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/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | norm_num | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ 4 ≤ 6 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | intro n | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ (0.8095 * (z.normSq.log * -0.9635).exp).hi ≠ nan →
potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : (0.8095 * (z.normSq.log * -0.9635).exp).hi ≠ nan
⊢ potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [Ne, Interval.hi_eq_nan] at n | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : (0.8095 * (z.normSq.log * -0.9635).exp).hi ≠ nan
⊢ potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | refine le_trans (potential_error_le_of_z6 _ z6 cz) ?_ | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 / Complex.abs z' ^ 1.927 ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | apply Interval.le_hi n | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 / Complex.abs z' ^ 1.927 ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 / Complex.abs z' ^ 1.927 ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [div_eq_mul_inv, ←Real.rpow_neg (Complex.abs.nonneg _), Real.rpow_def_of_pos (by linarith)] | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 / Complex.abs z' ^ 1.927 ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 * ((Complex.abs z').log * -1.927).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | have e : Real.log (Complex.abs z') * -1.927 = Real.log (Complex.abs z' ^ 2) * -0.9635 := by
rw [Real.log_pow, Nat.cast_two, mul_comm (2:ℝ), mul_assoc]; norm_num | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 * ((Complex.abs z').log * -1.927).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
⊢ 0.8095 * ((Complex.abs z').log * -1.927).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [e] | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
⊢ 0.8095 * ((Complex.abs z').log * -1.927).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp... | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
⊢ 0.8095 * ((Complex.abs z' ^ 2).log * -0.9635).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | mono | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
⊢ 0.8095 * ((Complex.abs z' ^ 2).log * -0.9635).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | linarith | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0 < Complex.abs z' | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [Real.log_pow, Nat.cast_two, mul_comm (2:ℝ), mul_assoc] | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635 | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ (Complex.abs z').log * -1.927 = (Complex.abs z').log * (2 * -0.9635) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | norm_num | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ (Complex.abs z').log * -1.927 = (Complex.abs z').log * (2 * -0.9635) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | have e : 1 / Complex.abs z' = Real.exp (-(Real.log (Complex.abs z' ^ 2) / 2)) := by
simp only [one_div, Real.log_pow, Nat.cast_ofNat, neg_mul, Real.rpow_neg zero_le_two,
Real.rpow_one, ←mul_assoc, mul_comm _ (2:ℝ)⁻¹]
rw [mul_div_cancel_left₀ _ two_ne_zero, Real.exp_neg, Real.exp_log (by linarith)] | case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ 1 / Complex.abs z' ∈ approx (-z.normSq.log.div2).exp | case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
⊢ 1 / Complex.abs z' ∈ approx (-z.normSq.log.div2).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [e] | case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
⊢ 1 / Complex.abs z' ∈ approx (-z.normSq.log.div2).exp | case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
⊢ (-((Complex.abs z' ^ 2).log / 2)).exp ∈ approx (-z.normSq.log.div2).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | mono | case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
⊢ (-((Complex.abs z' ^ 2).log / 2)).exp ∈ approx (-z.normSq.log.div2).exp | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | simp only [one_div, Real.log_pow, Nat.cast_ofNat, neg_mul, Real.rpow_neg zero_le_two,
Real.rpow_one, ←mul_assoc, mul_comm _ (2:ℝ)⁻¹] | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ (Complex.abs z')⁻¹ = (-(2 * (Complex.abs z').log / 2)).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [mul_div_cancel_left₀ _ two_ne_zero, Real.exp_neg, Real.exp_log (by linarith)] | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ (Complex.abs z')⁻¹ = (-(2 * (Complex.abs z').log / 2)).exp | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | linarith | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ 0 < Complex.abs z' | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | set s := superF 2 | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
⊢ ⋯.potential c' ↑z' ∈ approx (c.potential z n r).1 | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
⊢ s.potential c' ↑z' ∈ approx (c.potential z n r).1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [Box.potential] | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
⊢ s.potential c' ↑z' ∈ approx (c.potential z n r).1 | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
⊢ s.potential c' ↑z' ∈
approx
(let cs := c.normSq.hi;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.lar... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hcs : (normSq c).hi = cs | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
⊢ s.potential c' ↑z' ∈
approx
(let cs := c.normSq.hi;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.lar... | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
⊢ s.potential c' ↑z' ∈
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, Potential... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hi : iterate c z (cs.max 9) n = i | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
⊢ s.potential c' ↑z' ∈
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, Potential... | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
⊢ s.potential c' ↑z' ∈
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | by_cases csn : cs = nan | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
⊢ s.potential c' ↑z' ∈
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit... | case pos
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : cs = nan
⊢ s.potential c' ↑z' ∈
approx
(let cs := cs;
let i := iterate c z (cs.max 9)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [hi, Interval.hi_eq_nan, Floating.val_lt_val] | case neg
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
⊢ s.potential c' ↑z' ∈
approx
(let cs := cs;
let i := iterate c z (cs.max 9... | case neg
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
⊢ s.potential c' ↑z' ∈
approx
(match i.exit with
| Exit.nan => (nan, Potent... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hie : i.exit = ie | case neg
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
⊢ s.potential c' ↑z' ∈
approx
(match i.exit with
| Exit.nan => (nan, Potent... | case neg
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
ie : Exit
hie : i.exit = ie
⊢ s.potential c' ↑z' ∈
approx
(match ie with
| ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | induction ie | case neg
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
ie : Exit
hie : i.exit = ie
⊢ s.potential c' ↑z' ∈
approx
(match ie with
| ... | case neg.count
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
⊢ s.potential c' ↑z' ∈
approx
(match Exit.count wit... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [csn, Floating.nan_max, iterate_nan, Interval.approx_nan, mem_univ] | case pos
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : cs = nan
⊢ s.potential c' ↑z' ∈
approx
(let cs := cs;
let i := iterate c z (cs.max 9)... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hzs : (normSq i.z) = zs | case neg.count
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
⊢ s.potential c' ↑z' ∈
approx
(match Exit.count wit... | case neg.count
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
⊢ s.potential c' ↑z' ∈
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | by_cases bad : zs = nan ∨ (16 : Floating).val < zs.hi.val ∨ (16 : Floating).val < cs.val | case neg.count
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
⊢ s.potential c' ↑z' ∈
... | case pos
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : zs = nan ∨ 16.val < zs.hi... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [Floating.val_lt_val, bad, ↓reduceIte, Interval.approx_nan, mem_univ] | case pos
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : zs = nan ∨ 16.val < zs.hi... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [bad, ↓reduceIte] | case neg
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ¬(zs = nan ∨ 16.val < zs.... | case neg
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ¬(zs = nan ∨ 16.val < zs.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [not_or, not_lt, ←hzs] at bad | case neg
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ¬(zs = nan ∨ 16.val < zs.... | case neg
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ¬i.z.normSq = nan ∧ i.z.n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rcases bad with ⟨zsn, z4, c4⟩ | case neg
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ¬i.z.normSq = nan ∧ i.z.n... | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq =... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [Floating.val_ofNat] at c4 z4 | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq =... | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq =... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [← hcs, Nat.cast_ofNat, Interval.hi_eq_nan] at c4 z4 csn zsn | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq =... | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.no... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | apply Interval.mem_approx_iter_sqrt' s.potential_nonneg | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.no... | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.no... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [←s.potential_eqn_iter, f_f'_iter] | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.no... | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.no... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hw' : (f' 2 c')^[i.n] z' = w' | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.no... | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.no... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have le4 : Real.sqrt 16 ≤ 4 := by rw [Real.sqrt_le_iff]; norm_num | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.no... | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.no... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | apply approx_potential_small | case neg.intro.intro
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.no... | case neg.intro.intro.c4
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [Real.sqrt_le_iff] | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.normSq.hi.val ≤ 16
c4 :... | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.normSq.hi.val ≤ 16
c4 :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | norm_num | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z.normSq.hi.val ≤ 16
c4 :... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | exact le_trans (Box.abs_le_sqrt_normSq cm csn) (le_trans (Real.sqrt_le_sqrt c4) le4) | case neg.intro.intro.c4
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | refine le_trans (Box.abs_le_sqrt_normSq ?_ zsn) (le_trans (Real.sqrt_le_sqrt z4) le4) | case neg.intro.intro.z4
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z... | case neg.intro.intro.z4
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [←hw', ←hi] | case neg.intro.intro.z4
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z... | case neg.intro.intro.z4
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | exact mem_approx_iterate cm zm _ | case neg.intro.intro.z4
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ¬i.z.normSq = nan
z4 : i.z... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j | case neg.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
⊢ s.potential c' ↑z' ∈
approx
(match Exit.large wit... | case neg.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [hj] | case neg.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) ... | case neg.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hje : j.exit = je | case neg.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) ... | case neg.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | induction je | case neg.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) ... | case neg.large.count
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [Interval.approx_nan, mem_univ] | case neg.large.count
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hn : i.n + j.n = n | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | apply Interval.mem_approx_iter_sqrt' s.potential_nonneg | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [←s.potential_eqn_iter, f_f'_iter, ←hj] at hje ⊢ | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | generalize hw' : (f' 2 c')^[n] z' = w' | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have izm : (f' 2 c')^[i.n] z' ∈ approx i.z := by rw [←hi]; exact mem_approx_iterate cm zm _ | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have jl := iterate_large cm izm hje | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have jrn := ne_nan_of_iterate (hje.trans_ne (by decide)) | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [hj, ← Function.iterate_add_apply, add_comm _ i.n, hn, hw'] at jl | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [ne_eq, Floating.max_eq_nan, not_or] at jrn | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [Floating.val_max jrn.1 (Floating.max_ne_nan.mpr jrn.2),
Floating.val_max jrn.2.1 jrn.2.2, max_lt_iff, max_lt_iff, Floating.val_ofNat,
Nat.cast_eq_ofNat] at jl | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | apply approx_potential_large | case neg.large.large
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.m... | case neg.large.large.cz
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [←hi] | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : ... | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | exact mem_approx_iterate cm zm _ | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | decide | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | refine le_trans ?_ (le_trans (Real.sqrt_le_sqrt jl.2.1.le) ?_) | case neg.large.large.cz
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (c... | case neg.large.large.cz.refine_1
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r tru... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [← hcs, Interval.hi_eq_nan] at csn ⊢ | case neg.large.large.cz.refine_1
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r tru... | case neg.large.large.cz.refine_1
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 3... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | exact abs_le_sqrt_normSq cm csn | case neg.large.large.cz.refine_1
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 3... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [apply_nonneg, Real.sqrt_sq, le_refl] | case neg.large.large.cz.refine_2
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r tru... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | refine le_trans ?_ (le_trans (Real.sqrt_le_sqrt jl.2.2.le) ?_) | case neg.large.large.z6
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (c... | case neg.large.large.z6.refine_1
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r tru... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have e : (36 : ℝ) = 6 ^ 2 := by norm_num | case neg.large.large.z6.refine_1
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r tru... | case neg.large.large.z6.refine_1
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r tru... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [e, Real.sqrt_sq (by norm_num)] | case neg.large.large.z6.refine_1
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r tru... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | norm_num | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | norm_num | c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [apply_nonneg, Real.sqrt_sq, le_refl] | case neg.large.large.z6.refine_2
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r tru... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [←hw', ←hn, add_comm _ j.n, Function.iterate_add_apply, ←hj] | case neg.large.large.zm
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (c... | case neg.large.large.zm
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | exact mem_approx_iterate cm izm _ | case neg.large.large.zm
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n✝ : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n✝ = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (c... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [Interval.approx_nan, mem_univ] | case neg.large.nan
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 3... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [Interval.approx_nan, mem_univ] | case neg.nan
c' z' : ℂ
c z : Box
cm : c' ∈ approx c
zm : z' ∈ approx z
n : ℕ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
i : Iter
hi : iterate c z (cs.max 9) n = i
csn : ¬cs = nan
hie : i.exit = Exit.nan
⊢ s.potential c' ↑z' ∈
approx
(match Exit.nan with
... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential' | [214, 1] | [217, 83] | simp only [_root_.potential, RiemannSphere.fill_coe, mem_approx_potential cm cm] | c' : ℂ
c : Box
cm : c' ∈ approx c
n : ℕ
r : Floating
⊢ _root_.potential 2 ↑c' ∈ approx (c.potential c n r).1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_multibrotExt | [30, 1] | [33, 40] | rw [← ray_surj d] | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPathConnected (multibrotExt d) | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPathConnected (ray d '' ball 0 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_multibrotExt | [30, 1] | [33, 40] | apply IsPathConnected.image_of_continuousOn | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPathConnected (ray d '' ball 0 1) | case sc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPathConnected (ball 0 1)
case fc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ContinuousOn (ray d) (ball 0 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_multibrotExt | [30, 1] | [33, 40] | exact (convex_ball _ _).isPathConnected (Metric.nonempty_ball.mpr one_pos) | case sc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPathConnected (ball 0 1)
case fc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ContinuousOn (ray d) (ball 0 1) | case fc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ContinuousOn (ray d) (ball 0 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_multibrotExt | [30, 1] | [33, 40] | exact (rayHolomorphic d).continuousOn | case fc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ContinuousOn (ray d) (ball 0 1) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | rw [e] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
e : potential d ⁻¹' {p} = ray d '' sphere 0 p
⊢ IsPathConnected (potential d ⁻¹' {p}) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
e : potential d ⁻¹' {p} = ray d '' sphere 0 p
⊢ IsPathConnected (ray d '' sphere 0 p) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | apply (isPathConnected_sphere p0).image_of_continuousOn | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
e : potential d ⁻¹' {p} = ray d '' sphere 0 p
⊢ IsPathConnected (ray d '' sphere 0 p) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
e : potential d ⁻¹' {p} = ray d '' sphere 0 p
⊢ ContinuousOn (ray d) (sphere 0 p) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | exact (rayHolomorphic d).continuousOn.mono (Metric.sphere_subset_ball p1) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
e : potential d ⁻¹' {p} = ray d '' sphere 0 p
⊢ ContinuousOn (ray d) (sphere 0 p) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | apply Set.ext | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
⊢ potential d ⁻¹' {p} = ray d '' sphere 0 p | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
⊢ ∀ (x : 𝕊), x ∈ potential d ⁻¹' {p} ↔ x ∈ ray d '' sphere 0 p |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | intro c | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
⊢ ∀ (x : 𝕊), x ∈ potential d ⁻¹' {p} ↔ x ∈ ray d '' sphere 0 p | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
⊢ c ∈ potential d ⁻¹' {p} ↔ c ∈ ray d '' sphere 0 p |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | simp only [mem_preimage, mem_singleton_iff, ← abs_bottcher, mem_image, mem_sphere,
Complex.dist_eq, sub_zero] | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
⊢ c ∈ potential d ⁻¹' {p} ↔ c ∈ ray d '' sphere 0 p | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
⊢ Complex.abs (bottcher d c) = p ↔ ∃ x, Complex.abs x = p ∧ ray d x = c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | constructor | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
⊢ Complex.abs (bottcher d c) = p ↔ ∃ x, Complex.abs x = p ∧ ray d x = c | case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
⊢ Complex.abs (bottcher d c) = p → ∃ x, Complex.abs x = p ∧ ray d x = c
case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
⊢ (∃ x, Complex.abs x = p ∧ ray d x = c) → Complex.abs (bottcher d c) = p |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | intro h | case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
⊢ Complex.abs (bottcher d c) = p → ∃ x, Complex.abs x = p ∧ ray d x = c | case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
h : Complex.abs (bottcher d c) = p
⊢ ∃ x, Complex.abs x = p ∧ ray d x = c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | use bottcher d c | case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
h : Complex.abs (bottcher d c) = p
⊢ ∃ x, Complex.abs x = p ∧ ray d x = c | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
h : Complex.abs (bottcher d c) = p
⊢ Complex.abs (bottcher d c) = p ∧ ray d (bottcher d c) = c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | use h | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
h : Complex.abs (bottcher d c) = p
⊢ Complex.abs (bottcher d c) = p ∧ ray d (bottcher d c) = c | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
h : Complex.abs (bottcher d c) = p
⊢ ray d (bottcher d c) = c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | rw [ray_bottcher] | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
h : Complex.abs (bottcher d c) = p
⊢ ray d (bottcher d c) = c | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
h : Complex.abs (bottcher d c) = p
⊢ c ∈ multibrotExt d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | rw [← potential_lt_one, ← abs_bottcher, h] | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
h : Complex.abs (bottcher d c) = p
⊢ c ∈ multibrotExt d | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
h : Complex.abs (bottcher d c) = p
⊢ p < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | exact p1 | case right
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
h : Complex.abs (bottcher d c) = p
⊢ p < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | intro ⟨e, ep, ec⟩ | case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
⊢ (∃ x, Complex.abs x = p ∧ ray d x = c) → Complex.abs (bottcher d c) = p | case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
e : ℂ
ep : Complex.abs e = p
ec : ray d e = c
⊢ Complex.abs (bottcher d c) = p |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | rw [← ec, bottcher_ray] | case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
e : ℂ
ep : Complex.abs e = p
ec : ray d e = c
⊢ Complex.abs (bottcher d c) = p | case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
e : ℂ
ep : Complex.abs e = p
ec : ray d e = c
⊢ Complex.abs e = p
case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
e : ℂ
ep : Complex.abs e = p
ec : ray d e = c
⊢ e ∈ ball 0 1 |
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