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pkuAI4M
/
lean_github

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https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
simp only [mem_setOf, map_invβ‚€]
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ (Complex.abs c)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
apply inv_lt_inv_of_lt
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ (Complex.abs c)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹
case hb c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ 0 < max 16 (Complex.abs c / 2) case h c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ max 16 (Complex.abs c / 2) < Complex.abs c
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ (Complex.abs c)⁻¹ < (max 16 (Complex.abs c / 2))⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
bound
case hb c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ 0 < max 16 (Complex.abs c / 2) case h c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ max 16 (Complex.abs c / 2) < Complex.abs c
case h c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ max 16 (Complex.abs c / 2) < Complex.abs c
Please generate a tactic in lean4 to solve the state. STATE: case hb c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ 0 < max 16 (Complex.abs c / 2) case h c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ max 16 (Complex.abs c / 2) < Complex.abs c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
refine max_lt lo (half_lt_self (lt_trans (by norm_num) lo))
case h c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ max 16 (Complex.abs c / 2) < Complex.abs c
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ max 16 (Complex.abs c / 2) < Complex.abs c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
norm_num
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ 0 < 16
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ⊒ 0 < 16 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
norm_num
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} ⊒ 2 < 16
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} ⊒ 2 < 16 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
intro n
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d ⊒ βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d n : β„• ⊒ (f d c)^[n] ↑c β‰  0
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d ⊒ βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
contrapose mem
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d n : β„• ⊒ (f d c)^[n] ↑c β‰  0
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} n : β„• mem : Β¬(f d c)^[n] ↑c β‰  0 ⊒ Β¬c βˆ‰ multibrot d
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d n : β„• ⊒ (f d c)^[n] ↑c β‰  0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
simp only [not_not] at mem ⊒
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} n : β„• mem : Β¬(f d c)^[n] ↑c β‰  0 ⊒ Β¬c βˆ‰ multibrot d
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} n : β„• mem : (f d c)^[n] ↑c = 0 ⊒ c ∈ multibrot d
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} n : β„• mem : Β¬(f d c)^[n] ↑c β‰  0 ⊒ Β¬c βˆ‰ multibrot d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
exact multibrot_of_zero mem
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} n : β„• mem : (f d c)^[n] ↑c = 0 ⊒ c ∈ multibrot d
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} n : β„• mem : (f d c)^[n] ↑c = 0 ⊒ c ∈ multibrot d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
intro n
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 ⊒ βˆ€ (n : β„•), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• ⊒ ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 ⊒ βˆ€ (n : β„•), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
induction' n with n h
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• ⊒ ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
case zero c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 ⊒ ((f d c)^[0] ↑c)⁻¹ = ↑(g^[0] c⁻¹) case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹)
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• ⊒ ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
have cp : c β‰  0 := Complex.abs.ne_zero_iff.mp (lt_trans (by norm_num) lo).ne'
case zero c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 ⊒ ((f d c)^[0] ↑c)⁻¹ = ↑(g^[0] c⁻¹) case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹)
case zero c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 cp : c β‰  0 ⊒ ((f d c)^[0] ↑c)⁻¹ = ↑(g^[0] c⁻¹) case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹)
Please generate a tactic in lean4 to solve the state. STATE: case zero c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 ⊒ ((f d c)^[0] ↑c)⁻¹ = ↑(g^[0] c⁻¹) case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
simp only [Function.iterate_zero_apply, inv_coe cp, toComplex_coe]
case zero c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 cp : c β‰  0 ⊒ ((f d c)^[0] ↑c)⁻¹ = ↑(g^[0] c⁻¹) case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹)
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹)
Please generate a tactic in lean4 to solve the state. STATE: case zero c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 cp : c β‰  0 ⊒ ((f d c)^[0] ↑c)⁻¹ = ↑(g^[0] c⁻¹) case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
have e : (f d c)^[n] ↑c = ((g^[n] c⁻¹ : β„‚) : π•Š)⁻¹ := by rw [← h, inv_inv]
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹)
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹)
Please generate a tactic in lean4 to solve the state. STATE: case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
simp only [Function.iterate_succ_apply', e]
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹)
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ ⊒ (f d c (↑(g^[n] c⁻¹))⁻¹)⁻¹ = ↑(g (g^[n] c⁻¹))
Please generate a tactic in lean4 to solve the state. STATE: case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ ⊒ ((f d c)^[n + 1] ↑c)⁻¹ = ↑(g^[n + 1] c⁻¹) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
generalize hz : g^[n] c⁻¹ = z
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ ⊒ (f d c (↑(g^[n] c⁻¹))⁻¹)⁻¹ = ↑(g (g^[n] c⁻¹))
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ (f d c (↑z)⁻¹)⁻¹ = ↑(g z)
Please generate a tactic in lean4 to solve the state. STATE: case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ ⊒ (f d c (↑(g^[n] c⁻¹))⁻¹)⁻¹ = ↑(g (g^[n] c⁻¹)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
simp only [← hg, fl, extChartAt_inf, PartialEquiv.trans_apply, Equiv.toPartialEquiv_apply, invEquiv_apply, RiemannSphere.inv_inf, coePartialEquiv_symm_apply, toComplex_zero, sub_zero, Function.comp, add_zero, PartialEquiv.coe_trans_symm, PartialEquiv.symm_symm, coePartialEquiv_apply, Equiv.toPartialEquiv_symm_apply, invEquiv_symm]
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ (f d c (↑z)⁻¹)⁻¹ = ↑(g z)
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ (f d c (↑z)⁻¹)⁻¹ = ↑(f d c (↑z)⁻¹)⁻¹.toComplex
Please generate a tactic in lean4 to solve the state. STATE: case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ (f d c (↑z)⁻¹)⁻¹ = ↑(g z) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
rw [coe_toComplex]
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ (f d c (↑z)⁻¹)⁻¹ = ↑(f d c (↑z)⁻¹)⁻¹.toComplex
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ (f d c (↑z)⁻¹)⁻¹ β‰  ∞
Please generate a tactic in lean4 to solve the state. STATE: case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ (f d c (↑z)⁻¹)⁻¹ = ↑(f d c (↑z)⁻¹)⁻¹.toComplex TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
simp only [Ne, inv_eq_inf, ← hz, ← h, inv_inv, ← Function.iterate_succ_apply' (f d c)]
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ (f d c (↑z)⁻¹)⁻¹ β‰  ∞
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ Β¬(f d c)^[n.succ] ↑c = 0
Please generate a tactic in lean4 to solve the state. STATE: case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ (f d c (↑z)⁻¹)⁻¹ β‰  ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
apply nz
case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ Β¬(f d c)^[n.succ] ↑c = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case succ c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) e : (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ z : β„‚ hz : g^[n] c⁻¹ = z ⊒ Β¬(f d c)^[n.succ] ↑c = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
norm_num
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 ⊒ 0 < 16
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 ⊒ 0 < 16 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
rw [← h, inv_inv]
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• h : ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) ⊒ (f d c)^[n] ↑c = (↑(g^[n] c⁻¹))⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
rw [← hg]
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 iter : βˆ€ (n : β„•), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) n : β„• right✝ : (c, (f d c)^[n] ↑c) ∈ s.near eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹) e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n ⊒ bottcherNear g d (g^[n] c⁻¹) = bottcherNear g d c⁻¹ ^ d ^ n
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 iter : βˆ€ (n : β„•), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) n : β„• right✝ : (c, (f d c)^[n] ↑c) ∈ s.near eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹) e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n ⊒ bottcherNear (fl (f d) ∞ c) d ((fl (f d) ∞ c)^[n] c⁻¹) = bottcherNear (fl (f d) ∞ c) d c⁻¹ ^ d ^ n
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 iter : βˆ€ (n : β„•), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) n : β„• right✝ : (c, (f d c)^[n] ↑c) ∈ s.near eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹) e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n ⊒ bottcherNear g d (g^[n] c⁻¹) = bottcherNear g d c⁻¹ ^ d ^ n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
exact bottcherNear_eqn_iter (superNearF d c) ct n
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 iter : βˆ€ (n : β„•), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) n : β„• right✝ : (c, (f d c)^[n] ↑c) ∈ s.near eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹) e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n ⊒ bottcherNear (fl (f d) ∞ c) d ((fl (f d) ∞ c)^[n] c⁻¹) = bottcherNear (fl (f d) ∞ c) d c⁻¹ ^ d ^ n
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 iter : βˆ€ (n : β„•), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) n : β„• right✝ : (c, (f d c)^[n] ↑c) ∈ s.near eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹) e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n ⊒ bottcherNear (fl (f d) ∞ c) d ((fl (f d) ∞ c)^[n] c⁻¹) = bottcherNear (fl (f d) ∞ c) d c⁻¹ ^ d ^ n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
apply (pow_left_inj (Complex.abs.nonneg _) (Complex.abs.nonneg _) (pow_ne_zero n (d_ne_zero d))).mp
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• right✝ : (c, (f d c)^[n] ↑c) ∈ s.near eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n ⊒ Complex.abs (bottcher' d c) = Complex.abs (bottcherNear g d c⁻¹)
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• right✝ : (c, (f d c)^[n] ↑c) ∈ s.near eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n ⊒ Complex.abs (bottcher' d c) ^ d ^ n = Complex.abs (bottcherNear g d c⁻¹) ^ d ^ n
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• right✝ : (c, (f d c)^[n] ↑c) ∈ s.near eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n ⊒ Complex.abs (bottcher' d c) = Complex.abs (bottcherNear g d c⁻¹) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_bound
[513, 1]
[554, 59]
simp only [← Complex.abs.map_pow, eq]
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• right✝ : (c, (f d c)^[n] ↑c) ∈ s.near eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n ⊒ Complex.abs (bottcher' d c) ^ d ^ n = Complex.abs (bottcherNear g d c⁻¹) ^ d ^ n
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : β„‚ lo : 16 < Complex.abs c s : Super (f d) d ∞ := superF d g : β„‚ β†’ β„‚ hg : fl (f d) ∞ c = g ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹} mem : c βˆ‰ multibrot d nz : βˆ€ (n : β„•), (f d c)^[n] ↑c β‰  0 n : β„• right✝ : (c, (f d c)^[n] ↑c) ∈ s.near eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n ⊒ Complex.abs (bottcher' d c) ^ d ^ n = Complex.abs (bottcherNear g d c⁻¹) ^ d ^ n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
rw [Metric.tendsto_nhds]
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ Tendsto (bottcher' d) atInf (𝓝 0)
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ βˆ€ Ξ΅ > 0, βˆ€αΆ  (x : β„‚) in atInf, dist (bottcher' d x) 0 < Ξ΅
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ Tendsto (bottcher' d) atInf (𝓝 0) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
intro r rp
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ βˆ€ Ξ΅ > 0, βˆ€αΆ  (x : β„‚) in atInf, dist (bottcher' d x) 0 < Ξ΅
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ βˆ€αΆ  (x : β„‚) in atInf, dist (bottcher' d x) 0 < r
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ βˆ€ Ξ΅ > 0, βˆ€αΆ  (x : β„‚) in atInf, dist (bottcher' d x) 0 < Ξ΅ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
rw [atInf_basis.eventually_iff]
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ βˆ€αΆ  (x : β„‚) in atInf, dist (bottcher' d x) 0 < r
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ βˆƒ i, True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | i < β€–xβ€–} β†’ dist (bottcher' d x) 0 < r
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ βˆ€αΆ  (x : β„‚) in atInf, dist (bottcher' d x) 0 < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
use max 16 (3 / r)
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ βˆƒ i, True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | i < β€–xβ€–} β†’ dist (bottcher' d x) 0 < r
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | max 16 (3 / r) < β€–xβ€–} β†’ dist (bottcher' d x) 0 < r
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ βˆƒ i, True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | i < β€–xβ€–} β†’ dist (bottcher' d x) 0 < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
simp only [true_and_iff, mem_setOf, Complex.dist_eq, sub_zero, Complex.norm_eq_abs, max_lt_iff]
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | max 16 (3 / r) < β€–xβ€–} β†’ dist (bottcher' d x) 0 < r
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ βˆ€ ⦃x : ℂ⦄, 16 < Complex.abs x ∧ 3 / r < Complex.abs x β†’ Complex.abs (bottcher' d x) < r
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | max 16 (3 / r) < β€–xβ€–} β†’ dist (bottcher' d x) 0 < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
intro z ⟨lo, rz⟩
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ βˆ€ ⦃x : ℂ⦄, 16 < Complex.abs x ∧ 3 / r < Complex.abs x β†’ Complex.abs (bottcher' d x) < r
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 / r < Complex.abs z ⊒ Complex.abs (bottcher' d z) < r
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 ⊒ βˆ€ ⦃x : ℂ⦄, 16 < Complex.abs x ∧ 3 / r < Complex.abs x β†’ Complex.abs (bottcher' d x) < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
apply lt_of_le_of_lt (bottcher_bound lo)
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 / r < Complex.abs z ⊒ Complex.abs (bottcher' d z) < r
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 / r < Complex.abs z ⊒ 3 * Complex.abs z⁻¹ < r
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 / r < Complex.abs z ⊒ Complex.abs (bottcher' d z) < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
rw [div_lt_iff rp] at rz
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 / r < Complex.abs z ⊒ 3 * Complex.abs z⁻¹ < r
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 < Complex.abs z * r ⊒ 3 * Complex.abs z⁻¹ < r
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 / r < Complex.abs z ⊒ 3 * Complex.abs z⁻¹ < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
rw [map_invβ‚€, mul_inv_lt_iff (lt_trans (by norm_num) lo)]
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 < Complex.abs z * r ⊒ 3 * Complex.abs z⁻¹ < r
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 < Complex.abs z * r ⊒ 3 < Complex.abs z * r
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 < Complex.abs z * r ⊒ 3 * Complex.abs z⁻¹ < r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
exact rz
case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 < Complex.abs z * r ⊒ 3 < Complex.abs z * r
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 < Complex.abs z * r ⊒ 3 < Complex.abs z * r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_tendsto_zero
[557, 1]
[562, 96]
norm_num
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 < Complex.abs z * r ⊒ 0 < 16
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) r : ℝ rp : r > 0 z : β„‚ lo : 16 < Complex.abs z rz : 3 < Complex.abs z * r ⊒ 0 < 16 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_analytic
[565, 1]
[568, 28]
set s := superF d
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ AnalyticOn β„‚ (bottcher' d) (multibrot d)ᢜ
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ AnalyticOn β„‚ (bottcher' d) (multibrot d)ᢜ
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ AnalyticOn β„‚ (bottcher' d) (multibrot d)ᢜ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_analytic
[565, 1]
[568, 28]
intro c m
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ AnalyticOn β„‚ (bottcher' d) (multibrot d)ᢜ
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : β„‚ m : c ∈ (multibrot d)ᢜ ⊒ AnalyticAt β„‚ (bottcher' d) c
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ AnalyticOn β„‚ (bottcher' d) (multibrot d)ᢜ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_analytic
[565, 1]
[568, 28]
apply HolomorphicAt.analyticAt I I
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : β„‚ m : c ∈ (multibrot d)ᢜ ⊒ AnalyticAt β„‚ (bottcher' d) c
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : β„‚ m : c ∈ (multibrot d)ᢜ ⊒ HolomorphicAt I I (bottcher' d) c
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : β„‚ m : c ∈ (multibrot d)ᢜ ⊒ AnalyticAt β„‚ (bottcher' d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcher_analytic
[565, 1]
[568, 28]
exact (s.bottcher_holomorphicOn (c, c) (multibrotPost m)).compβ‚‚_of_eq holomorphicAt_id (holomorphic_coe _) rfl
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : β„‚ m : c ∈ (multibrot d)ᢜ ⊒ HolomorphicAt I I (bottcher' d) c
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : β„‚ m : c ∈ (multibrot d)ᢜ ⊒ HolomorphicAt I I (bottcher' d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherHolomorphic
[571, 1]
[579, 86]
intro c m
c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) ⊒ HolomorphicOn I I (bottcher d) (multibrotExt d)
c✝ : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d ⊒ HolomorphicAt I I (bottcher d) c
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) ⊒ HolomorphicOn I I (bottcher d) (multibrotExt d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherHolomorphic
[571, 1]
[579, 86]
induction c using OnePoint.rec
c✝ : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d ⊒ HolomorphicAt I I (bottcher d) c
case h₁ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ HolomorphicAt I I (bottcher d) ∞ case hβ‚‚ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ HolomorphicAt I I (bottcher d) ↑x✝
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d ⊒ HolomorphicAt I I (bottcher d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherHolomorphic
[571, 1]
[579, 86]
refine holomorphicAt_fill_inf ?_ bottcher_tendsto_zero
case h₁ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ HolomorphicAt I I (bottcher d) ∞
case h₁ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ βˆ€αΆ  (z : β„‚) in atInf, HolomorphicAt I I (bottcher' d) z
Please generate a tactic in lean4 to solve the state. STATE: case h₁ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ HolomorphicAt I I (bottcher d) ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherHolomorphic
[571, 1]
[579, 86]
rw [atInf_basis.eventually_iff]
case h₁ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ βˆ€αΆ  (z : β„‚) in atInf, HolomorphicAt I I (bottcher' d) z
case h₁ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ βˆƒ i, True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | i < β€–xβ€–} β†’ HolomorphicAt I I (bottcher' d) x
Please generate a tactic in lean4 to solve the state. STATE: case h₁ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ βˆ€αΆ  (z : β„‚) in atInf, HolomorphicAt I I (bottcher' d) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherHolomorphic
[571, 1]
[579, 86]
use 2
case h₁ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ βˆƒ i, True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | i < β€–xβ€–} β†’ HolomorphicAt I I (bottcher' d) x
case h c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | 2 < β€–xβ€–} β†’ HolomorphicAt I I (bottcher' d) x
Please generate a tactic in lean4 to solve the state. STATE: case h₁ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ βˆƒ i, True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | i < β€–xβ€–} β†’ HolomorphicAt I I (bottcher' d) x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherHolomorphic
[571, 1]
[579, 86]
simp only [true_and_iff, mem_setOf, Complex.norm_eq_abs]
case h c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | 2 < β€–xβ€–} β†’ HolomorphicAt I I (bottcher' d) x
case h c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ βˆ€ ⦃x : ℂ⦄, 2 < Complex.abs x β†’ HolomorphicAt I I (bottcher' d) x
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ True ∧ βˆ€ ⦃x : ℂ⦄, x ∈ {x | 2 < β€–xβ€–} β†’ HolomorphicAt I I (bottcher' d) x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherHolomorphic
[571, 1]
[579, 86]
intro z a
case h c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ βˆ€ ⦃x : ℂ⦄, 2 < Complex.abs x β†’ HolomorphicAt I I (bottcher' d) x
case h c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d z : β„‚ a : 2 < Complex.abs z ⊒ HolomorphicAt I I (bottcher' d) z
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d ⊒ βˆ€ ⦃x : ℂ⦄, 2 < Complex.abs x β†’ HolomorphicAt I I (bottcher' d) x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherHolomorphic
[571, 1]
[579, 86]
exact (bottcher_analytic _ (multibrot_two_lt a)).holomorphicAt I I
case h c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d z : β„‚ a : 2 < Complex.abs z ⊒ HolomorphicAt I I (bottcher' d) z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) m : ∞ ∈ multibrotExt d z : β„‚ a : 2 < Complex.abs z ⊒ HolomorphicAt I I (bottcher' d) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherHolomorphic
[571, 1]
[579, 86]
simp only [multibrotExt_coe] at m
case hβ‚‚ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ HolomorphicAt I I (bottcher d) ↑x✝
case hβ‚‚ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ m : x✝ βˆ‰ multibrot d ⊒ HolomorphicAt I I (bottcher d) ↑x✝
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ HolomorphicAt I I (bottcher d) ↑x✝ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherHolomorphic
[571, 1]
[579, 86]
exact holomorphicAt_fill_coe ((bottcher_analytic (d := d) _ m).holomorphicAt I I)
case hβ‚‚ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ m : x✝ βˆ‰ multibrot d ⊒ HolomorphicAt I I (bottcher d) ↑x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ c : β„‚ d✝ : β„• inst✝¹ : Fact (2 ≀ d✝) d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ m : x✝ βˆ‰ multibrot d ⊒ HolomorphicAt I I (bottcher d) ↑x✝ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
abs_bottcher
[589, 1]
[593, 68]
set s := superF d
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š ⊒ Complex.abs (bottcher d c) = potential d c
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š s : Super (f d) d ∞ := superF d ⊒ Complex.abs (bottcher d c) = potential d c
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š ⊒ Complex.abs (bottcher d c) = potential d c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
abs_bottcher
[589, 1]
[593, 68]
induction c using OnePoint.rec
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š s : Super (f d) d ∞ := superF d ⊒ Complex.abs (bottcher d c) = potential d c
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ Complex.abs (bottcher d ∞) = potential d ∞ case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ Complex.abs (bottcher d ↑x✝) = potential d ↑x✝
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š s : Super (f d) d ∞ := superF d ⊒ Complex.abs (bottcher d c) = potential d c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
abs_bottcher
[589, 1]
[593, 68]
simp only [bottcher, potential, fill_inf, Complex.abs.map_zero]
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ Complex.abs (bottcher d ∞) = potential d ∞
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ Complex.abs (bottcher d ∞) = potential d ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
abs_bottcher
[589, 1]
[593, 68]
simp only [bottcher, potential, fill_coe]
case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ Complex.abs (bottcher d ↑x✝) = potential d ↑x✝
case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ Complex.abs (bottcher' d x✝) = β‹―.potential x✝ ↑x✝
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ Complex.abs (bottcher d ↑x✝) = potential d ↑x✝ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
abs_bottcher
[589, 1]
[593, 68]
exact s.abs_bottcher
case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ Complex.abs (bottcher' d x✝) = β‹―.potential x✝ ↑x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ Complex.abs (bottcher' d x✝) = β‹―.potential x✝ ↑x✝ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_continuous
[595, 1]
[603, 49]
set s := superF d
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ Continuous (potential d)
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ Continuous (potential d)
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ Continuous (potential d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_continuous
[595, 1]
[603, 49]
rw [continuous_iff_continuousAt]
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ Continuous (potential d)
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ βˆ€ (x : π•Š), ContinuousAt (potential d) x
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ Continuous (potential d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_continuous
[595, 1]
[603, 49]
intro c
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ βˆ€ (x : π•Š), ContinuousAt (potential d) x
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : π•Š ⊒ ContinuousAt (potential d) c
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ βˆ€ (x : π•Š), ContinuousAt (potential d) x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_continuous
[595, 1]
[603, 49]
induction c using OnePoint.rec
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : π•Š ⊒ ContinuousAt (potential d) c
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ ContinuousAt (potential d) ∞ case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ ContinuousAt (potential d) ↑x✝
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : π•Š ⊒ ContinuousAt (potential d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_continuous
[595, 1]
[603, 49]
have e : potential d =αΆ [𝓝 ∞] fun c ↦ abs (bottcher d c) := by refine eventually_of_forall fun c ↦ ?_; rw [← abs_bottcher]
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ ContinuousAt (potential d) ∞
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d e : (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c) ⊒ ContinuousAt (potential d) ∞
Please generate a tactic in lean4 to solve the state. STATE: case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ ContinuousAt (potential d) ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_continuous
[595, 1]
[603, 49]
rw [continuousAt_congr e]
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d e : (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c) ⊒ ContinuousAt (potential d) ∞
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d e : (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c) ⊒ ContinuousAt (fun c => Complex.abs (bottcher d c)) ∞
Please generate a tactic in lean4 to solve the state. STATE: case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d e : (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c) ⊒ ContinuousAt (potential d) ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_continuous
[595, 1]
[603, 49]
exact Complex.continuous_abs.continuousAt.comp (bottcherHolomorphic d _ multibrotExt_inf).continuousAt
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d e : (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c) ⊒ ContinuousAt (fun c => Complex.abs (bottcher d c)) ∞
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d e : (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c) ⊒ ContinuousAt (fun c => Complex.abs (bottcher d c)) ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_continuous
[595, 1]
[603, 49]
refine eventually_of_forall fun c ↦ ?_
c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c)
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : π•Š ⊒ potential d c = (fun c => Complex.abs (bottcher d c)) c
Please generate a tactic in lean4 to solve the state. STATE: c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ (𝓝 ∞).EventuallyEq (potential d) fun c => Complex.abs (bottcher d c) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_continuous
[595, 1]
[603, 49]
rw [← abs_bottcher]
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : π•Š ⊒ potential d c = (fun c => Complex.abs (bottcher d c)) c
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d c : π•Š ⊒ potential d c = (fun c => Complex.abs (bottcher d c)) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_continuous
[595, 1]
[603, 49]
exact continuousAt_fill_coe ((Continuous.potential s).compβ‚‚ continuous_id continuous_coe).continuousAt
case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ ContinuousAt (potential d) ↑x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ ContinuousAt (potential d) ↑x✝ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
set s := superF d
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š ⊒ potential d c < 1 ↔ c ∈ multibrotExt d
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š s : Super (f d) d ∞ := superF d ⊒ potential d c < 1 ↔ c ∈ multibrotExt d
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š ⊒ potential d c < 1 ↔ c ∈ multibrotExt d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
induction c using OnePoint.rec
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š s : Super (f d) d ∞ := superF d ⊒ potential d c < 1 ↔ c ∈ multibrotExt d
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ potential d ∞ < 1 ↔ ∞ ∈ multibrotExt d case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ potential d ↑x✝ < 1 ↔ ↑x✝ ∈ multibrotExt d
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š s : Super (f d) d ∞ := superF d ⊒ potential d c < 1 ↔ c ∈ multibrotExt d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
simp only [potential, bottcher, fill_inf, zero_lt_one, multibrotExt_inf]
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ potential d ∞ < 1 ↔ ∞ ∈ multibrotExt d
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d ⊒ potential d ∞ < 1 ↔ ∞ ∈ multibrotExt d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
constructor
case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ potential d ↑x✝ < 1 ↔ ↑x✝ ∈ multibrotExt d
case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ potential d ↑x✝ < 1 β†’ ↑x✝ ∈ multibrotExt d case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ ↑x✝ ∈ multibrotExt d β†’ potential d ↑x✝ < 1
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ potential d ↑x✝ < 1 ↔ ↑x✝ ∈ multibrotExt d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
intro h
case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ potential d ↑x✝ < 1 β†’ ↑x✝ ∈ multibrotExt d
case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : potential d ↑x✝ < 1 ⊒ ↑x✝ ∈ multibrotExt d
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ potential d ↑x✝ < 1 β†’ ↑x✝ ∈ multibrotExt d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
contrapose h
case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : potential d ↑x✝ < 1 ⊒ ↑x✝ ∈ multibrotExt d
case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : ↑x✝ βˆ‰ multibrotExt d ⊒ Β¬potential d ↑x✝ < 1
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : potential d ↑x✝ < 1 ⊒ ↑x✝ ∈ multibrotExt d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
simp only [not_not, not_lt, multibrot_basin', potential, fill_coe, Super.basin, mem_setOf, not_exists, multibrotExt_coe] at h ⊒
case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : ↑x✝ βˆ‰ multibrotExt d ⊒ Β¬potential d ↑x✝ < 1
case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : βˆ€ (x : β„•), (x✝, (f d x✝)^[x] ↑x✝) βˆ‰ β‹―.near ⊒ 1 ≀ β‹―.potential x✝ ↑x✝
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : ↑x✝ βˆ‰ multibrotExt d ⊒ Β¬potential d ↑x✝ < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
rw [s.potential_eq_one]
case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : βˆ€ (x : β„•), (x✝, (f d x✝)^[x] ↑x✝) βˆ‰ β‹―.near ⊒ 1 ≀ β‹―.potential x✝ ↑x✝
case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : βˆ€ (x : β„•), (x✝, (f d x✝)^[x] ↑x✝) βˆ‰ β‹―.near ⊒ βˆ€ (n : β„•), (x✝, (f d x✝)^[n] ↑x✝) βˆ‰ s.near
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : βˆ€ (x : β„•), (x✝, (f d x✝)^[x] ↑x✝) βˆ‰ β‹―.near ⊒ 1 ≀ β‹―.potential x✝ ↑x✝ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
exact h
case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : βˆ€ (x : β„•), (x✝, (f d x✝)^[x] ↑x✝) βˆ‰ β‹―.near ⊒ βˆ€ (n : β„•), (x✝, (f d x✝)^[n] ↑x✝) βˆ‰ s.near
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚.mp c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ h : βˆ€ (x : β„•), (x✝, (f d x✝)^[x] ↑x✝) βˆ‰ β‹―.near ⊒ βˆ€ (n : β„•), (x✝, (f d x✝)^[n] ↑x✝) βˆ‰ s.near TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
intro m
case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ ↑x✝ ∈ multibrotExt d β†’ potential d ↑x✝ < 1
case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ potential d ↑x✝ < 1
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ ⊒ ↑x✝ ∈ multibrotExt d β†’ potential d ↑x✝ < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
rw [← abs_bottcher]
case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ potential d ↑x✝ < 1
case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ Complex.abs (bottcher d ↑x✝) < 1
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ potential d ↑x✝ < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
simp only [bottcher, fill_coe]
case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ Complex.abs (bottcher d ↑x✝) < 1
case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ Complex.abs (bottcher' d x✝) < 1
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ Complex.abs (bottcher d ↑x✝) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
simp only [multibrotExt_coe] at m
case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ Complex.abs (bottcher' d x✝) < 1
case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : x✝ βˆ‰ multibrot d ⊒ Complex.abs (bottcher' d x✝) < 1
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : ↑x✝ ∈ multibrotExt d ⊒ Complex.abs (bottcher' d x✝) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_lt_one
[605, 1]
[616, 48]
exact s.bottcher_lt_one (multibrotPost m)
case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : x✝ βˆ‰ multibrot d ⊒ Complex.abs (bottcher' d x✝) < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚.mpr c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) s : Super (f d) d ∞ := superF d x✝ : β„‚ m : x✝ βˆ‰ multibrot d ⊒ Complex.abs (bottcher' d x✝) < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_nonneg
[618, 1]
[621, 71]
induction c using OnePoint.rec
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š ⊒ 0 ≀ potential d c
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ 0 ≀ potential d ∞ case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ ⊒ 0 ≀ potential d ↑x✝
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š ⊒ 0 ≀ potential d c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_nonneg
[618, 1]
[621, 71]
simp only [potential, fill_inf, le_refl]
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ 0 ≀ potential d ∞
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ 0 ≀ potential d ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_nonneg
[618, 1]
[621, 71]
simp only [potential, fill_coe]
case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ ⊒ 0 ≀ potential d ↑x✝
case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ ⊒ 0 ≀ β‹―.potential x✝ ↑x✝
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ ⊒ 0 ≀ potential d ↑x✝ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_nonneg
[618, 1]
[621, 71]
exact (superF d).potential_nonneg
case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ ⊒ 0 ≀ β‹―.potential x✝ ↑x✝
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ ⊒ 0 ≀ β‹―.potential x✝ ↑x✝ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_eq_zero
[623, 1]
[626, 81]
induction c using OnePoint.rec
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š ⊒ potential d c = 0 ↔ c = ∞
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ potential d ∞ = 0 ↔ ∞ = ∞ case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ ⊒ potential d ↑x✝ = 0 ↔ ↑x✝ = ∞
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š ⊒ potential d c = 0 ↔ c = ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_eq_zero
[623, 1]
[626, 81]
simp only [potential, fill_inf, eq_self_iff_true]
case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ potential d ∞ = 0 ↔ ∞ = ∞
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h₁ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) ⊒ potential d ∞ = 0 ↔ ∞ = ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
potential_eq_zero
[623, 1]
[626, 81]
simp only [potential, fill_coe, (superF d).potential_eq_zero_of_onePreimage]
case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ ⊒ potential d ↑x✝ = 0 ↔ ↑x✝ = ∞
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hβ‚‚ c : β„‚ d : β„• inst✝ : Fact (2 ≀ d) x✝ : β„‚ ⊒ potential d ↑x✝ = 0 ↔ ↑x✝ = ∞ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
by_cases h : βˆƒαΆ  e in 𝓝 c, bottcher d e β‰  bottcher d c
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d ⊒ NontrivialHolomorphicAt (bottcher d) c
case pos c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ NontrivialHolomorphicAt (bottcher d) c case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : Β¬βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ NontrivialHolomorphicAt (bottcher d) c
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d ⊒ NontrivialHolomorphicAt (bottcher d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
exact { holomorphicAt := bottcherHolomorphic d _ m nonconst := h }
case pos c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ NontrivialHolomorphicAt (bottcher d) c case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : Β¬βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ NontrivialHolomorphicAt (bottcher d) c
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : Β¬βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ NontrivialHolomorphicAt (bottcher d) c
Please generate a tactic in lean4 to solve the state. STATE: case pos c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ NontrivialHolomorphicAt (bottcher d) c case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : Β¬βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ NontrivialHolomorphicAt (bottcher d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
exfalso
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : Β¬βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ NontrivialHolomorphicAt (bottcher d) c
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : Β¬βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : Β¬βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ NontrivialHolomorphicAt (bottcher d) c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
simp only [Filter.not_frequently, not_not] at h
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : Β¬βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ False
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = bottcher d c ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : Β¬βˆƒαΆ  (e : π•Š) in 𝓝 c, bottcher d e β‰  bottcher d c ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
set b := bottcher d c
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = bottcher d c ⊒ False
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = bottcher d c ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
have b1 : abs b < 1 := by simp only [abs_bottcher, potential_lt_one, m, b]
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b ⊒ False
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
set t := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  e in 𝓝 c, bottcher d e = b}
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 ⊒ False
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
have m0 : (0 : π•Š) ∈ multibrotExt d := haveI m : (0 : π•Š) ∈ t := by simp only [tu, mem_univ] m.1
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} tu : t = univ ⊒ False
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} tu : t = univ m0 : 0 ∈ multibrotExt d ⊒ False
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} tu : t = univ ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
simp only [← coe_zero, multibrotExt_coe, multibrot_zero, not_true] at m0
case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} tu : t = univ m0 : 0 ∈ multibrotExt d ⊒ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} tu : t = univ m0 : 0 ∈ multibrotExt d ⊒ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
simp only [abs_bottcher, potential_lt_one, m, b]
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b ⊒ Complex.abs b < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b ⊒ Complex.abs b < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
refine IsClopen.eq_univ ?_ ⟨c, m, h⟩
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} ⊒ t = univ
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} ⊒ IsClopen t
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} ⊒ t = univ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
bottcherNontrivial
[634, 1]
[670, 75]
constructor
c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} ⊒ IsClopen t
case left c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} ⊒ IsClosed t case right c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} ⊒ IsOpen t
Please generate a tactic in lean4 to solve the state. STATE: c✝ : β„‚ d : β„• inst✝ : Fact (2 ≀ d) c : π•Š m : c ∈ multibrotExt d b : β„‚ := bottcher d c h : βˆ€αΆ  (x : π•Š) in 𝓝 c, bottcher d x = b b1 : Complex.abs b < 1 t : Set π•Š := {c | c ∈ multibrotExt d ∧ βˆ€αΆ  (e : π•Š) in 𝓝 c, bottcher d e = b} ⊒ IsClopen t TACTIC: