nN : n β‰₯ N ⊒ (5 / 8) ^ n * Complex.abs z ≀ (5 / 8) ^ N * Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 N : β„• Nb : (5 / 8) ^ N * Complex.abs z < e n : β„• nN : n β‰₯ N ⊒ (5 / 8) ^ n * Complex.abs z ≀ (5 / 8) ^ N * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_tendsto
[425, 1]
[435, 8]
norm_num
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 ⊒ 5 / 8 < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t z0 : Β¬z = 0 e : ℝ ep : e > 0 xp : e / Complex.abs z > 0 ⊒ 5 / 8 < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_lt_one
[438, 1]
[448, 88]
rcases Metric.continuousAt_iff.mp (bottcherNear_analytic_z s _ s.t0).continuousAt 1 zero_lt_one with ⟨r, rp, rs⟩
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs (bottcherNear f d z) < 1
case intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, dist x 0 < r β†’ dist (bottcherNear f d x) (bottcherNear f d 0) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs (bottcherNear f d z) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_lt_one
[438, 1]
[448, 88]
simp only [Complex.dist_eq, sub_zero, bottcherNear_zero] at rs
case intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, dist x 0 < r β†’ dist (bottcherNear f d x) (bottcherNear f d 0) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1
case intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, dist x 0 < r β†’ dist (bottcherNear f d x) (bottcherNear f d 0) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_lt_one
[438, 1]
[448, 88]
have b' : βˆ€αΆ  n in atTop, abs (bottcherNear f d (f^[n] z)) < 1 := by refine (Metric.tendsto_nhds.mp (iterates_tendsto s zt) r rp).mp (Filter.eventually_of_forall fun n h ↦ ?_) rw [Complex.dist_eq, sub_zero] at h; exact rs h
case intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1
case intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_lt_one
[438, 1]
[448, 88]
rcases b'.exists with ⟨n, b⟩
case intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1
case intro.intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 n : β„• b : Complex.abs (bottcherNear f d (f^[n] z)) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_lt_one
[438, 1]
[448, 88]
contrapose b
case intro.intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 n : β„• b : Complex.abs (bottcherNear f d (f^[n] z)) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1
case intro.intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 n : β„• b : Β¬Complex.abs (bottcherNear f d z) < 1 ⊒ Β¬Complex.abs (bottcherNear f d (f^[n] z)) < 1
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 n : β„• b : Complex.abs (bottcherNear f d (f^[n] z)) < 1 ⊒ Complex.abs (bottcherNear f d z) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_lt_one
[438, 1]
[448, 88]
simp only [not_lt] at b ⊒
case intro.intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 n : β„• b : Β¬Complex.abs (bottcherNear f d z) < 1 ⊒ Β¬Complex.abs (bottcherNear f d (f^[n] z)) < 1
case intro.intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 n : β„• b : 1 ≀ Complex.abs (bottcherNear f d z) ⊒ 1 ≀ Complex.abs (bottcherNear f d (f^[n] z))
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 n : β„• b : Β¬Complex.abs (bottcherNear f d z) < 1 ⊒ Β¬Complex.abs (bottcherNear f d (f^[n] z)) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_lt_one
[438, 1]
[448, 88]
simp only [bottcherNear_eqn_iter s zt n, Complex.abs.map_pow, one_le_pow_of_one_le b]
case intro.intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 n : β„• b : 1 ≀ Complex.abs (bottcherNear f d z) ⊒ 1 ≀ Complex.abs (bottcherNear f d (f^[n] z))
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro.intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 b' : βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 n : β„• b : 1 ≀ Complex.abs (bottcherNear f d z) ⊒ 1 ≀ Complex.abs (bottcherNear f d (f^[n] z)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_lt_one
[438, 1]
[448, 88]
refine (Metric.tendsto_nhds.mp (iterates_tendsto s zt) r rp).mp (Filter.eventually_of_forall fun n h ↦ ?_)
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 ⊒ βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 n : β„• h : dist (f^[n] z) 0 < r ⊒ Complex.abs (bottcherNear f d (f^[n] z)) < 1
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 ⊒ βˆ€αΆ  (n : β„•) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_lt_one
[438, 1]
[448, 88]
rw [Complex.dist_eq, sub_zero] at h
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 n : β„• h : dist (f^[n] z) 0 < r ⊒ Complex.abs (bottcherNear f d (f^[n] z)) < 1
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 n : β„• h : Complex.abs (f^[n] z) < r ⊒ Complex.abs (bottcherNear f d (f^[n] z)) < 1
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 n : β„• h : dist (f^[n] z) 0 < r ⊒ Complex.abs (bottcherNear f d (f^[n] z)) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_lt_one
[438, 1]
[448, 88]
exact rs h
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 n : β„• h : Complex.abs (f^[n] z) < r ⊒ Complex.abs (bottcherNear f d (f^[n] z)) < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t r : ℝ rp : r > 0 rs : βˆ€ {x : β„‚}, Complex.abs x < r β†’ Complex.abs (bottcherNear f d x) < 1 n : β„• h : Complex.abs (f^[n] z) < r ⊒ Complex.abs (bottcherNear f d (f^[n] z)) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
simp only [bottcherNear, Complex.abs.map_mul]
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs (bottcherNear f d z) ≀ 3 * Complex.abs z
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs z * Complex.abs (∏' (n : β„•), term f d n z) ≀ 3 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs (bottcherNear f d z) ≀ 3 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
rw [mul_comm]
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs z * Complex.abs (∏' (n : β„•), term f d n z) ≀ 3 * Complex.abs z
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs (∏' (n : β„•), term f d n z) * Complex.abs z ≀ 3 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs z * Complex.abs (∏' (n : β„•), term f d n z) ≀ 3 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
refine mul_le_mul_of_nonneg_right ?_ (Complex.abs.nonneg _)
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs (∏' (n : β„•), term f d n z) * Complex.abs z ≀ 3 * Complex.abs z
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs (∏' (n : β„•), term f d n z) ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs (∏' (n : β„•), term f d n z) * Complex.abs z ≀ 3 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
rcases term_prod_exists s _ zt with ⟨p, h⟩
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs (∏' (n : β„•), term f d n z) ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : HasProd (fun n => term f d n z) p ⊒ Complex.abs (∏' (n : β„•), term f d n z) ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t ⊒ Complex.abs (∏' (n : β„•), term f d n z) ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
rw [h.tprod_eq]
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : HasProd (fun n => term f d n z) p ⊒ Complex.abs (∏' (n : β„•), term f d n z) ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : HasProd (fun n => term f d n z) p ⊒ Complex.abs p ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : HasProd (fun n => term f d n z) p ⊒ Complex.abs (∏' (n : β„•), term f d n z) ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
simp only [HasProd] at h
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : HasProd (fun n => term f d n z) p ⊒ Complex.abs p ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p) ⊒ Complex.abs p ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : HasProd (fun n => term f d n z) p ⊒ Complex.abs p ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
apply le_of_tendsto' (Filter.Tendsto.comp Complex.continuous_abs.continuousAt h)
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p) ⊒ Complex.abs p ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p) ⊒ βˆ€ (c : Finset β„•), (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) c ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p) ⊒ Complex.abs p ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
intro A
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p) ⊒ βˆ€ (c : Finset β„•), (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) c ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p) A : Finset β„• ⊒ (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) A ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p) ⊒ βˆ€ (c : Finset β„•), (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) c ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
clear h
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p) A : Finset β„• ⊒ (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) A ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• ⊒ (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) A ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p) A : Finset β„• ⊒ (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) A ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
simp only [Function.comp, Complex.abs.map_prod]
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• ⊒ (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) A ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• ⊒ (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) A ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
have tb : βˆ€ n, abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2 : ℝ) ^ n := by intro n calc abs (term f d n z) _ = abs (1 + (term f d n z - 1)) := by ring_nf _ ≀ Complex.abs 1 + abs (term f d n z - 1) := by bound _ = 1 + abs (term f d n z - 1) := by norm_num _ ≀ 1 + 1 / 2 * (1 / 2 : ℝ) ^ n := by bound [term_converges s n zt]
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
have p : βˆ€ n : β„•, 0 < (1 : ℝ) + 1 / 2 * (1 / 2 : ℝ) ^ n := fun _ ↦ add_pos (by bound) (by bound)
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
have lb : βˆ€ n : β„•, Real.log ((1 : ℝ) + 1 / 2 * (1 / 2 : ℝ) ^ n) ≀ 1 / 2 * (1 / 2 : ℝ) ^ n := fun n ↦ le_trans (Real.log_le_sub_one_of_pos (p n)) (le_of_eq (by ring))
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
refine le_trans (Finset.prod_le_prod (fun _ _ ↦ Complex.abs.nonneg _) fun n _ ↦ tb n) ?_
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => 1 + 1 / 2 * (1 / 2) ^ i) ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => Complex.abs (term f d i z)) ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
rw [← Real.exp_log (Finset.prod_pos fun n _ ↦ p n), Real.log_prod _ _ fun n _ ↦ (p n).ne']
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => 1 + 1 / 2 * (1 / 2) ^ i) ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.sum fun i => (1 + 1 / 2 * (1 / 2) ^ i).log).exp ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.prod fun i => 1 + 1 / 2 * (1 / 2) ^ i) ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
refine le_trans (Real.exp_le_exp.mpr (Finset.sum_le_sum fun n _ ↦ lb n)) ?_
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.sum fun i => (1 + 1 / 2 * (1 / 2) ^ i).log).exp ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i).exp ≀ 3
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.sum fun i => (1 + 1 / 2 * (1 / 2) ^ i).log).exp ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
refine le_trans (Real.exp_le_exp.mpr ?_) Real.exp_one_lt_3.le
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i).exp ≀ 3
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≀ 1
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i).exp ≀ 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
have geom := partial_scaled_geometric_bound (1 / 2) A one_half_pos.le one_half_lt_one
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≀ 1
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n geom : (A.sum fun n => ↑(1 / 2) * (1 / 2) ^ n) ≀ ↑(1 / 2) * (1 - 1 / 2)⁻¹ ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≀ 1
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≀ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
simp only [NNReal.coe_div, NNReal.coe_one, NNReal.coe_two] at geom
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n geom : (A.sum fun n => ↑(1 / 2) * (1 / 2) ^ n) ≀ ↑(1 / 2) * (1 - 1 / 2)⁻¹ ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≀ 1
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n geom : (A.sum fun x => 1 / 2 * (1 / 2) ^ x) ≀ 1 / 2 * (1 - 1 / 2)⁻¹ ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≀ 1
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n geom : (A.sum fun n => ↑(1 / 2) * (1 / 2) ^ n) ≀ ↑(1 / 2) * (1 - 1 / 2)⁻¹ ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≀ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
exact le_trans geom (by norm_num)
case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n geom : (A.sum fun x => 1 / 2 * (1 / 2) ^ x) ≀ 1 / 2 * (1 - 1 / 2)⁻¹ ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≀ 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n geom : (A.sum fun x => 1 / 2 * (1 / 2) ^ x) ≀ 1 / 2 * (1 - 1 / 2)⁻¹ ⊒ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≀ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
intro n
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• ⊒ βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• ⊒ βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
calc abs (term f d n z) _ = abs (1 + (term f d n z - 1)) := by ring_nf _ ≀ Complex.abs 1 + abs (term f d n z - 1) := by bound _ = 1 + abs (term f d n z - 1) := by norm_num _ ≀ 1 + 1 / 2 * (1 / 2 : ℝ) ^ n := by bound [term_converges s n zt]
