url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | linarith | c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ 0 < Complex.abs z' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
z : Box
cz : Complex.abs c' β€ Complex.abs z'
z6 : 6 β€ Complex.abs z'
zm : z' β approx z
β’ 0 < Complex.abs z'
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
β’ β―.potential c' βz' β approx (c.potential z n r).1 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β approx (c.potential z n r).1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
β’ β―.potential c' βz' β approx (c.potential z n r).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [Box.potential] | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β approx (c.potential z n r).1 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β approx (c.potential z n r).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
r : Floating
s : Super (f 2) 2 OnePoint.infty := superF 2
cs : Floating
hcs : c.normSq.hi = cs
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
ie : Exit
hie : i.exit = ie
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
ie : Exit
hie : i.exit = ie
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.large
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg.nan
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.nan
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
ie : Exit
hie : i.exit = ie
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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
β’ s.potential c' βz' β
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 β¨ 16 < zs β¨ 16 < cs then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
β’ s.potential c' βz' β
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 β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | by_cases bad : zs = nan β¨ (16 : Floating).val < zs.hi.val β¨ (16 : Floating).val < cs.val | case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
β’ s.potential c' βz' β
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 β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val
β’ s.potential c' βz' β
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 β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β
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 β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
β’ s.potential c' βz' β
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 β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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, βreduceIte, Interval.approx_nan, mem_univ] | case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val
β’ s.potential c' βz' β
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 β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val
β’ s.potential c' βz' β
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 β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [bad, βreduceIte] | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β
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 β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β
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 β¨ 16.val < zs.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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, βhzs] at bad | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬i.z.normSq = nan β§ i.z.normSq.hi.val β€ 16.val β§ cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬(zs = nan β¨ 16.val < zs.hi.val β¨ 16.val < cs.val)
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rcases bad with β¨zsn, z4, c4β© | case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬i.z.normSq = nan β§ i.z.normSq.hi.val β€ 16.val β§ cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16.val
c4 : cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
bad : Β¬i.z.normSq = nan β§ i.z.normSq.hi.val β€ 16.val β§ cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n)
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16.val
c4 : cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ β16
c4 : cs.val β€ β16
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16.val
c4 : cs.val β€ 16.val
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [β hcs, Nat.cast_ofNat, Interval.hi_eq_nan] at c4 z4 csn zsn | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ β16
c4 : cs.val β€ β16
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.count
zs : Interval
hzs : i.z.normSq = zs
zsn : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ β16
c4 : cs.val β€ β16
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n)
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n) | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' ^ 2 ^ i.n β approx potential_small | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' β approx (potential_small.iter_sqrt i.n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [βs.potential_eqn_iter, f_f'_iter] | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' ^ 2 ^ i.n β approx potential_small | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' β((f' 2 c')^[i.n] z') β approx potential_small | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' βz' ^ 2 ^ i.n β approx potential_small
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' β((f' 2 c')^[i.n] z') β approx potential_small | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ s.potential c' βw' β approx potential_small | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
β’ s.potential c' β((f' 2 c')^[i.n] z') β approx potential_small
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have le4 : Real.sqrt 16 β€ 4 := by rw [Real.sqrt_le_iff]; norm_num | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ s.potential c' βw' β approx potential_small | case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ s.potential c' βw' β approx potential_small | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ s.potential c' βw' β approx potential_small
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ s.potential c' βw' β approx potential_small | case neg.intro.intro.c4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs c' β€ 4
case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs w' β€ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ s.potential c' βw' β approx potential_small
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ β16 β€ 4 | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ 0 β€ 4 β§ 16 β€ 4 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ β16 β€ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | norm_num | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ 0 β€ 4 β§ 16 β€ 4 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
β’ 0 β€ 4 β§ 16 β€ 4 ^ 2
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs c' β€ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.c4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs c' β€ 4
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs w' β€ 4 | case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ w' β approx i.z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ Complex.abs w' β€ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [βhw', βhi] | case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ w' β approx i.z | case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ (f' 2 c')^[(iterate c z (cs.max 9) n).n] z' β approx (iterate c z (cs.max 9) n).z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ w' β approx i.z
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ (f' 2 c')^[(iterate c z (cs.max 9) n).n] z' β approx (iterate c z (cs.max 9) n).z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.z4
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬i.z.normSq = nan
z4 : i.z.normSq.hi.val β€ 16
c4 : c.normSq.hi.val β€ 16
csn : Β¬c.normSq = nan
w' : β
hw' : (f' 2 c')^[i.n] z' = w'
le4 : β16 β€ 4
β’ (f' 2 c')^[(iterate c z (cs.max 9) n).n] z' β approx (iterate c z (cs.max 9) n).z
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.large
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.large
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
approx
(match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
approx
(match je with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
β’ s.potential c' βz' β
approx
(match j.exit with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
approx
(match Exit.nan with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
approx
(match je with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.count
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
approx
(match Exit.count with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt (i.n + j.n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
approx
(match Exit.large with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt (i.n + j.n)) | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = 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 : β
hn : i.n + j.n = n
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt (i.n + j.n))
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = 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 : β
hn : i.n + j.n = n
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt n) | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = 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 : β
hn : i.n + j.n = n
β’ s.potential c' βz' ^ 2 ^ n β approx j.z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = 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 : β
hn : i.n + j.n = n
β’ s.potential c' βz' β approx (j.z.potential_large.iter_sqrt n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [βs.potential_eqn_iter, f_f'_iter, βhj] at hje β’ | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = 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 : β
hn : i.n + j.n = n
β’ s.potential c' βz' ^ 2 ^ n β approx j.z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
β’ s.potential c' β((f' 2 c')^[n] z') β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = 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 : β
hn : i.n + j.n = n
β’ s.potential c' βz' ^ 2 ^ n β approx j.z.potential_large
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
β’ s.potential c' β((f' 2 c')^[n] z') β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
β’ s.potential c' β((f' 2 c')^[n] z') β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
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' β approx i.z := by rw [βhi]; exact mem_approx_iterate cm zm _ | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β 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
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β 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
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β 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) β nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β 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
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [hj, β Function.iterate_add_apply, add_comm _ i.n, hn, hw'] at jl | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β 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) β nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jrn : (r.mul r true).max (cs.max 36) β nan
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β 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) β nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jrn : (r.mul r true).max (cs.max 36) β nan
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jrn : (r.mul r true).max (cs.max 36) β nan
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : ((r.mul r true).max (cs.max 36)).val < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large | case neg.large.large.cz
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ Complex.abs w'
case neg.large.large.z6
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ Complex.abs w'
case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ w' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ s.potential c' βw' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z.potential_large
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [βhi] | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ (f' 2 c')^[i.n] z' β approx i.z | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ (f' 2 c')^[(iterate c z (cs.max 9) nβ).n] z' β approx (iterate c z (cs.max 9) nβ).z | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ (f' 2 c')^[i.n] z' β approx i.z
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ (f' 2 c')^[(iterate c z (cs.max 9) nβ).n] z' β approx (iterate c z (cs.max 9) nβ).z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
β’ (f' 2 c')^[(iterate c z (cs.max 9) nβ).n] z' β approx (iterate c z (cs.max 9) nβ).z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | decide | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β 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
β’ Exit.large β Exit.nan | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β 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
β’ Exit.large β Exit.nan
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ Complex.abs w' | case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ cs.val.sqrt
case neg.large.large.cz.refine_2
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w' | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.cz
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ Complex.abs w'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | simp only [β hcs, Interval.hi_eq_nan] at csn β’ | case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ cs.val.sqrt | case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
csn : Β¬c.normSq = nan
β’ Complex.abs c' β€ c.normSq.hi.val.sqrt | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ Complex.abs c' β€ cs.val.sqrt
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
csn : Β¬c.normSq = nan
β’ Complex.abs c' β€ c.normSq.hi.val.sqrt | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.cz.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
csn : Β¬c.normSq = nan
β’ Complex.abs c' β€ c.normSq.hi.val.sqrt
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.cz.refine_2
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w'
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ Complex.abs w' | case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ β36
case neg.large.large.z6.refine_2
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w' | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.z6
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ Complex.abs w'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | have e : (36 : β) = 6 ^ 2 := by norm_num | case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ β36 | case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
e : 36 = 6 ^ 2
β’ 6 β€ β36 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 6 β€ β36
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
e : 36 = 6 ^ 2
β’ 6 β€ β36 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.z6.refine_1
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
e : 36 = 6 ^ 2
β’ 6 β€ β36
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | norm_num | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 36 = 6 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ 36 = 6 ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | norm_num | c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
e : 36 = 6 ^ 2
β’ 0 β€ 6 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
e : 36 = 6 ^ 2
β’ 0 β€ 6
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.z6.refine_2
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (Complex.abs w' ^ 2).sqrt β€ Complex.abs w'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.mem_approx_potential | [153, 1] | [212, 46] | rw [βhw', βhn, add_comm _ j.n, Function.iterate_add_apply, βhj] | case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ w' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z | case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z') β
approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ w' β approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z') β
approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.large.zm
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
nβ : β
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) nβ = i
csn : Β¬cs = nan
hie : i.exit = Exit.large
j : Iter
hj : iterate c i.z ((r.mul r true).max (cs.max 36)) 1000 = j
n : β
hn : i.n + j.n = n
hje : (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).exit = Exit.large
w' : β
hw' : (f' 2 c')^[n] z' = w'
izm : (f' 2 c')^[i.n] z' β approx i.z
jl : (r.mul r true).val < Complex.abs w' ^ 2 β§ cs.val < Complex.abs w' ^ 2 β§ 36 < Complex.abs w' ^ 2
jrn : Β¬r.mul r true = nan β§ Β¬cs = nan β§ Β¬36 = nan
β’ (f' 2 c')^[(iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).n] ((f' 2 c')^[i.n] z') β
approx (iterate c i.z ((r.mul r true).max (cs.max 36)) 1000).z
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.large.nan
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = 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
β’ s.potential c' βz' β
approx
(match Exit.nan with
| Exit.large => (j.z.potential_large.iter_sqrt (i.n + j.n), PotentialMode.large)
| x => (nan, PotentialMode.nan)).1
TACTIC:
|
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' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.nan
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.nan
c' z' : β
c z : Box
cm : c' β approx c
zm : z' β approx z
n : β
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 : Β¬cs = nan
hie : i.exit = Exit.nan
β’ s.potential c' βz' β
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 β¨ 16.val < i.z.normSq.hi.val β¨ 16.val < cs.val then (nan, PotentialMode.nan)
else (potential_small.iter_sqrt i.n, PotentialMode.small)).1
TACTIC:
|
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' : β
c : Box
cm : c' β approx c
n : β
r : Floating
β’ _root_.potential 2 βc' β approx (c.potential c n r).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c' : β
c : Box
cm : c' β approx c
n : β
r : Floating
β’ _root_.potential 2 βc' β approx (c.potential c n r).1
TACTIC:
|
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 β¦ (fa _ m).holomorphicAt I I | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
β’ NontrivialHolomorphicAt f z | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
β’ NontrivialHolomorphicAt f z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
β’ NontrivialHolomorphicAt f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have zs : z β closedBall z r := mem_closedBall_self rp.le | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
β’ NontrivialHolomorphicAt f z | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ NontrivialHolomorphicAt f z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
β’ NontrivialHolomorphicAt f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | use fh _ zs | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ NontrivialHolomorphicAt f z | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ βαΆ (w : β) in π z, f w β f z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ NontrivialHolomorphicAt f z
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ βαΆ (w : β) in π z, f w β f z | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : Β¬βαΆ (w : β) in π z, f w β f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ | Please generate a tactic in lean4 to solve the state.
STATE:
case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
ef : β w β sphere z r, e β€ βf w - f zβ
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
β’ βαΆ (w : β) in π z, f w β f z
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : Β¬βαΆ (w : β) in π z, f w β f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ | Please generate a tactic in lean4 to solve the state.
STATE:
case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : Β¬βαΆ (w : β) in π z, f w β f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e | Please generate a tactic in lean4 to solve the state.
STATE:
case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ Β¬β w β sphere z r, e β€ βf w - f zβ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | nontrivial_local_of_global | [46, 1] | [61, 29] | have zrs : z + r β 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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e | case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e | Please generate a tactic in lean4 to solve the state.
STATE:
case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ βf (z + βr) - f zβ < e | Please generate a tactic in lean4 to solve the state.
STATE:
case nonconst
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ β x, β (_ : x β sphere z r), βf x - f zβ < e
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ βf (z + βr) - f zβ < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
zrs : z + βr β sphere z r
β’ βf (z + βr) - f zβ < e
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ z + βr β sphere z r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
z : β
e r : β
fa : AnalyticOn β f (closedBall z r)
rp : 0 < r
ep : 0 < e
fh : HolomorphicOn π(β, β) π(β, β) f (closedBall z r)
zs : z β closedBall z r
ef : βαΆ (x : β) in π z, f x = f z
β’ z + βr β sphere z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have fn : β d, d β u β βαΆ w in π z, f d w β f d z := by
refine fun d m β¦ (nontrivial_local_of_global (fa.along_snd.mono ?_) rp ep (ef d m)).nonconst
simp only [β closedBall_prod_same, mem_prod_eq, setOf_mem_eq, iff_true_iff.mpr m,
true_and_iff, subset_refl] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have op : β d, d β u β ball (f d z) (e / 2) β 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 β fun w β¦ (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 [β closedBall_prod_same, mem_prod_eq, du, wr, true_and_iff, du] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z)
TACTIC:
|
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
β¨s, sp, shβ© | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z)
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z) | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β u_1 β π c, β v β π (f c z), u_1 ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r β π (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | refine β¨u β© ball c s, Filter.inter_mem un (Metric.ball_mem_nhds c (by linarith)), ?_β© | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β u_1 β π c, β v β π (f c z), u_1 ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β v β π (f c z), (u β© ball c s) ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β u_1 β π c, β v β π (f c z), u_1 ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β v β π (f c z), (u β© ball c s) ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (u β© ball c s) ΓΛ’ ball (f c z) (e / 4) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ β v β π (f c z), (u β© ball c s) ΓΛ’ v β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | intro β¨d, wβ© m | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (u β© ball c s) ΓΛ’ ball (f c z) (e / 4) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d, w) β (u β© ball c s) ΓΛ’ ball (f c z) (e / 4)
β’ (d, w) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
β’ (u β© ball c s) ΓΛ’ ball (f c z) (e / 4) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r
TACTIC:
|
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 β’ | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d, w) β (u β© ball c s) ΓΛ’ ball (f c z) (e / 4)
β’ (d, w) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d, w) β (u β© ball c s) ΓΛ’ ball (f c z) (e / 4)
β’ (d, w) β (fun p => (p.1, f p.1 p.2)) '' u ΓΛ’ closedBall z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | have wm : w β ball (f d z) (e / 2) := by
simp only [mem_ball] at m β’
specialize @sh β¨d, zβ©; 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)
_ β€ 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]
_ β€ e / 4 + e / 4 := by linarith [sh]
_ = e / 2 := by ring | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w)
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w)
TACTIC:
|
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 β¨y, yr, ywβ© | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | case right.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | useβ¨d, yβ© | case right.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w) | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ ((d, y).1 β u β§ (d, y).2 β closedBall z r) β§ ((d, y).1, f (d, y).1 (d, y).2) = (d, w) | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ β x, (x.1 β u β§ x.2 β closedBall z r) β§ (x.1, f x.1 x.2) = (d, w)
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ ((d, y).1 β u β§ (d, y).2 β closedBall z r) β§ ((d, y).1, f (d, y).1 (d, y).2) = (d, w) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
s : β
sp : s > 0
sh : β {x : β Γ β}, dist x (c, z) < s β dist (uncurry f x) (uncurry f (c, z)) < e / 4
d w : β
m : (d β u β§ dist d c < s) β§ w β ball (f c z) (e / 4)
wm : w β ball (f d z) (e / 2)
op : w β f d '' closedBall z r
y : β
yr : y β closedBall z r
yw : f d y = w
β’ ((d, y).1 β u β§ (d, y).2 β closedBall z r) β§ ((d, y).1, f (d, y).1 (d, y).2) = (d, w)
TACTIC:
|
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 β¦ (nontrivial_local_of_global (fa.along_snd.mono ?_) rp ep (ef d m)).nonconst | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
β’ β d β u, βαΆ (w : β) in π z, f d w β f d z | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
d : β
m : d β u
β’ closedBall z r β {y | (d, y) β u ΓΛ’ closedBall z r} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
β’ β d β u, βαΆ (w : β) in π z, f d w β f d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [β closedBall_prod_same, mem_prod_eq, setOf_mem_eq, iff_true_iff.mpr m,
true_and_iff, subset_refl] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
d : β
m : d β u
β’ closedBall z r β {y | (d, y) β u ΓΛ’ closedBall z r} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
d : β
m : d β u
β’ closedBall z r β {y | (d, y) β u ΓΛ’ closedBall z r}
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
β’ β d β u, ball (f d z) (e / 2) β f d '' closedBall z r | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ ball (f d z) (e / 2) β f d '' closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
β’ β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ ball (f d z) (e / 2) β f d '' closedBall z r | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ DiffContOnCl β (f d) (ball z r) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ ball (f d z) (e / 2) β f d '' closedBall z r
TACTIC:
|
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 β fun w β¦ (d, w) := rfl | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ DiffContOnCl β (f d) (ball z r) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (f d) (ball z r) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
β’ DiffContOnCl β (f d) (ball z r)
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (f d) (ball z r) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (uncurry f β fun w => (d, w)) (ball z r) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (f d) (ball z r)
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (uncurry f β fun w => (d, w)) (ball z r) | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DifferentiableOn β (uncurry f β fun w => (d, w)) (closure (ball z r)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DiffContOnCl β (uncurry f β fun w => (d, w)) (ball z r)
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DifferentiableOn β (uncurry f β fun w => (d, w)) (closure (ball z r)) | case h.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ AnalyticOn β (uncurry f β fun w => (d, w)) (closure (ball z r)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ DifferentiableOn β (uncurry f β fun w => (d, w)) (closure (ball z r))
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ AnalyticOn β (uncurry f β fun w => (d, w)) (closure (ball z r)) | case h.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ MapsTo (fun w => (d, w)) (closure (ball z r)) (u ΓΛ’ closedBall z r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ AnalyticOn β (uncurry f β fun w => (d, w)) (closure (ball z r))
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ MapsTo (fun w => (d, w)) (closure (ball z r)) (u ΓΛ’ closedBall z r) | case h.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
w : β
wr : w β closure (ball z r)
β’ (fun w => (d, w)) w β u ΓΛ’ closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
β’ MapsTo (fun w => (d, w)) (closure (ball z r)) (u ΓΛ’ closedBall z r)
TACTIC:
|
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
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
w : β
wr : w β closure (ball z r)
β’ (fun w => (d, w)) w β u ΓΛ’ closedBall z r | case h.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
w : β
wr : w β closedBall z r
β’ (fun w => (d, w)) w β u ΓΛ’ closedBall z r | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
w : β
wr : w β closure (ball z r)
β’ (fun w => (d, w)) w β u ΓΛ’ closedBall z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | simp only [β closedBall_prod_same, mem_prod_eq, du, wr, true_and_iff, du] | case h.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
w : β
wr : w β closedBall z r
β’ (fun w => (d, w)) w β u ΓΛ’ closedBall z r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
eβ r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < eβ
un : u β π c
ef : β d β u, β w β sphere z r, eβ β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
d : β
du : d β u
e : f d = uncurry f β fun w => (d, w)
w : β
wr : w β closedBall z r
β’ (fun w => (d, w)) w β u ΓΛ’ closedBall z r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/OpenMapping.lean | AnalyticOn.ball_subset_image_closedBall_param | [65, 1] | [101, 101] | linarith | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
β’ e / 4 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β β β
c z : β
e r : β
u : Set β
fa : AnalyticOn β (uncurry f) (u ΓΛ’ closedBall z r)
rp : 0 < r
ep : 0 < e
un : u β π c
ef : β d β u, β w β sphere z r, e β€ βf d w - f d zβ
fn : β d β u, βαΆ (w : β) in π z, f d w β f d z
op : β d β u, ball (f d z) (e / 2) β f d '' closedBall z r
β’ e / 4 > 0
TACTIC:
|
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