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/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | calc dist (f w) (f z)
_ β€ dist (f w) (f (z.fst, w.snd)) + dist (f (z.fst, w.snd)) (f z) := dist_triangle _ _ _
_ β€ e / 4 + e / 4 := by linarith
_ = e / 2 := by ring
_ < e := by linarith | case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : β Γ β
aβ : w β s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) β€ e / 4
s1 : dist (f (z.1, w.2)) (f z) β€ e / 4
β’ dist (f w) (f z) < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : β Γ β
aβ : w β s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) β€ e / 4
s1 : dist (f (z.1, w.2)) (f z) β€ e / 4
β’ dist (f w) (f z) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | exact osgood o c h.fa0 h.fa1 | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
c : ContinuousOn f s
β’ AnalyticOn β f s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
c : ContinuousOn f s
β’ AnalyticOn β f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | have fz : β z, z β s β f z = 0 := fun z zs β¦
norm_eq_zero.mp (le_antisymm (_root_.trans (fb z zs) bp) (norm_nonneg (f z))) | case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : b β€ 0
β’ ContinuousOn f s | case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : b β€ 0
fz : β z β s, f z = 0
β’ ContinuousOn f s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : b β€ 0
β’ ContinuousOn f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | rw [continuousOn_congr fz] | case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : b β€ 0
fz : β z β s, f z = 0
β’ ContinuousOn f s | case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : b β€ 0
fz : β z β s, f z = 0
β’ ContinuousOn (fun x => 0) s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : b β€ 0
fz : β z β s, f z = 0
β’ ContinuousOn f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | exact continuousOn_const | case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : b β€ 0
fz : β z β s, f z = 0
β’ ContinuousOn (fun x => 0) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : b β€ 0
fz : β z β s, f z = 0
β’ ContinuousOn (fun x => 0) s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | bound | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
β’ min (r / 2) (e * r / b / 24) > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
β’ min (r / 2) (e * r / b / 24) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | linarith | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : β Γ β
aβ : w β s
wz : dist w z < min (r / 2) (e * r / b / 24)
β’ r / 2 β€ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : β Γ β
aβ : w β s
wz : dist w z < min (r / 2) (e * r / b / 24)
β’ r / 2 β€ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | linarith | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : β Γ β
aβ : w β s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) β€ e / 4
s1 : dist (f (z.1, w.2)) (f z) β€ e / 4
β’ dist (f w) (f (z.1, w.2)) + dist (f (z.1, w.2)) (f z) β€ e / 4 + e / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : β Γ β
aβ : w β s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) β€ e / 4
s1 : dist (f (z.1, w.2)) (f z) β€ e / 4
β’ dist (f w) (f (z.1, w.2)) + dist (f (z.1, w.2)) (f z) β€ e / 4 + e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | ring | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : β Γ β
aβ : w β s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) β€ e / 4
s1 : dist (f (z.1, w.2)) (f z) β€ e / 4
β’ e / 4 + e / 4 = e / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : β Γ β
aβ : w β s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) β€ e / 4
s1 : dist (f (z.1, w.2)) (f z) β€ e / 4
β’ e / 4 + e / 4 = e / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | of_bounded | [155, 1] | [176, 25] | linarith | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : β Γ β
aβ : w β s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) β€ e / 4
s1 : dist (f (z.1, w.2)) (f z) β€ e / 4
β’ e / 2 < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
rβ r0 r1 bβ eβ : β
h : Har f s
o : IsOpen s
b : β
fb : β z β s, βf zβ β€ b
bp : 0 < b
z : β Γ β
zs : z β s
r : β
rp : r > 0
rs : ball z r β s
e : β
ep : e > 0
up : min (r / 2) (e * r / b / 24) > 0
w : β Γ β
aβ : w β s
wz : dist w z < min (r / 2) (e * r / b / 24)
s0 : dist (f w) (f (z.1, w.2)) β€ e / 4
s1 : dist (f (z.1, w.2)) (f z) β€ e / 4
β’ e / 2 < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | set s := interior (β k, f k) | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
hU : (interior (β k, f k)).Nonempty
β’ β k, (interior (f k)).Nonempty | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
β’ β k, (interior (f k)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
hU : (interior (β k, f k)).Nonempty
β’ β k, (interior (f k)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | set f' : Option B β Set A := fun k β¦
match k with
| none => sαΆ
| some k => f k | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
β’ β k, (interior (f k)).Nonempty | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
β’ β k, (interior (f k)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
