url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | simp only [Function.iterate_succ_apply'] | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
n : ℕ
h : (c, (f d c)^[n] 0) ∈ s.near
⊢ (c, (f d c)^[n + 1] 0) ∈ ⋯.near | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
n : ℕ
h : (c, (f d c)^[n] 0) ∈ s.near
⊢ (c, f d c ((f d c)^[n] 0)) ∈ ⋯.near |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_basin | [287, 1] | [296, 88] | exact s.stays_near h | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : ∀ (n : ℕ), (f d c)^[n] ↑c = (f d c)^[n + 1] 0
n : ℕ
h : (c, (f d c)^[n] 0) ∈ s.near
⊢ (c, f d c ((f d c)^[n] 0)) ∈ ⋯.near | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | set s := superF d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ⋯.p c = ⋯.potential c 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ s.p c = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | simp only [Super.p, e, csInf_pair] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : s.ps c = {1, s.potential c 0}
⊢ s.p c = s.potential c 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : s.ps c = {1, s.potential c 0}
⊢ 1 ⊓ s.potential c 0 = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact inf_of_le_right s.potential_le_one | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
e : s.ps c = {1, s.potential c 0}
⊢ 1 ⊓ s.potential c 0 = s.potential c 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | apply Set.ext | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ s.ps c = {1, s.potential c 0} | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ∀ (x : ℝ), x ∈ s.ps c ↔ x ∈ {1, s.potential c 0} |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | intro p | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
⊢ ∀ (x : ℝ), x ∈ s.ps c ↔ x ∈ {1, s.potential c 0} | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
⊢ p ∈ s.ps c ↔ p ∈ {1, s.potential c 0} |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | simp only [Super.ps, mem_singleton_iff, mem_setOf, critical_f, Ne, mem_insert_iff,
mem_singleton_iff] | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
⊢ p ∈ s.ps c ↔ p ∈ {1, s.potential c 0} | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
⊢ (p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)) ↔ p = 1 ∨ p = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | constructor | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
⊢ (p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)) ↔ p = 1 ∨ p = s.potential c 0 | case h.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
⊢ (p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)) → p = 1 ∨ p = s.potential c 0
case h.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
⊢ p = 1 ∨ p = s.potential c 0 → p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potent... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | intro h | case h.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
⊢ (p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)) → p = 1 ∨ p = s.potential c 0 | case h.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
⊢ p = 1 ∨ p = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | cases' h with h h | case h.mp
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
⊢ p = 1 ∨ p = s.potential c 0 | case h.mp.inl
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1
⊢ p = 1 ∨ p = s.potential c 0
case h.mp.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
⊢ p = 1 ∨ p = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | left | case h.mp.inl
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1
⊢ p = 1 ∨ p = s.potential c 0
case h.mp.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
⊢ p = 1 ∨ p = s.potential c 0 | case h.mp.inl.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1
⊢ p = 1
case h.mp.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
⊢ p = 1 ∨ p = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact h | case h.mp.inl.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1
⊢ p = 1
case h.mp.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
⊢ p = 1 ∨ p = s.potential c 0 | case h.mp.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
⊢ p = 1 ∨ p = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | right | case h.mp.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
⊢ p = 1 ∨ p = s.potential c 0 | case h.mp.inr.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
⊢ p = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | rcases h with ⟨p0, z, e, h⟩ | case h.mp.inr.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
⊢ p = s.potential c 0 | case h.mp.inr.h.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.potential c z = p
h : z = 0 ∨ z = ∞
⊢ p = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | cases' h with h h | case h.mp.inr.h.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.potential c z = p
h : z = 0 ∨ z = ∞
⊢ p = s.potential c 0 | case h.mp.inr.h.intro.intro.intro.inl
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.potential c z = p
h : z = 0
⊢ p = s.potential c 0
case h.mp.inr.h.intro.intro.intro.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | rw [h] at e | case h.mp.inr.h.intro.intro.intro.inl
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.potential c z = p
h : z = 0
⊢ p = s.potential c 0
case h.mp.inr.h.intro.intro.intro.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.p... | case h.mp.inr.h.intro.intro.intro.inl
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.potential c 0 = p
h : z = 0
⊢ p = s.potential c 0
case h.mp.inr.h.intro.intro.intro.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact e.symm | case h.mp.inr.h.intro.intro.intro.inl
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.potential c 0 = p
h : z = 0
⊢ p = s.potential c 0
case h.mp.inr.h.intro.intro.intro.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.p... | case h.mp.inr.h.intro.intro.intro.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.potential c z = p
h : z = ∞
⊢ p = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | rw [h, s.potential_a] at e | case h.mp.inr.h.intro.intro.intro.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : s.potential c z = p
h : z = ∞
⊢ p = s.potential c 0 | case h.mp.inr.h.intro.intro.intro.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : 0 = p
h : z = ∞
⊢ p = s.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exfalso | case h.mp.inr.h.intro.intro.intro.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : 0 = p
h : z = ∞
⊢ p = s.potential c 0 | case h.mp.inr.h.intro.intro.intro.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : 0 = p
h : z = ∞
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact p0 e.symm | case h.mp.inr.h.intro.intro.intro.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
p0 : ¬p = 0
z : 𝕊
e : 0 = p
h : z = ∞
⊢ False | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | intro h | case h.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
⊢ p = 1 ∨ p = s.potential c 0 → p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) | case h.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1 ∨ p = s.potential c 0
⊢ p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | cases' h with h h | case h.mpr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1 ∨ p = s.potential c 0
⊢ p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) | case h.mpr.inl
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1
⊢ p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
case h.mpr.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | left | case h.mpr.inl
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1
⊢ p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞)
case h.mpr.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = ... | case h.mpr.inl.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1
⊢ p = 1
case h.mpr.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact h | case h.mpr.inl.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = 1
⊢ p = 1
case h.mpr.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) | case h.mpr.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | right | case h.mpr.inr
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ p = 1 ∨ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) | case h.mpr.inr.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | constructor | case h.mpr.inr.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ ¬p = 0 ∧ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) | case h.mpr.inr.h.left
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ ¬p = 0
case h.mpr.inr.h.right
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | simp only [h, ← ne_eq, s.potential_ne_zero] | case h.mpr.inr.h.left
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ ¬p = 0 | case h.mpr.inr.h.left
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ 0 ≠ ∞ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | exact inf_ne_zero.symm | case h.mpr.inr.h.left
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ 0 ≠ ∞ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_p | [299, 1] | [312, 79] | use 0, h.symm, Or.inl rfl | case h.mpr.inr.h.right
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
p : ℝ
h : p = s.potential c 0
⊢ ∃ z, s.potential c z = p ∧ (z = 0 ∨ z = ∞) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotPost | [315, 1] | [320, 40] | set s := superF d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
m : c ∉ multibrot d
⊢ Postcritical ⋯ c ↑c | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
m : c ∉ multibrot d
s : Super (f d) d ∞ := superF d
⊢ Postcritical s c ↑c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotPost | [315, 1] | [320, 40] | simp only [Postcritical, multibrot_p, ← f_0 d, s.potential_eqn] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
m : c ∉ multibrot d
s : Super (f d) d ∞ := superF d
⊢ Postcritical s c ↑c | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
m : c ∉ multibrot d
s : Super (f d) d ∞ := superF d
⊢ s.potential c 0 ^ d < ⋯.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotPost | [315, 1] | [320, 40] | simp only [multibrot_basin, not_not] at m | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
m : c ∉ multibrot d
s : Super (f d) d ∞ := superF d
⊢ s.potential c 0 ^ d < ⋯.potential c 0 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
m : (c, 0) ∈ ⋯.basin
⊢ s.potential c 0 ^ d < ⋯.potential c 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrotPost | [315, 1] | [320, 40] | exact pow_lt_self_of_lt_one ((s.potential_pos c).mpr inf_ne_zero.symm)
(s.potential_lt_one m) (d_gt_one d) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
s : Super (f d) d ∞ := superF d
m : (c, 0) ∈ ⋯.basin
⊢ s.potential c 0 ^ d < ⋯.potential c 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | induction' n with n h | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (b - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | case zero
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (b - 1) ^ 0 * Complex.abs z ≤ Complex.abs ((f' d c)^[0] z)
case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | simp only [Nat.zero_eq, pow_zero, one_mul, Function.iterate_zero_apply, le_refl] | case zero
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (b - 1) ^ 0 * Complex.abs z ≤ Complex.abs ((f' d c)^[0] z) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | simp only [Function.iterate_succ_apply'] | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs ((f' d c)^[n + 1] z) | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c ((f' d c)^[n] z)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | generalize hw : (f' d c)^[n] z = w | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c ((f' d c)^[n] z)) | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
w : ℂ
hw : (f' d c)^[n] z = w
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [hw] at h | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
w : ℂ
hw : (f' d c)^[n] z = w
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c w) | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
hw : (f' d c)^[n] z = w
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | clear hw | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
hw : (f' d c)^[n] z = w
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c w) | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | have z1 : 1 ≤ abs z := le_trans (by norm_num) (le_trans b2 bz) | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c w) | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | have b1 : 1 ≤ b - 1 := by linarith | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c w) | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | have b0 : 0 ≤ b - 1 := by linarith | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs (f' d c w) | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | have nd : n + 1 ≤ n * d + 1 := by bound | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
⊢ (b - 1) ^ (n + 1) * Complex.abs z ≤ Complex.abs ... | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ (b - 1) ^ (n + 1) * Compl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | calc abs (w ^ d + c)
_ ≥ abs (w ^ d) - abs c := by bound
_ = abs w ^ d - abs c := by rw [Complex.abs.map_pow]
_ ≥ ((b-1) ^ n * abs z) ^ d - abs c := by bound
_ = (b-1) ^ (n*d) * abs z ^ d - abs c := by rw [mul_pow, pow_mul]
_ ≥ (b-1) ^ (n*d) * abs z ^ 2 - abs c := by bound
_ = (b-1) ^ (n*d) * (abs z * abs z... | case succ
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ (b - 1) ^ (n + 1) * Compl... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
⊢ 1 ≤ 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
⊢ 1 ≤ b - 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
⊢ 0 ≤ b - 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
⊢ n + 1 ≤ n * d + 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ Complex.abs (w ^ d + c) ≥ Complex.a... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [Complex.abs.map_pow] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ Complex.abs (w ^ d) - Complex.abs c... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ Complex.abs w ^ d - Complex.abs c ≥... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [mul_pow, pow_mul] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ ((b - 1) ^ n * Complex.abs z) ^ d -... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ (b - 1) ^ (n * d) * Complex.abs z ^... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [pow_two] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ (b - 1) ^ (n * d) * Complex.abs z ^... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ (b - 1) ^ (n * d) * (Complex.abs z ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | ring | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ (b - 1) ^ (n * d) * (b * Complex.ab... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [pow_succ] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ (b - 1) ^ (n * d) * (b - 1) * Compl... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ (b - 1) ^ (n * d + 1) * Complex.abs... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | rw [one_mul] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ (b - 1) ^ (n + 1) * Complex.abs z +... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_large | [342, 1] | [365, 46] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b2 : 2 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
w : ℂ
h : (b - 1) ^ n * Complex.abs z ≤ Complex.abs w
z1 : 1 ≤ Complex.abs z
b1 : 1 ≤ b - 1
b0 : 0 ≤ b - 1
nd : n + 1 ≤ n * d + 1
⊢ (b - 1) ^ (n + 1) * Complex.abs z +... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_small | [367, 1] | [371, 57] | calc abs (z^d + c)
_ ≤ abs (z^d) + abs c := by bound
_ ≤ abs z ^ d + abs c := by rw [Complex.abs.map_pow] | c✝ : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
c z : ℂ
⊢ Complex.abs (f' d c z) ≤ Complex.abs z ^ d + Complex.abs c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_small | [367, 1] | [371, 57] | bound | c✝ : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
c z : ℂ
⊢ Complex.abs (z ^ d + c) ≤ Complex.abs (z ^ d) + Complex.abs c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | iter_small | [367, 1] | [371, 57] | rw [Complex.abs.map_pow] | c✝ : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
c z : ℂ
⊢ Complex.abs (z ^ d) + Complex.abs c ≤ Complex.abs z ^ d + Complex.abs c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_f'_iter | [378, 1] | [381, 32] | induction' n with n h | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
z : ℂ
⊢ (f d c)^[n] ↑z = ↑((f' d c)^[n] z) | case zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (f d c)^[0] ↑z = ↑((f' d c)^[0] z)
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
n : ℕ
h : (f d c)^[n] ↑z = ↑((f' d c)^[n] z)
⊢ (f d c)^[n + 1] ↑z = ↑((f' d c)^[n + 1] z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_f'_iter | [378, 1] | [381, 32] | simp only [Function.iterate_zero, id] | case zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (f d c)^[0] ↑z = ↑((f' d c)^[0] z)
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
n : ℕ
h : (f d c)^[n] ↑z = ↑((f' d c)^[n] z)
⊢ (f d c)^[n + 1] ↑z = ↑((f' d c)^[n + 1] z) | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
n : ℕ
h : (f d c)^[n] ↑z = ↑((f' d c)^[n] z)
⊢ (f d c)^[n + 1] ↑z = ↑((f' d c)^[n + 1] z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_f'_iter | [378, 1] | [381, 32] | simp only [h, Function.iterate_succ_apply'] | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
n : ℕ
h : (f d c)^[n] ↑z = ↑((f' d c)^[n] z)
⊢ (f d c)^[n + 1] ↑z = ↑((f' d c)^[n + 1] z) | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
n : ℕ
h : (f d c)^[n] ↑z = ↑((f' d c)^[n] z)
⊢ f d c ↑((f' d c)^[n] z) = ↑(f' d c ((f' d c)^[n] z)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | f_f'_iter | [378, 1] | [381, 32] | simp only [f, lift', rec_coe] | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
n : ℕ
h : (f d c)^[n] ↑z = ↑((f' d c)^[n] z)
⊢ f d c ↑((f' d c)^[n] z) = ↑(f' d c ((f' d c)^[n] z)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_coe | [383, 1] | [384, 90] | simp only [multibrot, mem_setOf, f_f'_iter, not_iff_not, tendsto_inf_iff_tendsto_atInf] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ c ∈ multibrot d ↔ ¬Tendsto (fun n => (f' d c)^[n] c) atTop atInf | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | simp only [multibrot_coe, not_le, not_not, (superF d).basin_iff_attracts, Attracts, f_f'_iter,
tendsto_inf_iff_tendsto_atInf, tendsto_atInf_iff_norm_tendsto_atTop,
Complex.norm_eq_abs] at z2 ⊢ | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z2 : 2 < Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (c, ↑z) ∈ ⋯.basin | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z2 : 2 < Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | apply Filter.tendsto_atTop_mono (iter_large d (abs z) z2.le (le_refl _) cz) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z2 : 2 < Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z2 : 2 < Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (Complex.abs z - 1) ^ n * Complex.abs z) atTop atTop |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | refine Filter.Tendsto.atTop_mul (by linarith) ?_ tendsto_const_nhds | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z2 : 2 < Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (Complex.abs z - 1) ^ n * Complex.abs z) atTop atTop | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z2 : 2 < Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (Complex.abs z - 1) ^ n) atTop atTop |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | apply tendsto_pow_atTop_atTop_of_one_lt | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z2 : 2 < Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (Complex.abs z - 1) ^ n) atTop atTop | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z2 : 2 < Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 1 < Complex.abs z - 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | linarith | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z2 : 2 < Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 1 < Complex.abs z - 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_two_lt | [387, 1] | [393, 52] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z2 : 2 < Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < Complex.abs z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_le_two | [396, 1] | [399, 26] | contrapose m | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
m : (c, ↑z) ∉ ⋯.basin
cz : Complex.abs c ≤ Complex.abs z
⊢ Complex.abs z ≤ 2 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
cz : Complex.abs c ≤ Complex.abs z
m : ¬Complex.abs z ≤ 2
⊢ ¬(c, ↑z) ∉ ⋯.basin |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_le_two | [396, 1] | [399, 26] | simp only [not_le, not_not] at m ⊢ | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
cz : Complex.abs c ≤ Complex.abs z
m : ¬Complex.abs z ≤ 2
⊢ ¬(c, ↑z) ∉ ⋯.basin | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
cz : Complex.abs c ≤ Complex.abs z
m : 2 < Complex.abs z
⊢ (c, ↑z) ∈ ⋯.basin |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | julia_le_two | [396, 1] | [399, 26] | exact julia_two_lt m cz | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
cz : Complex.abs c ≤ Complex.abs z
m : 2 < Complex.abs z
⊢ (c, ↑z) ∈ ⋯.basin | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_le_two | [411, 1] | [413, 35] | rw [multibrot_basin' (d := d)] at m | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
m : c ∈ multibrot d
⊢ Complex.abs c ≤ 2 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
m : (c, ↑c) ∉ ⋯.basin
⊢ Complex.abs c ≤ 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_le_two | [411, 1] | [413, 35] | exact julia_le_two m (le_refl _) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
m : (c, ↑c) ∉ ⋯.basin
⊢ Complex.abs c ≤ 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_subset_closedBall | [416, 1] | [417, 93] | intro c m | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ multibrot d ⊆ closedBall 0 2 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
m : c ∈ multibrot d
⊢ c ∈ closedBall 0 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_subset_closedBall | [416, 1] | [417, 93] | simp only [mem_closedBall, Complex.dist_eq, sub_zero] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
m : c ∈ multibrot d
⊢ c ∈ closedBall 0 2 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
m : c ∈ multibrot d
⊢ Complex.abs c ≤ 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_subset_closedBall | [416, 1] | [417, 93] | exact multibrot_le_two m | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
m : c ∈ multibrot d
⊢ Complex.abs c ≤ 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_two_lt | [420, 1] | [421, 77] | contrapose a | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a : 2 < Complex.abs c
⊢ c ∉ multibrot d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a : ¬c ∉ multibrot d
⊢ ¬2 < Complex.abs c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_two_lt | [420, 1] | [421, 77] | simp only [not_lt, not_not] at a ⊢ | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a : ¬c ∉ multibrot d
⊢ ¬2 < Complex.abs c | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a : c ∈ multibrot d
⊢ Complex.abs c ≤ 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_two_lt | [420, 1] | [421, 77] | exact multibrot_le_two a | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a : c ∈ multibrot d
⊢ Complex.abs c ≤ 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | generalize hg : (fun n ↦ (f' d c)^[n] c) = g | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
h : (f d c)^[a] ↑c = (f d c)^[b] ↑c
⊢ c ∈ multibrot d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
h : (f d c)^[a] ↑c = (f d c)^[b] ↑c
g : ℕ → ℂ
hg : (fun n => (f' d c)^[n] c) = g
⊢ c ∈ multibrot d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | replace hg : ∀ n, (f' d c)^[n] c = g n := fun n ↦ by rw [← hg] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
h : (f d c)^[a] ↑c = (f d c)^[b] ↑c
g : ℕ → ℂ
hg : (fun n => (f' d c)^[n] c) = g
⊢ c ∈ multibrot d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
h : (f d c)^[a] ↑c = (f d c)^[b] ↑c
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
⊢ c ∈ multibrot d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | simp only [f_f'_iter, ← coe_zero, coe_eq_coe, hg] at h | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
h : (f d c)^[a] ↑c = (f d c)^[b] ↑c
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
⊢ c ∈ multibrot d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
⊢ c ∈ multibrot d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | simp only [multibrot_coe, atInf_basis.tendsto_right_iff, true_imp_iff, not_forall,
Filter.not_eventually, mem_setOf, not_lt, Complex.norm_eq_abs, hg] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
⊢ c ∈ multibrot d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
⊢ ∃ x, ∃ᶠ (x_1 : ℕ) in atTop, Complex.abs (g x_1) ≤ x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | use partialSups (fun k ↦ Complex.abs (g k)) b | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
⊢ ∃ x, ∃ᶠ (x_1 : ℕ) in atTop, Complex.abs (g x_1) ≤ x | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
⊢ ∃ᶠ (x : ℕ) in atTop, Complex.abs (g x) ≤ (partialSups fun k => Complex.abs (g k)) b |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | apply Filter.frequently_of_forall | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
⊢ ∃ᶠ (x : ℕ) in atTop, Complex.abs (g x) ≤ (partialSups fun k => Complex.abs (g k)) b | case h.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
⊢ ∀ (x : ℕ), Complex.abs (g x) ≤ (partialSups fun k => Complex.abs (g k)) b |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | intro k | case h.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
⊢ ∀ (x : ℕ), Complex.abs (g x) ≤ (partialSups fun k => Complex.abs (g k)) b | case h.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
k : ℕ
⊢ Complex.abs (g k) ≤ (partialSups fun k => Complex.abs (g k)) b |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | rcases lo k with ⟨l, lb, kl⟩ | case h.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
k : ℕ
⊢ Complex.abs (g k) ≤ (partialSups fun k => Complex.abs (g k)) b | case h.h.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
k l : ℕ
lb : l ≤ b
kl : g k = g l
⊢ Complex.abs (g k) ≤ (partialSups fun k => Complex.abs (g k)) b |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | rw [kl] | case h.h.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
k l : ℕ
lb : l ≤ b
kl : g k = g l
⊢ Complex.abs (g k) ≤ (partialSups fun k => Complex.abs (g k)) b | case h.h.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
k l : ℕ
lb : l ≤ b
kl : g k = g l
⊢ Complex.abs (g l) ≤ (partialSups fun k => Complex.abs (g k)) b |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | exact le_partialSups_of_le (fun k ↦ abs (g k)) lb | case h.h.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
lo : ∀ (n : ℕ), ∃ k ≤ b, g n = g k
k l : ℕ
lb : l ≤ b
kl : g k = g l
⊢ Complex.abs (g l) ≤ (partialSups fun k => Complex.abs (g k)) b | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | rw [← hg] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
h : (f d c)^[a] ↑c = (f d c)^[b] ↑c
g : ℕ → ℂ
hg : (fun n => (f' d c)^[n] c) = g
n : ℕ
⊢ (f' d c)^[n] c = g n | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | intro n | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
⊢ ∀ (n : ℕ), ∃ k ≤ b, g n = g k | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n : ℕ
⊢ ∃ k ≤ b, g n = g k |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | induction' n with n h | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n : ℕ
⊢ ∃ k ≤ b, g n = g k | case zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
⊢ ∃ k ≤ b, g 0 = g k
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h✝ : g a = g b
n : ℕ
h : ∃ k ≤ b, g n = g k
⊢ ∃ k ≤ b, g (n + 1) =... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | use 0, Nat.zero_le _ | case zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
⊢ ∃ k ≤ b, g 0 = g k | no goals |
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