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
ring_nf
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ Complex.abs (term f d n z) = Complex.abs (1 + (term f d n z - 1))
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ Complex.abs (term f d n z) = Complex.abs (1 + (term f d n z - 1)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
bound
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ Complex.abs (1 + (term f d n z - 1)) ≀ Complex.abs 1 + Complex.abs (term f d n z - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ Complex.abs (1 + (term f d n z - 1)) ≀ Complex.abs 1 + Complex.abs (term f d n z - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
norm_num
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ Complex.abs 1 + Complex.abs (term f d n z - 1) = 1 + Complex.abs (term f d n z - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ Complex.abs 1 + Complex.abs (term f d n z - 1) = 1 + Complex.abs (term f d n z - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
bound [term_converges s n zt]
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ 1 + Complex.abs (term f d n z - 1) ≀ 1 + 1 / 2 * (1 / 2) ^ n
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• n : β„• ⊒ 1 + Complex.abs (term f d n z - 1) ≀ 1 + 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
bound
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n x✝ : β„• ⊒ 0 < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n x✝ : β„• ⊒ 0 < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
bound
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n x✝ : β„• ⊒ 0 < 1 / 2 * (1 / 2) ^ x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n x✝ : β„• ⊒ 0 < 1 / 2 * (1 / 2) ^ x✝ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
ring
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n n : β„• ⊒ 1 + 1 / 2 * (1 / 2) ^ n - 1 = 1 / 2 * (1 / 2) ^ n
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n n : β„• ⊒ 1 + 1 / 2 * (1 / 2) ^ n - 1 = 1 / 2 * (1 / 2) ^ n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_le
[451, 1]
[474, 36]
norm_num
f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n geom : (A.sum fun x => 1 / 2 * (1 / 2) ^ x) ≀ 1 / 2 * (1 - 1 / 2)⁻¹ ⊒ 1 / 2 * (1 - 1 / 2)⁻¹ ≀ 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ d : β„• z : β„‚ t : Set β„‚ s : SuperNear f d t zt : z ∈ t p✝ : β„‚ A : Finset β„• tb : βˆ€ (n : β„•), Complex.abs (term f d n z) ≀ 1 + 1 / 2 * (1 / 2) ^ n p : βˆ€ (n : β„•), 0 < 1 + 1 / 2 * (1 / 2) ^ n lb : βˆ€ (n : β„•), (1 + 1 / 2 * (1 / 2) ^ n).log ≀ 1 / 2 * (1 / 2) ^ n geom : (A.sum fun x => 1 / 2 * (1 / 2) ^ x) ≀ 1 / 2 * (1 - 1 / 2)⁻¹ ⊒ 1 / 2 * (1 - 1 / 2)⁻¹ ≀ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.ou
[505, 1]
[509, 36]
have e : u = Prod.fst '' t := by ext c; simp only [Set.mem_image, Prod.exists, exists_and_right, exists_eq_right] exact ⟨fun m ↦ ⟨0, (s.s m).t0⟩, fun h ↦ Exists.elim h fun z m ↦ s.tc m⟩
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t ⊒ IsOpen u
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t e : u = Prod.fst '' t ⊒ IsOpen u
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t ⊒ IsOpen u TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.ou
[505, 1]
[509, 36]
rw [e]
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t e : u = Prod.fst '' t ⊒ IsOpen u
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t e : u = Prod.fst '' t ⊒ IsOpen (Prod.fst '' t)
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t e : u = Prod.fst '' t ⊒ IsOpen u TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.ou
[505, 1]
[509, 36]
exact isOpenMap_fst _ s.o
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t e : u = Prod.fst '' t ⊒ IsOpen (Prod.fst '' t)
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t e : u = Prod.fst '' t ⊒ IsOpen (Prod.fst '' t) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.ou
[505, 1]
[509, 36]
ext c
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t ⊒ u = Prod.fst '' t
case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t c : β„‚ ⊒ c ∈ u ↔ c ∈ Prod.fst '' t
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t ⊒ u = Prod.fst '' t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.ou
[505, 1]
[509, 36]
simp only [Set.mem_image, Prod.exists, exists_and_right, exists_eq_right]
case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t c : β„‚ ⊒ c ∈ u ↔ c ∈ Prod.fst '' t
case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t c : β„‚ ⊒ c ∈ u ↔ βˆƒ x, (c, x) ∈ t
Please generate a tactic in lean4 to solve the state. STATE: case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t c : β„‚ ⊒ c ∈ u ↔ c ∈ Prod.fst '' t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.ou
[505, 1]
[509, 36]
exact ⟨fun m ↦ ⟨0, (s.s m).t0⟩, fun h ↦ Exists.elim h fun z m ↦ s.tc m⟩
case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t c : β„‚ ⊒ c ∈ u ↔ βˆƒ x, (c, x) ∈ t
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t c : β„‚ ⊒ c ∈ u ↔ βˆƒ x, (c, x) ∈ t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.superAtC
[512, 1]
[521, 42]
intro c m
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t ⊒ βˆ€ {c : β„‚}, c ∈ u β†’ SuperAt (f c) d
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t c : β„‚ m : c ∈ u ⊒ SuperAt (f c) d
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t ⊒ βˆ€ {c : β„‚}, c ∈ u β†’ SuperAt (f c) d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.superAtC
[512, 1]
[521, 42]
have s := s.s m
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t c : β„‚ m : c ∈ u ⊒ SuperAt (f c) d
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s✝ : SuperNearC f d u t c : β„‚ m : c ∈ u s : SuperNear (f c) d {z | (c, z) ∈ t} ⊒ SuperAt (f c) d
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s : SuperNearC f d u t c : β„‚ m : c ∈ u ⊒ SuperAt (f c) d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.superAtC
[512, 1]
[521, 42]
exact { d2 := s.d2 fa0 := s.fa0 fd := s.fd fc := s.fc }
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s✝ : SuperNearC f d u t c : β„‚ m : c ∈ u s : SuperNear (f c) d {z | (c, z) ∈ t} ⊒ SuperAt (f c) d
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t : Set (β„‚ Γ— β„‚) s✝ : SuperNearC f d u t c : β„‚ m : c ∈ u s : SuperNear (f c) d {z | (c, z) ∈ t} ⊒ SuperAt (f c) d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
refine Pair.hartogs o ?_ ?_
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u ⊒ AnalyticOn β„‚ (g2 f d) t
case refine_1 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u ⊒ βˆ€ (c0 c1 : β„‚), (c0, c1) ∈ t β†’ AnalyticAt β„‚ (fun z0 => g2 f d (z0, c1)) c0 case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u ⊒ βˆ€ (c0 c1 : β„‚), (c0, c1) ∈ t β†’ AnalyticAt β„‚ (fun z1 => g2 f d (c0, z1)) c1
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u ⊒ AnalyticOn β„‚ (g2 f d) t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
intro c z m
case refine_1 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u ⊒ βˆ€ (c0 c1 : β„‚), (c0, c1) ∈ t β†’ AnalyticAt β„‚ (fun z0 => g2 f d (z0, c1)) c0
case refine_1 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (fun z0 => g2 f d (z0, z)) c
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u ⊒ βˆ€ (c0 c1 : β„‚), (c0, c1) ∈ t β†’ AnalyticAt β„‚ (fun z0 => g2 f d (z0, c1)) c0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
simp only [g2, g]
case refine_1 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (fun z0 => g2 f d (z0, z)) c
case refine_1 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (fun z0 => g2 f d (z0, z)) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
by_cases zero : z = 0
case refine_1 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c
case pos f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : z = 0 ⊒ AnalyticAt β„‚ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ AnalyticAt β„‚ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
simp only [zero, eq_self_iff_true, if_true]
case pos f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : z = 0 ⊒ AnalyticAt β„‚ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c
case pos f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : z = 0 ⊒ AnalyticAt β„‚ (fun z0 => 1) c
Please generate a tactic in lean4 to solve the state. STATE: case pos f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : z = 0 ⊒ AnalyticAt β„‚ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
exact analyticAt_const
case pos f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : z = 0 ⊒ AnalyticAt β„‚ (fun z0 => 1) c
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : z = 0 ⊒ AnalyticAt β„‚ (fun z0 => 1) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
simp only [zero, if_false]
case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ AnalyticAt β„‚ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c
case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ AnalyticAt β„‚ (fun z0 => f z0 z / z ^ d) c
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ AnalyticAt β„‚ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
refine AnalyticAt.div ?_ analyticAt_const (pow_ne_zero _ zero)
case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ AnalyticAt β„‚ (fun z0 => f z0 z / z ^ d) c
case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ AnalyticAt β„‚ (fun z0 => f z0 z) c
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ AnalyticAt β„‚ (fun z0 => f z0 z / z ^ d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
refine (fa _ ?_).compβ‚‚ (analyticAt_id _ _) analyticAt_const
case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ AnalyticAt β„‚ (fun z0 => f z0 z) c
case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ (c, z) ∈ t
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ AnalyticAt β„‚ (fun z0 => f z0 z) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
exact m
case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ (c, z) ∈ t
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t zero : Β¬z = 0 ⊒ (c, z) ∈ t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
intro c z m
case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u ⊒ βˆ€ (c0 c1 : β„‚), (c0, c1) ∈ t β†’ AnalyticAt β„‚ (fun z1 => g2 f d (c0, z1)) c1
case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (fun z1 => g2 f d (c, z1)) z
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u ⊒ βˆ€ (c0 c1 : β„‚), (c0, c1) ∈ t β†’ AnalyticAt β„‚ (fun z1 => g2 f d (c0, z1)) c1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
apply (s.s (tc m)).ga_of_fa
case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (fun z1 => g2 f d (c, z1)) z
case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (f (c, z).1) z
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (fun z1 => g2 f d (c, z1)) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
refine (fa _ ?_).compβ‚‚ analyticAt_const (analyticAt_id _ _)
case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (f (c, z).1) z
case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ (c, id z) ∈ t
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ AnalyticAt β„‚ (f (c, z).1) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperAtC.ga_of_fa
[527, 1]
[537, 73]
exact m
case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ (c, id z) ∈ t
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) s : SuperAtC f d u t : Set (β„‚ Γ— β„‚) o : IsOpen t fa : AnalyticOn β„‚ (uncurry f) t tc : βˆ€ {p : β„‚ Γ— β„‚}, p ∈ t β†’ p.1 ∈ u c z : β„‚ m : (c, z) ∈ t ⊒ (c, id z) ∈ t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
set tu := ⋃ i, t i
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) ⊒ SuperNearC f d (⋃ i, u i) (⋃ i, t i)
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i ⊒ SuperNearC f d (⋃ i, u i) tu
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) ⊒ SuperNearC f d (⋃ i, u i) (⋃ i, t i) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
have o : IsOpen tu := isOpen_iUnion fun i ↦ (s i).o
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i ⊒ SuperNearC f d (⋃ i, u i) tu
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu ⊒ SuperNearC f d (⋃ i, u i) tu
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i ⊒ SuperNearC f d (⋃ i, u i) tu TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
exact { o tc := by intro p m; rcases Set.mem_iUnion.mp m with ⟨i, m⟩ exact Set.subset_iUnion _ i ((s i).tc m) fa := by intro p m; rcases Set.mem_iUnion.mp m with ⟨i, m⟩; exact (s i).fa _ m s := by intro c m; rcases Set.mem_iUnion.mp m with ⟨i, m⟩; have s := (s i).s m exact { d2 := s.d2 fa0 := s.fa0 fd := s.fd fc := s.fc o := o.snd_preimage c t0 := Set.subset_iUnion _ i s.t0 t2 := by intro z m; rcases sm m with ⟨u, m, _, s⟩; exact s.t2 m fa := by intro z m; rcases sm m with ⟨u, m, _, s⟩; exact s.fa _ m ft := by intro z m; rcases sm m with ⟨u, m, us, s⟩; exact us (s.ft m) gs' := by intro z z0 m; rcases sm m with ⟨u, m, _, s⟩; exact s.gs' z0 m } }
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu sm : βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u ⊒ SuperNearC f d (⋃ i, u i) tu
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu sm : βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u ⊒ SuperNearC f d (⋃ i, u i) tu TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
intro c z m
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu ⊒ βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m : (c, z) ∈ tu ⊒ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu ⊒ βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
rcases Set.mem_iUnion.mp m with ⟨i, m⟩
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m : (c, z) ∈ tu ⊒ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
case intro f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m : (c, z) ∈ tu ⊒ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
use{z | (c, z) ∈ t i}
case intro f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ z ∈ {z | (c, z) ∈ t i} ∧ {z | (c, z) ∈ t i} βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d {z | (c, z) ∈ t i}
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
simp only [Set.mem_setOf_eq, m, Set.mem_iUnion, Set.setOf_subset_setOf, true_and, tu]
case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ z ∈ {z | (c, z) ∈ t i} ∧ {z | (c, z) ∈ t i} βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d {z | (c, z) ∈ t i}
case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ (βˆ€ (a : β„‚), (c, a) ∈ t i β†’ βˆƒ i, (c, a) ∈ t i) ∧ SuperNear (f c) d {z | (c, z) ∈ t i}
Please generate a tactic in lean4 to solve the state. STATE: case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ z ∈ {z | (c, z) ∈ t i} ∧ {z | (c, z) ∈ t i} βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d {z | (c, z) ∈ t i} TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
constructor
case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ (βˆ€ (a : β„‚), (c, a) ∈ t i β†’ βˆƒ i, (c, a) ∈ t i) ∧ SuperNear (f c) d {z | (c, z) ∈ t i}
case h.left f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ βˆ€ (a : β„‚), (c, a) ∈ t i β†’ βˆƒ i, (c, a) ∈ t i case h.right f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ SuperNear (f c) d {z | (c, z) ∈ t i}
Please generate a tactic in lean4 to solve the state. STATE: case h f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ (βˆ€ (a : β„‚), (c, a) ∈ t i β†’ βˆƒ i, (c, a) ∈ t i) ∧ SuperNear (f c) d {z | (c, z) ∈ t i} TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
exact fun z m ↦ ⟨i, m⟩
case h.left f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ βˆ€ (a : β„‚), (c, a) ∈ t i β†’ βˆƒ i, (c, a) ∈ t i
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.left f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ βˆ€ (a : β„‚), (c, a) ∈ t i β†’ βˆƒ i, (c, a) ∈ t i TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
exact (s i).s ((s i).tc m)
case h.right f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ SuperNear (f c) d {z | (c, z) ∈ t i}
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.right f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu c z : β„‚ m✝ : (c, z) ∈ tu i : I m : (c, z) ∈ t i ⊒ SuperNear (f c) d {z | (c, z) ∈ t i} TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
intro p m
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu sm : βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u ⊒ βˆ€ {p : β„‚ Γ— β„‚}, p ∈ tu β†’ p.1 ∈ ⋃ i, u i
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu sm : βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u p : β„‚ Γ— β„‚ m : p ∈ tu ⊒ p.1 ∈ ⋃ i, u i
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu sm : βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u ⊒ βˆ€ {p : β„‚ Γ— β„‚}, p ∈ tu β†’ p.1 ∈ ⋃ i, u i TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
rcases Set.mem_iUnion.mp m with ⟨i, m⟩
f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu sm : βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u p : β„‚ Γ— β„‚ m : p ∈ tu ⊒ p.1 ∈ ⋃ i, u i
case intro f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu sm : βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u p : β„‚ Γ— β„‚ m✝ : p ∈ tu i : I m : p ∈ t i ⊒ p.1 ∈ ⋃ i, u i
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu sm : βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u p : β„‚ Γ— β„‚ m : p ∈ tu ⊒ p.1 ∈ ⋃ i, u i TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNearC.union
[544, 1]
[573, 88]
exact Set.subset_iUnion _ i ((s i).tc m)
case intro f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu sm : βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u p : β„‚ Γ— β„‚ m✝ : p ∈ tu i : I m : p ∈ t i ⊒ p.1 ∈ ⋃ i, u i
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ β†’ β„‚ d : β„• u✝ : Set β„‚ t✝ : Set (β„‚ Γ— β„‚) I : Type u : I β†’ Set β„‚ t : I β†’ Set (β„‚ Γ— β„‚) s : βˆ€ (i : I), SuperNearC f d (u i) (t i) tu : Set (β„‚ Γ— β„‚) := ⋃ i, t i o : IsOpen tu sm : βˆ€ {c z : β„‚}, (c, z) ∈ tu β†’ βˆƒ u, z ∈ u ∧ u βŠ† {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u p : β„‚ Γ— β„‚ m✝ : p ∈ tu i : I m : p ∈ t i ⊒ p.1 ∈ ⋃ i, u i TACTIC: