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stringlengths 7
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stringlengths 1
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stringlengths 6
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stringlengths 6
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stringlengths 1
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stringlengths 3
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| state_after
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|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | linarith | c' z' : โ
z : Box
cz : Complex.abs c' โค Complex.abs z'
z6 : 6 โค Complex.abs z'
zm : z' โ approx z
โข 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
r : Floating
โข โฏ.potential c' โz' โ approx (c.potential z n r).1 | c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
โข s.potential c' โz' โ 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
โข s.potential c' โz' โ approx (c.potential z n r).1 | c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
โข s.potential c' โz' โ
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.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan โจ 16 < zs โจ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
โข s.potential c' โz' โ
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.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan โจ 16 < zs โจ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
โข s.potential c' โz' โ
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan โจ 16 < zs โจ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
โข s.potential c' โz' โ
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan โจ 16 < zs โจ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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
โข s.potential c' โz' โ
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan โจ 16 < zs โจ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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
โข s.potential c' โz' โ
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan โจ 16 < zs โจ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case pos
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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
โข s.potential c' โz' โ
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan โจ 16 < zs โจ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
โข s.potential c' โz' โ
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan โจ 16 < zs โจ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
โข s.potential c' โz' โ
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan โจ 16 < zs โจ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
โข s.potential c' โz' โ
approx
(match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
โข s.potential c' โz' โ
approx
(match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
ie : Exit
hie : i.exit = ie
โข s.potential c' โz' โ
approx
(match ie with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
ie : Exit
hie : i.exit = ie
โข s.potential c' โz' โ
approx
(match ie with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.count
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
โข s.potential c' โz' โ
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
โข s.potential c' โz' โ
approx
(match Exit.large with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg.nan
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.nan
โข s.potential c' โz' โ
approx
(match Exit.nan with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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
โข s.potential c' โz' โ
approx
(let cs := cs;
let i := iterate c z (cs.max 9) n;
match i.exit with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
let rc := (r.mul r true).max (cs.max 36);
let j := iterate c i.z rc 1000;
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
let zs := i.z.normSq.hi;
if zs = nan โจ 16 < zs โจ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
โข s.potential c' โz' โ
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.count
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
โข s.potential c' โz' โ
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | by_cases bad : zs = nan โจ (16 : Floating).val < zs.hi.val โจ (16 : Floating).val < cs.val | case neg.count
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
โข s.potential c' โz' โ
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case pos
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val
โข s.potential c' โz' โ
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ยฌ(zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val)
โข s.potential c' โz' โ
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 |
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, โreduceIte, Interval.approx_nan, mem_univ] | case pos
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val
โข s.potential c' โz' โ
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [bad, โreduceIte] | case neg
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ยฌ(zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val)
โข s.potential c' โz' โ
approx
(match Exit.count with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ยฌ(zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val)
โข s.potential c' โz' โ approx (potential_small.iter_sqrt i.n) |
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, โhzs] at bad | case neg
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ยฌ(zs = nan โจ 16.val < zs.hi.val โจ 16.val < cs.val)
โข s.potential c' โz' โ approx (potential_small.iter_sqrt i.n) | case neg
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ยฌi.z.normSq = nan โง i.z.normSq.hi.val โค 16.val โง cs.val โค 16.val
โข s.potential c' โz' โ approx (potential_small.iter_sqrt i.n) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rcases bad with โจzsn, z4, c4โฉ | case neg
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : ยฌi.z.normSq = nan โง i.z.normSq.hi.val โค 16.val โง cs.val โค 16.val
โข s.potential c' โz' โ approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16.val
c4 : cs.val โค 16.val
โข s.potential c' โz' โ approx (potential_small.iter_sqrt i.n) |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16.val
c4 : cs.val โค 16.val
โข s.potential c' โz' โ approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค โ16
c4 : cs.val โค โ16
โข s.potential c' โz' โ approx (potential_small.iter_sqrt i.n) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [โ hcs, Nat.cast_ofNat, Interval.hi_eq_nan] at c4 z4 csn zsn | case neg.intro.intro
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค โ16
c4 : cs.val โค โ16
โข s.potential c' โz' โ approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
โข s.potential c' โz' โ approx (potential_small.iter_sqrt i.n) |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
โข s.potential c' โz' โ approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
โข s.potential c' โz' ^ 2 ^ i.n โ approx potential_small |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [โs.potential_eqn_iter, f_f'_iter] | case neg.intro.intro
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
โข s.potential c' โz' ^ 2 ^ i.n โ approx potential_small | case neg.intro.intro
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
โข s.potential c' โ((f' 2 c')^[i.n] z') โ approx potential_small |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
โข s.potential c' โ((f' 2 c')^[i.n] z') โ approx potential_small | case neg.intro.intro
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
โข s.potential c' โw' โ approx potential_small |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have le4 : Real.sqrt 16 โค 4 := by rw [Real.sqrt_le_iff]; norm_num | case neg.intro.intro
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
โข s.potential c' โw' โ approx potential_small | case neg.intro.intro
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
le4 : โ16 โค 4
โข s.potential c' โw' โ approx potential_small |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
le4 : โ16 โค 4
โข s.potential c' โw' โ approx potential_small | case neg.intro.intro.c4
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
le4 : โ16 โค 4
โข Complex.abs c' โค 4
case neg.intro.intro.z4
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
le4 : โ16 โค 4
โข Complex.abs w' โค 4 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
โข โ16 โค 4 | c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
โข 0 โค 4 โง 16 โค 4 ^ 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | norm_num | c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
โข 0 โค 4 โง 16 โค 4 ^ 2 | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
le4 : โ16 โค 4
โข Complex.abs c' โค 4 | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
le4 : โ16 โค 4
โข Complex.abs w' โค 4 | case neg.intro.intro.z4
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
le4 : โ16 โค 4
โข w' โ approx i.z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [โhw', โhi] | case neg.intro.intro.z4
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
le4 : โ16 โค 4
โข w' โ approx i.z | case neg.intro.intro.z4
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
le4 : โ16 โค 4
โข (f' 2 c')^[(iterate c z (cs.max 9) n).n] z' โ approx (iterate c z (cs.max 9) n).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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌi.z.normSq = nan
z4 : i.z.normSq.hi.val โค 16
c4 : c.normSq.hi.val โค 16
csn : ยฌc.normSq = nan
w' : โ
hw' : (f' 2 c')^[i.n] z' = w'
le4 : โ16 โค 4
โข (f' 2 c')^[(iterate c z (cs.max 9) n).n] z' โ approx (iterate c z (cs.max 9) n).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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
โข s.potential c' โz' โ
approx
(match Exit.large with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
โข s.potential c' โz' โ
approx
(match Exit.large with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
โข s.potential c' โz' โ
approx
(match Exit.large with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
โข s.potential c' โz' โ
approx
(match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
โข s.potential c' โz' โ
approx
(match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | case neg.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
je : Exit
hje : j.exit = je
โข s.potential c' โz' โ
approx
(match je with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
je : Exit
hje : j.exit = je
โข s.potential c' โz' โ
approx
(match je with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | case neg.large.count
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.count
โข s.potential c' โz' โ
approx
(match Exit.count with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
โข s.potential c' โz' โ
approx
(match Exit.large with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
case neg.large.nan
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.nan
โข s.potential c' โz' โ
approx
(match Exit.nan with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.count
โข s.potential c' โz' โ
approx
(match Exit.count with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
โข s.potential c' โz' โ
approx
(match Exit.large with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
โข s.potential c' โz' โ approx (j.z.potential_large.iter_sqrt (i.n + j.n)) |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
โข s.potential c' โz' โ approx (j.z.potential_large.iter_sqrt (i.n + j.n)) | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
n : โ
hn : i.n + j.n = n
โข s.potential c' โz' โ approx (j.z.potential_large.iter_sqrt n) |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
n : โ
hn : i.n + j.n = n
โข s.potential c' โz' โ approx (j.z.potential_large.iter_sqrt n) | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
n : โ
hn : i.n + j.n = n
โข s.potential c' โz' ^ 2 ^ n โ approx j.z.potential_large |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [โs.potential_eqn_iter, f_f'_iter, โhj] at hje โข | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.large
n : โ
hn : i.n + j.n = n
โข s.potential c' โz' ^ 2 ^ n โ approx j.z.potential_large | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
โข s.potential c' โ((f' 2 c')^[n] z') โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
โข s.potential c' โ((f' 2 c')^[n] z') โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large |
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' โ approx i.z := by rw [โhi]; exact mem_approx_iterate cm zm _ | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
jrn : (r.mul r true).max (cs.max 36) โ nan
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [hj, โ Function.iterate_add_apply, add_comm _ i.n, hn, hw'] at jl | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
jrn : (r.mul r true).max (cs.max 36) โ nan
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jrn : (r.mul r true).max (cs.max 36) โ nan
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jrn : (r.mul r true).max (cs.max 36) โ nan
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข s.potential c' โw' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large.cz
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข Complex.abs c' โค Complex.abs w'
case neg.large.large.z6
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข 6 โค Complex.abs w'
case neg.large.large.zm
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข w' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [โhi] | c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
โข (f' 2 c')^[i.n] z' โ approx i.z | c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
โข (f' 2 c')^[(iterate c z (cs.max 9) nโ).n] z' โ approx (iterate c z (cs.max 9) nโ).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 _ | c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
โข (f' 2 c')^[(iterate c z (cs.max 9) nโ).n] z' โ approx (iterate c z (cs.max 9) nโ).z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | decide | c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl :
((r.mul r true).max (cs.max 36)).val <
Complex.abs ((f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z')) ^ 2
โข Exit.large โ Exit.nan | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข Complex.abs c' โค Complex.abs w' | case neg.large.large.cz.refine_1
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข Complex.abs c' โค cs.val.sqrt
case neg.large.large.cz.refine_2
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข (Complex.abs w' ^ 2).sqrt โค Complex.abs w' |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [โ hcs, Interval.hi_eq_nan] at csn โข | case neg.large.large.cz.refine_1
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข Complex.abs c' โค cs.val.sqrt | case neg.large.large.cz.refine_1
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
csn : ยฌc.normSq = nan
โข Complex.abs c' โค c.normSq.hi.val.sqrt |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
csn : ยฌc.normSq = nan
โข Complex.abs c' โค c.normSq.hi.val.sqrt | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข (Complex.abs w' ^ 2).sqrt โค Complex.abs w' | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข 6 โค Complex.abs w' | case neg.large.large.z6.refine_1
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข 6 โค โ36
case neg.large.large.z6.refine_2
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข (Complex.abs w' ^ 2).sqrt โค Complex.abs w' |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have e : (36 : โ) = 6 ^ 2 := by norm_num | case neg.large.large.z6.refine_1
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข 6 โค โ36 | case neg.large.large.z6.refine_1
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
e : 36 = 6 ^ 2
โข 6 โค โ36 |
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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
e : 36 = 6 ^ 2
โข 6 โค โ36 | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข 36 = 6 ^ 2 | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
e : 36 = 6 ^ 2
โข 0 โค 6 | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข (Complex.abs w' ^ 2).sqrt โค Complex.abs w' | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [โhw', โhn, add_comm _ j.n, Function.iterate_add_apply, โhj] | case neg.large.large.zm
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข w' โ approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z | case neg.large.large.zm
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข (f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z') โ
approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).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 izm _ | case neg.large.large.zm
c' z' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
nโ : โ
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) nโ = i
csn : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : โ
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : โ
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' โ approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 โง cs.val < Complex.abs w' ^ 2 โง 36 < Complex.abs w' ^ 2
jrn : ยฌr.mul r true = nan โง ยฌcs = nan โง ยฌ36 = nan
โข (f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z') โ
approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
hje : j.exit = Exit.nan
โข s.potential c' โz' โ
approx
(match Exit.nan with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | 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' : โ
c z : Box
cm : c' โ approx c
zm : z' โ approx z
n : โ
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 : ยฌcs = nan
hie : i.exit = Exit.nan
โข s.potential c' โz' โ
approx
(match Exit.nan with
| Exit.nan => (nan, PotentialMode.nan)
| Exit.large =>
match (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit with
| Exit.large =>
((iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large.iter_sqrt
(i.n + (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n),
PotentialMode.large)
| x => (nan, PotentialMode.nan)
| Exit.count =>
if i.z.normSq = nan โจ 16.val < i.z.normSq.hi.val โจ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | 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' : โ
c : Box
cm : c' โ approx c
n : โ
r : Floating
โข _root_.potential 2 โc' โ approx (c.potential c n r).1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have fh : HolomorphicOn I I f (closedBall z r) := fun _ m โฆ (fa _ m).holomorphicAt I I | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : โ w โ sphere z r, e โค โf w - f zโ
โข NontrivialHolomorphicAt f z | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : โ w โ sphere z r, e โค โf w - f zโ
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
โข NontrivialHolomorphicAt f z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have zs : z โ closedBall z r := mem_closedBall_self rp.le | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : โ w โ sphere z r, e โค โf w - f zโ
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
โข NontrivialHolomorphicAt f z | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : โ w โ sphere z r, e โค โf w - f zโ
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
โข NontrivialHolomorphicAt f z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | use fh _ zs | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : โ w โ sphere z r, e โค โf w - f zโ
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
โข NontrivialHolomorphicAt f z | case nonconst
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : โ w โ sphere z r, e โค โf w - f zโ
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
โข โแถ (w : โ) in ๐ z, f w โ f z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | contrapose ef | case nonconst
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : โ w โ sphere z r, e โค โf w - f zโ
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
โข โแถ (w : โ) in ๐ z, f w โ f z | case nonconst
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : ยฌโแถ (w : โ) in ๐ z, f w โ f z
โข ยฌโ w โ sphere z r, e โค โf w - f zโ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [Filter.not_frequently, not_not] at ef | case nonconst
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : ยฌโแถ (w : โ) in ๐ z, f w โ f z
โข ยฌโ w โ sphere z r, e โค โf w - f zโ | case nonconst
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : โแถ (x : โ) in ๐ z, f x = f z
โข ยฌโ w โ sphere z r, e โค โf w - f zโ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [not_forall, not_le] | case nonconst
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : โแถ (x : โ) in ๐ z, f x = f z
โข ยฌโ w โ sphere z r, e โค โf w - f zโ | case nonconst
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : โแถ (x : โ) in ๐ z, f x = f z
โข โ x, โ (_ : x โ sphere z r), โf x - f zโ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have zrs : z + r โ sphere z r := by
simp only [mem_sphere, Complex.dist_eq, add_sub_cancel_left, Complex.abs_ofReal, abs_of_pos rp] | case nonconst
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : โแถ (x : โ) in ๐ z, f x = f z
โข โ x, โ (_ : x โ sphere z r), โf x - f zโ < e | case nonconst
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : โแถ (x : โ) in ๐ z, f x = f z
zrs : z + โr โ sphere z r
โข โ x, โ (_ : x โ sphere z r), โf x - f zโ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | use z + r, zrs | case nonconst
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : โแถ (x : โ) in ๐ z, f x = f z
zrs : z + โr โ sphere z r
โข โ x, โ (_ : x โ sphere z r), โf x - f zโ < e | case h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : โแถ (x : โ) in ๐ z, f x = f z
zrs : z + โr โ sphere z r
โข โf (z + โr) - f zโ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [fh.const_of_locally_const' zs (convex_closedBall z r).isPreconnected ef (z + r)
(Metric.sphere_subset_closedBall zrs),
sub_self, norm_zero, ep] | case h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : โแถ (x : โ) in ๐ z, f x = f z
zrs : z + โr โ sphere z r
โข โf (z + โr) - f zโ < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | simp only [mem_sphere, Complex.dist_eq, add_sub_cancel_left, Complex.abs_ofReal, abs_of_pos rp] | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ
z : โ
e r : โ
fa : AnalyticOn โ f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn ๐(โ, โ) ๐(โ, โ) f (closedBall z r)
zs : z โ closedBall z r
ef : โแถ (x : โ) in ๐ z, f x = f z
โข z + โr โ sphere z r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have fn : โ d, d โ u โ โแถ w in ๐ z, f d w โ f d z := by
refine fun d m โฆ (nontrivial_local_of_global (fa.along_snd.mono ?_) rp ep (ef d m)).nonconst
simp only [โ closedBall_prod_same, mem_prod_eq, setOf_mem_eq, iff_true_iff.mpr m,
true_and_iff, subset_refl] | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
โข (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r โ ๐ (c, f c z) | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
โข (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r โ ๐ (c, f c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have op : โ d, d โ u โ ball (f d z) (e / 2) โ f d '' closedBall z r := by
intro d du; refine DiffContOnCl.ball_subset_image_closedBall ?_ rp (ef d du) (fn d du)
have e : f d = uncurry f โ fun w โฆ (d, w) := rfl
rw [e]; apply DifferentiableOn.diffContOnCl; apply AnalyticOn.differentiableOn
refine fa.comp (analyticOn_const.prod (analyticOn_id _)) ?_
intro w wr; simp only [closure_ball _ rp.ne'] at wr
simp only [โ closedBall_prod_same, mem_prod_eq, du, wr, true_and_iff, du] | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
โข (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r โ ๐ (c, f c z) | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
โข (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r โ ๐ (c, f c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rcases Metric.continuousAt_iff.mp
(fa (c, z) (mk_mem_prod (mem_of_mem_nhds un) (mem_closedBall_self rp.le))).continuousAt
(e / 4) (by linarith) with
โจs, sp, shโฉ | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
โข (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r โ ๐ (c, f c z) | case intro.intro
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
โข (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r โ ๐ (c, f c z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rw [mem_nhds_prod_iff] | case intro.intro
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
โข (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r โ ๐ (c, f c z) | case intro.intro
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
โข โ u_1 โ ๐ c, โ v โ ๐ (f c z), u_1 รหข v โ (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine โจu โฉ ball c s, Filter.inter_mem un (Metric.ball_mem_nhds c (by linarith)), ?_โฉ | case intro.intro
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
โข โ u_1 โ ๐ c, โ v โ ๐ (f c z), u_1 รหข v โ (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r | case intro.intro
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
โข โ v โ ๐ (f c z), (u โฉ ball c s) รหข v โ (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | use ball (f c z) (e / 4), Metric.ball_mem_nhds _ (by linarith) | case intro.intro
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
โข โ v โ ๐ (f c z), (u โฉ ball c s) รหข v โ (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r | case right
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
โข (u โฉ ball c s) รหข ball (f c z) (e / 4) โ (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | intro โจd, wโฉ m | case right
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
โข (u โฉ ball c s) รหข ball (f c z) (e / 4) โ (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r | case right
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d, w) โ (u โฉ ball c s) รหข ball (f c z) (e / 4)
โข (d, w) โ (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [mem_inter_iff, mem_prod_eq, mem_image, @mem_ball _ _ c, lt_min_iff] at m op โข | case right
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d, w) โ (u โฉ ball c s) รหข ball (f c z) (e / 4)
โข (d, w) โ (fun p => (p.1, f p.1 p.2)) '' u รหข closedBall z r | case right
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d โ u โง dist d c < s) โง w โ ball (f c z) (e / 4)
โข โ x, (x.1 โ u โง x.2 โ closedBall z r) โง (x.1, f x.1 x.2) = (d, w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have wm : w โ ball (f d z) (e / 2) := by
simp only [mem_ball] at m โข
specialize @sh โจd, zโฉ; simp only [Prod.dist_eq, dist_self, Function.uncurry] at sh
specialize sh (max_lt m.1.2 sp); rw [dist_comm] at sh
calc dist w (f d z)
_ โค dist w (f c z) + dist (f c z) (f d z) := by bound
_ < e / 4 + dist (f c z) (f d z) := by linarith [m.2]
_ โค e / 4 + e / 4 := by linarith [sh]
_ = e / 2 := by ring | case right
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d โ u โง dist d c < s) โง w โ ball (f c z) (e / 4)
โข โ x, (x.1 โ u โง x.2 โ closedBall z r) โง (x.1, f x.1 x.2) = (d, w) | case right
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d โ u โง dist d c < s) โง w โ ball (f c z) (e / 4)
wm : w โ ball (f d z) (e / 2)
โข โ x, (x.1 โ u โง x.2 โ closedBall z r) โง (x.1, f x.1 x.2) = (d, w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | specialize op d m.1.1 wm | case right
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d โ u โง dist d c < s) โง w โ ball (f c z) (e / 4)
wm : w โ ball (f d z) (e / 2)
โข โ x, (x.1 โ u โง x.2 โ closedBall z r) โง (x.1, f x.1 x.2) = (d, w) | case right
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d โ u โง dist d c < s) โง w โ ball (f c z) (e / 4)
wm : w โ ball (f d z) (e / 2)
op : w โ f d '' closedBall z r
โข โ x, (x.1 โ u โง x.2 โ closedBall z r) โง (x.1, f x.1 x.2) = (d, w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rcases (mem_image _ _ _).mp op with โจy, yr, ywโฉ | case right
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d โ u โง dist d c < s) โง w โ ball (f c z) (e / 4)
wm : w โ ball (f d z) (e / 2)
op : w โ f d '' closedBall z r
โข โ x, (x.1 โ u โง x.2 โ closedBall z r) โง (x.1, f x.1 x.2) = (d, w) | case right.intro.intro
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d โ u โง dist d c < s) โง w โ ball (f c z) (e / 4)
wm : w โ ball (f d z) (e / 2)
op : w โ f d '' closedBall z r
y : โ
yr : y โ closedBall z r
yw : f d y = w
โข โ x, (x.1 โ u โง x.2 โ closedBall z r) โง (x.1, f x.1 x.2) = (d, w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | useโจd, yโฉ | case right.intro.intro
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d โ u โง dist d c < s) โง w โ ball (f c z) (e / 4)
wm : w โ ball (f d z) (e / 2)
op : w โ f d '' closedBall z r
y : โ
yr : y โ closedBall z r
yw : f d y = w
โข โ x, (x.1 โ u โง x.2 โ closedBall z r) โง (x.1, f x.1 x.2) = (d, w) | case h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d โ u โง dist d c < s) โง w โ ball (f c z) (e / 4)
wm : w โ ball (f d z) (e / 2)
op : w โ f d '' closedBall z r
y : โ
yr : y โ closedBall z r
yw : f d y = w
โข ((d, y).1 โ u โง (d, y).2 โ closedBall z r) โง ((d, y).1, f (d, y).1 (d, y).2) = (d, w) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [mem_prod_eq, Prod.ext_iff, yw, and_true_iff, eq_self_iff_true, true_and_iff, yr, m.1.1] | case h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
s : โ
sp : s > 0
sh : โ {x : โ ร โ}, dist x (c, z) < s โ dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : โ
m : (d โ u โง dist d c < s) โง w โ ball (f c z) (e / 4)
wm : w โ ball (f d z) (e / 2)
op : w โ f d '' closedBall z r
y : โ
yr : y โ closedBall z r
yw : f d y = w
โข ((d, y).1 โ u โง (d, y).2 โ closedBall z r) โง ((d, y).1, f (d, y).1 (d, y).2) = (d, w) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine fun d m โฆ (nontrivial_local_of_global (fa.along_snd.mono ?_) rp ep (ef d m)).nonconst | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
โข โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
d : โ
m : d โ u
โข closedBall z r โ {y | (d, y) โ u รหข closedBall z r} |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [โ closedBall_prod_same, mem_prod_eq, setOf_mem_eq, iff_true_iff.mpr m,
true_and_iff, subset_refl] | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
d : โ
m : d โ u
โข closedBall z r โ {y | (d, y) โ u รหข closedBall z r} | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | intro d du | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
โข โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
โข ball (f d z) (e / 2) โ f d '' closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine DiffContOnCl.ball_subset_image_closedBall ?_ rp (ef d du) (fn d du) | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
โข ball (f d z) (e / 2) โ f d '' closedBall z r | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
โข DiffContOnCl โ (f d) (ball z r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have e : f d = uncurry f โ fun w โฆ (d, w) := rfl | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
โข DiffContOnCl โ (f d) (ball z r) | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
โข DiffContOnCl โ (f d) (ball z r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | rw [e] | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
โข DiffContOnCl โ (f d) (ball z r) | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
โข DiffContOnCl โ (uncurry f โ fun w => (d, w)) (ball z r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | apply DifferentiableOn.diffContOnCl | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
โข DiffContOnCl โ (uncurry f โ fun w => (d, w)) (ball z r) | case h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
โข DifferentiableOn โ (uncurry f โ fun w => (d, w)) (closure (ball z r)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | apply AnalyticOn.differentiableOn | case h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
โข DifferentiableOn โ (uncurry f โ fun w => (d, w)) (closure (ball z r)) | case h.h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
โข AnalyticOn โ (uncurry f โ fun w => (d, w)) (closure (ball z r)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine fa.comp (analyticOn_const.prod (analyticOn_id _)) ?_ | case h.h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
โข AnalyticOn โ (uncurry f โ fun w => (d, w)) (closure (ball z r)) | case h.h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
โข MapsTo (fun w => (d, w)) (closure (ball z r)) (u รหข closedBall z r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | intro w wr | case h.h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
โข MapsTo (fun w => (d, w)) (closure (ball z r)) (u รหข closedBall z r) | case h.h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
w : โ
wr : w โ closure (ball z r)
โข (fun w => (d, w)) w โ u รหข closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [closure_ball _ rp.ne'] at wr | case h.h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
w : โ
wr : w โ closure (ball z r)
โข (fun w => (d, w)) w โ u รหข closedBall z r | case h.h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
w : โ
wr : w โ closedBall z r
โข (fun w => (d, w)) w โ u รหข closedBall z r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [โ closedBall_prod_same, mem_prod_eq, du, wr, true_and_iff, du] | case h.h
X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
eโ r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < eโ
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, eโ โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
d : โ
du : d โ u
e : f d = uncurry f โ fun w => (d, w)
w : โ
wr : w โ closedBall z r
โข (fun w => (d, w)) w โ u รหข closedBall z r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | linarith | X : Type
instโโถ : TopologicalSpace X
S : Type
instโโต : TopologicalSpace S
instโโด : ChartedSpace โ S
cms : AnalyticManifold ๐(โ, โ) S
T : Type
instโยณ : TopologicalSpace T
instโยฒ : ChartedSpace โ T
cmt : AnalyticManifold ๐(โ, โ) T
U : Type
instโยน : TopologicalSpace U
instโ : ChartedSpace โ U
cmu : AnalyticManifold ๐(โ, โ) U
f : โ โ โ โ โ
c z : โ
e r : โ
u : Set โ
fa : AnalyticOn โ (uncurry f) (u รหข closedBall z r)
rp : 0 < r
ep : 0 < e
un : u โ ๐ c
ef : โ d โ u, โ w โ sphere z r, e โค โf d w - f d zโ
fn : โ d โ u, โแถ (w : โ) in ๐ z, f d w โ f d z
op : โ d โ u, ball (f d z) (e / 2) โ f d '' closedBall z r
โข e / 4 > 0 | no goals |
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