Datasets:
internlm
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Lean-Github

Dataset card Data Studio Files Community
1
url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
full_name
stringlengths
1
94
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stringlengths
6
2.09M
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
iter_error_le_log_log_abs
[245, 1]
[259, 83]
positivity
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0 ≤ 1.38
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
iter_error_le_log_log_abs
[245, 1]
[259, 83]
positivity
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0 ≤ Complex.abs z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
set s := superF d
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ |⋯.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have z3 : 3 ≤ abs z := le_trans (by norm_num) z4
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have z0 : 0 < abs z := lt_of_lt_of_le (by norm_num) z3
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have l2 : 0 < log (abs z) := Real.log_pos (by linarith)
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have h := log_neg_log_potential_approx d z3 cz
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have p0 : 0 < s.potential c z := potential_pos
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z p0 : 0 < s.potential c ↑z ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have lp0 : 0 < -log (s.potential c z) := neg_pos.mpr (Real.log_neg p0 (potential_lt_one_of_two_lt (by linarith) cz))
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z p0 : 0 < s.potential c ↑z ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z p0 : 0 < s.potential c ↑z lp0 : 0 < -(s.potential c ↑z).log ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
generalize s.potential c z = p at h p0 lp0
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z p0 : 0 < s.potential c ↑z lp0 : 0 < -(s.potential c ↑z).log ⊢ |s.potential c ↑z - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ h : |(-p.log).log - (Complex.abs z).log.log| ≤ iter_error d c z p0 : 0 < p lp0 : 0 < -p.log ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
generalize hr : iter_error d c z = r at h
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ h : |(-p.log).log - (Complex.abs z).log.log| ≤ iter_error d c z p0 : 0 < p lp0 : 0 < -p.log ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have r0 : 0 ≤ r := le_trans (abs_nonneg _) h
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
set t := Ici (log (log (abs z)) - r)
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have yt : log (-log p) ∈ t := by simp only [abs_le, neg_le_sub_iff_le_add, tsub_le_iff_right, add_comm r] at h simp only [mem_Ici, tsub_le_iff_right, h, t]
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have lt : log (log (abs z)) ∈ t := by simp only [mem_Ici, tsub_le_iff_right, le_add_iff_nonneg_right, r0, t]
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
generalize hb : dene (log (log (abs z)) - r) = b
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have b0 : 0 ≤ b := by rw [←hb]; exact dene_nonneg
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have bound : ∀ x, x ∈ t → ‖deriv ene x‖ ≤ b := by intro x m simp only [Real.dist_eq, mem_Ici, ←hr, t] at m simp only [deriv_ene, norm_neg, Real.norm_of_nonneg dene_nonneg, ←hb, ←hr] apply dene_anti (sub_nonneg.mpr (iter_error_le_log_log_abs d z4 cz)) m
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
have m := Convex.norm_image_sub_le_of_norm_deriv_le (fun x _ ↦ (hasDerivAt_ene x).differentiableAt) bound (convex_Ici _) lt yt
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : ‖ene (-p.log).log - ene (Complex.abs z).log.log‖ ≤ b * ‖(-p.log).log - (Complex.abs z).log.log‖ ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
simp only [Real.norm_eq_abs] at m
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : ‖ene (-p.log).log - ene (Complex.abs z).log.log‖ ≤ b * ‖(-p.log).log - (Complex.abs z).log.log‖ ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log| ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
replace m := le_trans m (mul_le_mul_of_nonneg_left h (by bound))
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log| ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * r ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
simp only [ene, Real.exp_log lp0, neg_neg, Real.exp_log p0, Real.exp_log l2, Real.exp_neg, Real.exp_log z0, inv_eq_one_div] at m
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * r ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : |p - 1 / Complex.abs z| ≤ b * r ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
refine le_trans m (le_of_eq ?_)
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : |p - 1 / Complex.abs z| ≤ b * r ⊢ |p - 1 / Complex.abs z| ≤ potential_error d c z
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : |p - 1 / Complex.abs z| ≤ b * r ⊢ b * r = potential_error d c z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
simp only [←hr, ←hb, potential_error]
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : |p - 1 / Complex.abs z| ≤ b * r ⊢ b * r = potential_error d c z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
norm_num
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d ⊢ 3 ≤ 4
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
norm_num
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z ⊢ 0 < 3
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
linarith
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z ⊢ 1 < Complex.abs z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
linarith
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log h : |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z p0 : 0 < s.potential c ↑z ⊢ 2 < Complex.abs z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
simp only [abs_le, neg_le_sub_iff_le_add, tsub_le_iff_right, add_comm r] at h
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) ⊢ (-p.log).log ∈ t
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) h : (Complex.abs z).log.log ≤ (-p.log).log + r ∧ (-p.log).log ≤ (Complex.abs z).log.log + r ⊢ (-p.log).log ∈ t
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
simp only [mem_Ici, tsub_le_iff_right, h, t]
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) h : (Complex.abs z).log.log ≤ (-p.log).log + r ∧ (-p.log).log ≤ (Complex.abs z).log.log + r ⊢ (-p.log).log ∈ t
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
simp only [mem_Ici, tsub_le_iff_right, le_add_iff_nonneg_right, r0, t]
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t ⊢ (Complex.abs z).log.log ∈ t
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
rw [←hb]
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b ⊢ 0 ≤ b
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b ⊢ 0 ≤ dene ((Complex.abs z).log.log - r)
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
exact dene_nonneg
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b ⊢ 0 ≤ dene ((Complex.abs z).log.log - r)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
intro x m
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b ⊢ ∀ x ∈ t, ‖deriv ene x‖ ≤ b
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b x : ℝ m : x ∈ t ⊢ ‖deriv ene x‖ ≤ b
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
simp only [Real.dist_eq, mem_Ici, ←hr, t] at m
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b x : ℝ m : x ∈ t ⊢ ‖deriv ene x‖ ≤ b
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b x : ℝ m : (Complex.abs z).log.log - iter_error d c z ≤ x ⊢ ‖deriv ene x‖ ≤ b
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
simp only [deriv_ene, norm_neg, Real.norm_of_nonneg dene_nonneg, ←hb, ←hr]
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b x : ℝ m : (Complex.abs z).log.log - iter_error d c z ≤ x ⊢ ‖deriv ene x‖ ≤ b
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b x : ℝ m : (Complex.abs z).log.log - iter_error d c z ≤ x ⊢ dene x ≤ dene ((Complex.abs z).log.log - iter_error d c z)
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
apply dene_anti (sub_nonneg.mpr (iter_error_le_log_log_abs d z4 cz)) m
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b x : ℝ m : (Complex.abs z).log.log - iter_error d c z ≤ x ⊢ dene x ≤ dene ((Complex.abs z).log.log - iter_error d c z)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Potential.lean
potential_approx
[262, 1]
[295, 40]
bound
c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z s : Super (f d) d OnePoint.infty := superF d z3 : 3 ≤ Complex.abs z z0 : 0 < Complex.abs z l2 : 0 < (Complex.abs z).log p : ℝ p0 : 0 < p lp0 : 0 < -p.log r : ℝ hr : iter_error d c z = r h : |(-p.log).log - (Complex.abs z).log.log| ≤ r r0 : 0 ≤ r t : Set ℝ := Ici ((Complex.abs z).log.log - r) yt : (-p.log).log ∈ t lt : (Complex.abs z).log.log ∈ t b : ℝ hb : dene ((Complex.abs z).log.log - r) = b b0 : 0 ≤ b bound : ∀ x ∈ t, ‖deriv ene x‖ ≤ b m : |ene (-p.log).log - ene (Complex.abs z).log.log| ≤ b * |(-p.log).log - (Complex.abs z).log.log| ⊢ 0 ≤ b
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_iff_disjoint_range
[27, 1]
[30, 83]
simp only [Late, ge_iff_le, Finset.disjoint_iff_ne, Finset.mem_range, ne_eq]
m : ℕ A : Finset ℕ ⊢ Late A m ↔ Disjoint A (Finset.range m)
m : ℕ A : Finset ℕ ⊢ (∀ n ∈ A, m ≤ n) ↔ ∀ a ∈ A, ∀ b < m, ¬a = b
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_iff_disjoint_range
[27, 1]
[30, 83]
constructor
m : ℕ A : Finset ℕ ⊢ (∀ n ∈ A, m ≤ n) ↔ ∀ a ∈ A, ∀ b < m, ¬a = b
case mp m : ℕ A : Finset ℕ ⊢ (∀ n ∈ A, m ≤ n) → ∀ a ∈ A, ∀ b < m, ¬a = b case mpr m : ℕ A : Finset ℕ ⊢ (∀ a ∈ A, ∀ b < m, ¬a = b) → ∀ n ∈ A, m ≤ n
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_iff_disjoint_range
[27, 1]
[30, 83]
intro h n na b bm
case mp m : ℕ A : Finset ℕ ⊢ (∀ n ∈ A, m ≤ n) → ∀ a ∈ A, ∀ b < m, ¬a = b
case mp m : ℕ A : Finset ℕ h : ∀ n ∈ A, m ≤ n n : ℕ na : n ∈ A b : ℕ bm : b < m ⊢ ¬n = b
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_iff_disjoint_range
[27, 1]
[30, 83]
linarith [h _ na]
case mp m : ℕ A : Finset ℕ h : ∀ n ∈ A, m ≤ n n : ℕ na : n ∈ A b : ℕ bm : b < m ⊢ ¬n = b
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_iff_disjoint_range
[27, 1]
[30, 83]
intro h n na
case mpr m : ℕ A : Finset ℕ ⊢ (∀ a ∈ A, ∀ b < m, ¬a = b) → ∀ n ∈ A, m ≤ n
case mpr m : ℕ A : Finset ℕ h : ∀ a ∈ A, ∀ b < m, ¬a = b n : ℕ na : n ∈ A ⊢ m ≤ n
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_iff_disjoint_range
[27, 1]
[30, 83]
specialize h n na n
case mpr m : ℕ A : Finset ℕ h : ∀ a ∈ A, ∀ b < m, ¬a = b n : ℕ na : n ∈ A ⊢ m ≤ n
case mpr m : ℕ A : Finset ℕ n : ℕ na : n ∈ A h : n < m → ¬n = n ⊢ m ≤ n
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_iff_disjoint_range
[27, 1]
[30, 83]
simpa [not_true, imp_false, not_lt] using h
case mpr m : ℕ A : Finset ℕ n : ℕ na : n ∈ A h : n < m → ¬n = n ⊢ m ≤ n
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sdiff_late
[32, 1]
[38, 17]
intro Bm n nAB
m : ℕ B A : Finset ℕ ⊢ B ≥ Finset.range m → Late (A \ B) m
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A \ B ⊢ n ≥ m
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sdiff_late
[32, 1]
[38, 17]
rw [Finset.mem_sdiff] at nAB
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A \ B ⊢ n ≥ m
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A ∧ n ∉ B ⊢ n ≥ m
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sdiff_late
[32, 1]
[38, 17]
by_contra h
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A ∧ n ∉ B ⊢ n ≥ m
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A ∧ n ∉ B h : ¬n ≥ m ⊢ False
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sdiff_late
[32, 1]
[38, 17]
simp only [not_le] at h
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A ∧ n ∉ B h : ¬n ≥ m ⊢ False
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A ∧ n ∉ B h : n < m ⊢ False
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sdiff_late
[32, 1]
[38, 17]
have nr := Finset.mem_range.mpr h
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A ∧ n ∉ B h : n < m ⊢ False
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A ∧ n ∉ B h : n < m nr : n ∈ Finset.range m ⊢ False
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sdiff_late
[32, 1]
[38, 17]
have nB := Finset.mem_of_subset Bm nr
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A ∧ n ∉ B h : n < m nr : n ∈ Finset.range m ⊢ False
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A ∧ n ∉ B h : n < m nr : n ∈ Finset.range m nB : n ∈ B ⊢ False
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
sdiff_late
[32, 1]
[38, 17]
exact nAB.2 nB
m : ℕ B A : Finset ℕ Bm : B ≥ Finset.range m n : ℕ nAB : n ∈ A ∧ n ∉ B h : n < m nr : n ∈ Finset.range m nB : n ∈ B ⊢ False
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
partial_geometric_bound
[41, 1]
[44, 57]
intro n _
a : ℝ N : Finset ℕ a0 : 0 ≤ a a1 : a < 1 ⊢ ∀ n ∉ N, 0 ≤ a ^ n
a : ℝ N : Finset ℕ a0 : 0 ≤ a a1 : a < 1 n : ℕ a✝ : n ∉ N ⊢ 0 ≤ a ^ n
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
partial_geometric_bound
[41, 1]
[44, 57]
bound
a : ℝ N : Finset ℕ a0 : 0 ≤ a a1 : a < 1 n : ℕ a✝ : n ∉ N ⊢ 0 ≤ a ^ n
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
partial_scaled_geometric_bound
[46, 1]
[49, 42]
rw [←Finset.mul_sum]
a : ℝ c : ℝ≥0 N : Finset ℕ a0 : 0 ≤ a a1 : a < 1 ⊢ (N.sum fun n => ↑c * a ^ n) ≤ ↑c * (1 - a)⁻¹
a : ℝ c : ℝ≥0 N : Finset ℕ a0 : 0 ≤ a a1 : a < 1 ⊢ (↑c * N.sum fun i => a ^ i) ≤ ↑c * (1 - a)⁻¹
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
partial_scaled_geometric_bound
[46, 1]
[49, 42]
bound [partial_geometric_bound N a0 a1]
a : ℝ c : ℝ≥0 N : Finset ℕ a0 : 0 ≤ a a1 : a < 1 ⊢ (↑c * N.sum fun i => a ^ i) ≤ ↑c * (1 - a)⁻¹
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
set Ns := Finset.image (fun n ↦ n - m) N
m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ ⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m)
m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N ⊢ N.sum f = Ns.sum fun n => f (n + m)
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
rw [NNs]
m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N NNs : N = Finset.image (fun n => n + m) Ns ⊢ N.sum f = Ns.sum fun n => f (n + m)
m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N NNs : N = Finset.image (fun n => n + m) Ns ⊢ (Finset.image (fun n => n + m) Ns).sum f = Ns.sum fun n => f (n + m)
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
apply Finset.sum_image
m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N NNs : N = Finset.image (fun n => n + m) Ns ⊢ (Finset.image (fun n => n + m) Ns).sum f = Ns.sum fun n => f (n + m)
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N NNs : N = Finset.image (fun n => n + m) Ns ⊢ ∀ x ∈ Ns, ∀ y ∈ Ns, x + m = y + m → x = y
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
intro a _ b _
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N NNs : N = Finset.image (fun n => n + m) Ns ⊢ ∀ x ∈ Ns, ∀ y ∈ Ns, x + m = y + m → x = y
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N NNs : N = Finset.image (fun n => n + m) Ns a : ℕ a✝¹ : a ∈ Ns b : ℕ a✝ : b ∈ Ns ⊢ a + m = b + m → a = b
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
exact Nat.add_right_cancel
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N NNs : N = Finset.image (fun n => n + m) Ns a : ℕ a✝¹ : a ∈ Ns b : ℕ a✝ : b ∈ Ns ⊢ a + m = b + m → a = b
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
apply Finset.ext
m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N ⊢ N = Finset.image (fun n => n + m) Ns
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N ⊢ ∀ (a : ℕ), a ∈ N ↔ a ∈ Finset.image (fun n => n + m) Ns
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
intro k
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N ⊢ ∀ (a : ℕ), a ∈ N ↔ a ∈ Finset.image (fun n => n + m) Ns
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ⊢ k ∈ N ↔ k ∈ Finset.image (fun n => n + m) Ns
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
rw [Finset.image_image, Finset.mem_image]
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ⊢ k ∈ N ↔ k ∈ Finset.image (fun n => n + m) Ns
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ⊢ k ∈ N ↔ ∃ a ∈ N, ((fun n => n + m) ∘ fun n => n - m) a = k
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
simp
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ⊢ k ∈ N ↔ ∃ a ∈ N, ((fun n => n + m) ∘ fun n => n - m) a = k
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ⊢ k ∈ N ↔ ∃ a ∈ N, a - m + m = k
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
apply Iff.intro
case a m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ⊢ k ∈ N ↔ ∃ a ∈ N, a - m + m = k
case a.mp m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ⊢ k ∈ N → ∃ a ∈ N, a - m + m = k case a.mpr m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ⊢ (∃ a ∈ N, a - m + m = k) → k ∈ N
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
intro kN
case a.mp m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ⊢ k ∈ N → ∃ a ∈ N, a - m + m = k
case a.mp m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ kN : k ∈ N ⊢ ∃ a ∈ N, a - m + m = k
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
exists k
case a.mp m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ kN : k ∈ N ⊢ ∃ a ∈ N, a - m + m = k
case a.mp m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ kN : k ∈ N ⊢ k ∈ N ∧ k - m + m = k
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
apply And.intro
case a.mp m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ kN : k ∈ N ⊢ k ∈ N ∧ k - m + m = k
case a.mp.left m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ kN : k ∈ N ⊢ k ∈ N case a.mp.right m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ kN : k ∈ N ⊢ k - m + m = k
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
assumption
case a.mp.left m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ kN : k ∈ N ⊢ k ∈ N case a.mp.right m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ kN : k ∈ N ⊢ k - m + m = k
case a.mp.right m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ kN : k ∈ N ⊢ k - m + m = k
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
exact Nat.sub_add_cancel (h k kN)
case a.mp.right m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ kN : k ∈ N ⊢ k - m + m = k
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
intro ha
case a.mpr m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ⊢ (∃ a ∈ N, a - m + m = k) → k ∈ N
case a.mpr m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ha : ∃ a ∈ N, a - m + m = k ⊢ k ∈ N
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
rcases ha with ⟨a, aN, ak⟩
case a.mpr m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k : ℕ ha : ∃ a ∈ N, a - m + m = k ⊢ k ∈ N
case a.mpr.intro.intro m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k a : ℕ aN : a ∈ N ak : a - m + m = k ⊢ k ∈ N
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
rw [Nat.sub_add_cancel (h a aN)] at ak
case a.mpr.intro.intro m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k a : ℕ aN : a ∈ N ak : a - m + m = k ⊢ k ∈ N
case a.mpr.intro.intro m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k a : ℕ aN : a ∈ N ak : a = k ⊢ k ∈ N
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
rw [← ak]
case a.mpr.intro.intro m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k a : ℕ aN : a ∈ N ak : a = k ⊢ k ∈ N
case a.mpr.intro.intro m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k a : ℕ aN : a ∈ N ak : a = k ⊢ a ∈ N
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum
[52, 1]
[66, 44]
assumption
case a.mpr.intro.intro m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ Ns : Finset ℕ := Finset.image (fun n => n - m) N k a : ℕ aN : a ∈ N ak : a = k ⊢ a ∈ N
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum'
[69, 1]
[72, 28]
exists Finset.image (fun n ↦ n - m) N
m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ ⊢ ∃ M, N.sum f = M.sum fun n => f (n + m)
m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ ⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m)
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_series_sum'
[69, 1]
[72, 28]
exact late_series_sum h f
m : ℕ N : Finset ℕ h : Late N m f : ℕ → ℝ ⊢ N.sum f = (Finset.image (fun n => n - m) N).sum fun n => f (n + m)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
rcases late_series_sum' h (fun n ↦ a^n) with ⟨M,L⟩
m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 ⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹
case intro m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m) ⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
rw [L]
case intro m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m) ⊢ (N.sum fun n => a ^ n) ≤ a ^ m * (1 - a)⁻¹
case intro m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m) ⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
clear L
case intro m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ L : (N.sum fun n => a ^ n) = M.sum fun n => a ^ (n + m) ⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
case intro m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ ⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
have pa : (fun n ↦ a^(n + m)) = (fun n ↦ a^n * a^m) := by apply funext; intro n; rw [pow_add]
case intro m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ ⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
case intro m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m ⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
calc M.sum (fun n ↦ a^(n + m)) = M.sum (fun n ↦ a^n * a^m) := by rw [ pa ] _ = M.sum (fun n ↦ a^n) * a^m := (Finset.sum_mul _ _ _).symm _ ≤ (1 - a)⁻¹ * a^m := by bound [partial_geometric_bound M a0 a1] _ = a^m * (1 - a)⁻¹ := by ring
case intro m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m ⊢ (M.sum fun n => a ^ (n + m)) ≤ a ^ m * (1 - a)⁻¹
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
apply funext
m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ ⊢ (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m
case h m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ ⊢ ∀ (x : ℕ), a ^ (x + m) = a ^ x * a ^ m
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
intro n
case h m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ ⊢ ∀ (x : ℕ), a ^ (x + m) = a ^ x * a ^ m
case h m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ n : ℕ ⊢ a ^ (n + m) = a ^ n * a ^ m
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
rw [pow_add]
case h m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ n : ℕ ⊢ a ^ (n + m) = a ^ n * a ^ m
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
rw [ pa ]
m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m ⊢ (M.sum fun n => a ^ (n + m)) = M.sum fun n => a ^ n * a ^ m
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
bound [partial_geometric_bound M a0 a1]
m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m ⊢ (M.sum fun n => a ^ n) * a ^ m ≤ (1 - a)⁻¹ * a ^ m
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
late_geometric_bound
[74, 1]
[83, 35]
ring
m : ℕ a : ℝ N : Finset ℕ h : Late N m a0 : 0 ≤ a a1 : a < 1 M : Finset ℕ pa : (fun n => a ^ (n + m)) = fun n => a ^ n * a ^ m ⊢ (1 - a)⁻¹ * a ^ m = a ^ m * (1 - a)⁻¹
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
apply Finset.ext
A B : Finset ℕ ⊢ A = A \ B ∪ A ∩ B
case a A B : Finset ℕ ⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A \ B ∪ A ∩ B
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
simp only [Finset.mem_union, Finset.mem_sdiff, Finset.mem_inter]
case a A B : Finset ℕ ⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A \ B ∪ A ∩ B
case a A B : Finset ℕ ⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A ∧ a ∉ B ∨ a ∈ A ∧ a ∈ B
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
intro x
case a A B : Finset ℕ ⊢ ∀ (a : ℕ), a ∈ A ↔ a ∈ A ∧ a ∉ B ∨ a ∈ A ∧ a ∈ B
case a A B : Finset ℕ x : ℕ ⊢ x ∈ A ↔ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
constructor
case a A B : Finset ℕ x : ℕ ⊢ x ∈ A ↔ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
case a.mp A B : Finset ℕ x : ℕ ⊢ x ∈ A → x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B case a.mpr A B : Finset ℕ x : ℕ ⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B → x ∈ A
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
intro a
case a.mp A B : Finset ℕ x : ℕ ⊢ x ∈ A → x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
case a.mp A B : Finset ℕ x : ℕ a : x ∈ A ⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
by_cases b : x ∈ B
case a.mp A B : Finset ℕ x : ℕ a : x ∈ A ⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
case pos A B : Finset ℕ x : ℕ a : x ∈ A b : x ∈ B ⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B case neg A B : Finset ℕ x : ℕ a : x ∈ A b : x ∉ B ⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
right
case pos A B : Finset ℕ x : ℕ a : x ∈ A b : x ∈ B ⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
case pos.h A B : Finset ℕ x : ℕ a : x ∈ A b : x ∈ B ⊢ x ∈ A ∧ x ∈ B
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
use a,b
case pos.h A B : Finset ℕ x : ℕ a : x ∈ A b : x ∈ B ⊢ x ∈ A ∧ x ∈ B
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
left
case neg A B : Finset ℕ x : ℕ a : x ∈ A b : x ∉ B ⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B
case neg.h A B : Finset ℕ x : ℕ a : x ∈ A b : x ∉ B ⊢ x ∈ A ∧ x ∉ B
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
use a,b
case neg.h A B : Finset ℕ x : ℕ a : x ∈ A b : x ∉ B ⊢ x ∈ A ∧ x ∉ B
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
finset_partition
[85, 1]
[95, 21]
intro h
case a.mpr A B : Finset ℕ x : ℕ ⊢ x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B → x ∈ A
case a.mpr A B : Finset ℕ x : ℕ h : x ∈ A ∧ x ∉ B ∨ x ∈ A ∧ x ∈ B ⊢ x ∈ A