β’ β k, (interior (f k)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | have hc' : β k, IsClosed (f' k) := by
simp only [s, Option.forall, isClosed_compl_iff, isOpen_interior, true_and, f']
exact fun k β¦ hc k | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
β’ β k, (interior (f k)).Nonempty | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
β’ β k, (interior (f k)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
β’ β k, (interior (f k)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | have d := dense_iUnion_interior_of_closed hc' hU' | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
β’ β k, (interior (f k)).Nonempty | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
β’ β k, (interior (f k)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
β’ β k, (interior (f k)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | rcases Dense.exists_mem_open d isOpen_interior hU with β¨x, xi, xsβ© | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
β’ β k, (interior (f k)).Nonempty | case intro.intro
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
x : A
xi : x β β i, interior (f' i)
xs : x β interior (β k, f k)
β’ β k, (interior (f k)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
β’ β k, (interior (f k)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | rcases Set.mem_iUnion.mp xi with β¨k, xkβ© | case intro.intro
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
x : A
xi : x β β i, interior (f' i)
xs : x β interior (β k, f k)
β’ β k, (interior (f k)).Nonempty | case intro.intro.intro
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
x : A
xi : x β β i, interior (f' i)
xs : x β interior (β k, f k)
k : Option B
xk : x β interior (f' k)
β’ β k, (interior (f k)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
x : A
xi : x β β i, interior (f' i)
xs : x β interior (β k, f k)
β’ β k, (interior (f k)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | match k with
| none => simp only [s, interior_compl, Set.mem_compl_iff, subset_closure xs,
not_true_eq_false, f'] at xk
| some k => exact β¨k, Set.nonempty_of_mem xkβ© | case intro.intro.intro
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
x : A
xi : x β β i, interior (f' i)
xs : x β interior (β k, f k)
k : Option B
xk : x β interior (f' k)
β’ β k, (interior (f k)).Nonempty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
x : A
xi : x β β i, interior (f' i)
xs : x β interior (β k, f k)
k : Option B
xk : x β interior (f' k)
β’ β k, (interior (f k)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | simp only [s, Option.forall, isClosed_compl_iff, isOpen_interior, true_and, f'] | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
β’ β (k : Option B), IsClosed (f' k) | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
β’ β (x : B), IsClosed (f x) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
β’ β (k : Option B), IsClosed (f' k)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | exact fun k β¦ hc k | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
β’ β (x : B), IsClosed (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
β’ β (x : B), IsClosed (f x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | apply Set.ext | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
β’ β k, f' k = univ | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
β’ β (x : A), x β β k, f' k β x β univ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
β’ β k, f' k = univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | intro x | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
β’ β (x : A), x β β k, f' k β x β univ | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
β’ x β β k, f' k β x β univ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
β’ β (x : A), x β β k, f' k β x β univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | refine β¨fun _ β¦ Set.mem_univ _, ?_β© | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
β’ x β β k, f' k β x β univ | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
β’ x β univ β x β β k, f' k | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
β’ x β β k, f' k β x β univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | intro _ | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
β’ x β univ β x β β k, f' k | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
β’ x β β k, f' k | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
β’ x β univ β x β β k, f' k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | rw [Set.mem_iUnion] | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
β’ x β β k, f' k | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
β’ β i, x β f' i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
β’ x β β k, f' k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | by_cases m : x β s | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
β’ β i, x β f' i | case pos
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
β’ β i, x β f' i
case neg
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
β’ β i, x β f' i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
β’ β i, x β f' i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | rcases Set.mem_iUnion.mp (interior_subset m) with β¨k, mkβ© | case pos
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
β’ β i, x β f' i | case pos.intro
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
k : B
mk : x β f k
β’ β i, x β f' i | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
β’ β i, x β f' i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | use some k | case pos.intro
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
k : B
mk : x β f k
β’ β i, x β f' i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
k : B
mk : x β f k
β’ β i, x β f' i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | use none | case neg
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
β’ β i, x β f' i | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
β’ x β f' none | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
β’ β i, x β f' i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | exact m | case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
β’ x β f' none | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
x : A
aβ : x β univ
m : x β s
β’ x β f' none
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | simp only [s, interior_compl, Set.mem_compl_iff, subset_closure xs,
not_true_eq_false, f'] at xk | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
x : A
xi : x β β i, interior (f' i)
xs : x β interior (β k, f k)
k : Option B
xk : x β interior (f' none)
β’ β k, (interior (f k)).Nonempty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
x : A
xi : x β β i, interior (f' i)
xs : x β interior (β k, f k)
k : Option B
xk : x β interior (f' none)
β’ β k, (interior (f k)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | NonemptyInterior.nonempty_interior_of_iUnion_of_closed | [182, 1] | [205, 48] | exact β¨k, Set.nonempty_of_mem xkβ© | E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
x : A
xi : x β β i, interior (f' i)
xs : x β interior (β k, f k)
kβ : Option B
k : B
xk : x β interior (f' (some k))
β’ β k, (interior (f k)).Nonempty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββΆ : NormedAddCommGroup E
instββ΅ : NormedSpace β E
instββ΄ : CompleteSpace E
instβΒ³ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ² : TopologicalSpace A
instβΒΉ : BaireSpace A
instβ : Encodable B
f : B β Set A
hc : β (k : B), IsClosed (f k)
s : Set A := interior (β k, f k)
hU : s.Nonempty
f' : Option B β Set A :=
fun k =>
match k with
| none => sαΆ
| some k => f k
hc' : β (k : Option B), IsClosed (f' k)
hU' : β k, f' k = univ
d : Dense (β i, interior (f' i))
x : A
xi : x β β i, interior (f' i)
xs : x β interior (β k, f k)
kβ : Option B
k : B
xk : x β interior (f' (some k))
β’ β k, (interior (f k)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | forall_const_and_distrib | [208, 1] | [210, 80] | have d := @forall_and A (fun _ β¦ p) q | E : Type
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace β E
instβΒ² : CompleteSpace E
instβΒΉ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A : Type
instβ : Nonempty A
p : Prop
q : A β Prop
β’ (β (x : A), p β§ q x) β p β§ β (x : A), q x | E : Type
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace β E
instβΒ² : CompleteSpace E
instβΒΉ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A : Type
instβ : Nonempty A
p : Prop
q : A β Prop
d : (β (x : A), p β§ q x) β (A β p) β§ β (x : A), q x
β’ (β (x : A), p β§ q x) β p β§ β (x : A), q x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace β E
instβΒ² : CompleteSpace E
instβΒΉ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A : Type
instβ : Nonempty A
p : Prop
q : A β Prop
β’ (β (x : A), p β§ q x) β p β§ β (x : A), q x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | forall_const_and_distrib | [208, 1] | [210, 80] | simp only [forall_const] at d | E : Type
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace β E
instβΒ² : CompleteSpace E
instβΒΉ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A : Type
instβ : Nonempty A
p : Prop
q : A β Prop
d : (β (x : A), p β§ q x) β (A β p) β§ β (x : A), q x
β’ (β (x : A), p β§ q x) β p β§ β (x : A), q x | E : Type
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace β E
instβΒ² : CompleteSpace E
instβΒΉ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A : Type
instβ : Nonempty A
p : Prop
q : A β Prop
d : (β (x : A), p β§ q x) β p β§ β (x : A), q x
β’ (β (x : A), p β§ q x) β p β§ β (x : A), q x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace β E
instβΒ² : CompleteSpace E
instβΒΉ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A : Type
instβ : Nonempty A
p : Prop
q : A β Prop
d : (β (x : A), p β§ q x) β (A β p) β§ β (x : A), q x
β’ (β (x : A), p β§ q x) β p β§ β (x : A), q x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | forall_const_and_distrib | [208, 1] | [210, 80] | exact d | E : Type
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace β E
instβΒ² : CompleteSpace E
instβΒΉ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A : Type
instβ : Nonempty A
p : Prop
q : A β Prop
d : (β (x : A), p β§ q x) β p β§ β (x : A), q x
β’ (β (x : A), p β§ q x) β p β§ β (x : A), q x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ΄ : NormedAddCommGroup E
instβΒ³ : NormedSpace β E
instβΒ² : CompleteSpace E
instβΒΉ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A : Type
instβ : Nonempty A
p : Prop
q : A β Prop
d : (β (x : A), p β§ q x) β p β§ β (x : A), q x
β’ (β (x : A), p β§ q x) β p β§ β (x : A), q x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | rw [Set.setOf_and] | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
β’ IsClosed {x | x β s β§ f x β€ g x} | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
β’ IsClosed ({a | a β s} β© {a | f a β€ g a}) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
β’ IsClosed {x | x β s β§ f x β€ g x}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | simp only [Set.setOf_mem_eq] | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
β’ IsClosed ({a | a β s} β© {a | f a β€ g a}) | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
β’ IsClosed (s β© {a | f a β€ g a}) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
β’ IsClosed ({a | a β s} β© {a | f a β€ g a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | set t := {p : B Γ B | p.fst β€ p.snd} | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
β’ IsClosed (s β© {a | f a β€ g a}) | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
β’ IsClosed (s β© {a | f a β€ g a}) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
β’ IsClosed (s β© {a | f a β€ g a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | set fg := fun x β¦ (f x, g x) | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
β’ IsClosed (s β© {a | f a β€ g a}) | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
β’ IsClosed (s β© {a | f a β€ g a}) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
β’ IsClosed (s β© {a | f a β€ g a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | have e : {x | f x β€ g x} = fg β»ΒΉ' t := by
apply Set.ext; intro x; simp only [Set.preimage_setOf_eq, t] | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
β’ IsClosed (s β© {a | f a β€ g a}) | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b eβ : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
e : {x | f x β€ g x} = fg β»ΒΉ' t
β’ IsClosed (s β© {a | f a β€ g a}) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
β’ IsClosed (s β© {a | f a β€ g a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | rw [e] | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b eβ : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
e : {x | f x β€ g x} = fg β»ΒΉ' t
β’ IsClosed (s β© {a | f a β€ g a}) | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b eβ : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
e : {x | f x β€ g x} = fg β»ΒΉ' t
β’ IsClosed (s β© fg β»ΒΉ' t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b eβ : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
e : {x | f x β€ g x} = fg β»ΒΉ' t
β’ IsClosed (s β© {a | f a β€ g a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | exact ContinuousOn.preimage_isClosed_of_isClosed (ContinuousOn.prod fc gc) sc
OrderClosedTopology.isClosed_le' | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b eβ : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
e : {x | f x β€ g x} = fg β»ΒΉ' t
β’ IsClosed (s β© fg β»ΒΉ' t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b eβ : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
e : {x | f x β€ g x} = fg β»ΒΉ' t
β’ IsClosed (s β© fg β»ΒΉ' t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | apply Set.ext | E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
β’ {x | f x β€ g x} = fg β»ΒΉ' t | case h
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
β’ β (x : A), x β {x | f x β€ g x} β x β fg β»ΒΉ' t | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
β’ {x | f x β€ g x} = fg β»ΒΉ' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | intro x | case h
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
β’ β (x : A), x β {x | f x β€ g x} β x β fg β»ΒΉ' t | case h
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
x : A
β’ x β {x | f x β€ g x} β x β fg β»ΒΉ' t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
β’ β (x : A), x β {x | f x β€ g x} β x β fg β»ΒΉ' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | simp only [Set.preimage_setOf_eq, t] | case h
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
x : A
β’ x β {x | f x β€ g x} β x β fg β»ΒΉ' t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instββ· : NormedAddCommGroup E
instββΆ : NormedSpace β E
instββ΅ : CompleteSpace E
instββ΄ : SecondCountableTopology E
fβ : β Γ β β E
sβ : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
A B : Type
instβΒ³ : TopologicalSpace A
instβΒ² : TopologicalSpace B
instβΒΉ : Preorder B
instβ : OrderClosedTopology B
s : Set A
f g : A β B
sc : IsClosed s
fc : ContinuousOn f s
gc : ContinuousOn g s
t : Set (B Γ B) := {p | p.1 β€ p.2}
fg : A β B Γ B := fun x => (f x, g x)
x : A
β’ x β {x | f x β€ g x} β x β fg β»ΒΉ' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | set re := min r e | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have esub : closedBall c0 re β closedBall c0 r :=
Metric.closedBall_subset_closedBall (min_le_left _ _) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | generalize hS : (fun b : β β¦
{z0 | z0 β closedBall c0 re β§ β z1, z1 β closedBall c1 r β βf (z0, z1)β β€ b}) = S | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have hU : (interior (β b, S b)).Nonempty := by
refine Set.nonempty_of_mem
(mem_interior.mpr β¨ball c0 re, ?_, Metric.isOpen_ball, Metric.mem_ball_self (lt_min rp ep)β©)
rw [Set.subset_def]; intro z0 z0s; rw [Set.mem_iUnion]
have z0s' := esub (mem_open_closed z0s)
rcases (isCompact_closedBall _ _).bddAbove_image (h.on1 z0s').continuousOn.norm with β¨b, fbβ©
simp only [mem_upperBounds, Set.forall_mem_image] at fb
use Nat.ceil b; rw [β hS]; simp only [Set.mem_setOf]
refine β¨mem_open_closed z0s, ?_β©
simp only [Metric.mem_closedBall] at fb β’; intro z1 z1r
exact _root_.trans (fb z1r) (Nat.le_ceil _) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rcases NonemptyInterior.nonempty_interior_of_iUnion_of_closed hc hU with β¨b, biβ© | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | case intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
bi : (interior (S b)).Nonempty
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rcases bi with β¨c0', c0'sβ© | case intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
bi : (interior (S b)).Nonempty
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | case intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
bi : (interior (S b)).Nonempty
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | use c0' | case intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
β’ β c0', β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rcases mem_interior.mp c0's with β¨s', s's, so, c0s'β© | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | case h.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rcases Metric.isOpen_iff.mp so c0' c0s' with β¨t, tp, tsβ© | case h.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have tr : ball c0' t β closedBall c0 re := by
rw [Set.subset_def]; intro z0 z0t
have z0b := _root_.trans ts s's z0t
rw [β hS] at z0b; simp only [Set.setOf_and, Set.setOf_mem_eq, Set.mem_inter_iff] at z0b
exact z0b.left | case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have c0e : c0' β closedBall c0 e :=
_root_.trans tr (Metric.closedBall_subset_closedBall (min_le_right _ _))
(Metric.mem_ball_self tp) | case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have fb : β z, z β ball c0' t ΓΛ’ ball c1 r β βf zβ β€ b := by
intro z zs; rw [Set.mem_prod] at zs
have zb := _root_.trans ts s's zs.left
rw [β hS] at zb
simp only [Metric.mem_ball, Metric.mem_closedBall, le_min_iff, Set.mem_setOf_eq] at zb zs
have zb' := zb.right z.snd zs.right.le
simp only [Prod.mk.eta] at zb'; exact zb' | case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | use t, tp, c0e | case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ β t > 0, c0' β closedBall c0 e β§ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | refine of_bounded (h.mono ?_) (IsOpen.prod isOpen_ball isOpen_ball) fb | case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r) | case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ ball c0' t ΓΛ’ ball c1 r β closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ AnalyticOn β f (ball c0' t ΓΛ’ ball c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [β closedBall_prod_same] | case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ ball c0' t ΓΛ’ ball c1 r β closedBall (c0, c1) r | case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ ball c0' t ΓΛ’ ball c1 r β closedBall c0 r ΓΛ’ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ ball c0' t ΓΛ’ ball c1 r β closedBall (c0, c1) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | exact Set.prod_mono (_root_.trans tr esub) Metric.ball_subset_closedBall | case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ ball c0' t ΓΛ’ ball c1 r β closedBall c0 r ΓΛ’ closedBall c1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
fb : β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
β’ ball c0' t ΓΛ’ ball c1 r β closedBall c0 r ΓΛ’ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro b | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
β’ β (b : β), IsClosed (S b) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed (S b) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
β’ β (b : β), IsClosed (S b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [β hS] | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed (S b) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed ((fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) b) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed (S b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [β forall_const_and_distrib] | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed ((fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) b) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed {z0 | β (x : β), z0 β closedBall c0 re β§ (x β closedBall c1 r β βf (z0, x)β β€ βb)} | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed ((fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.setOf_forall] | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed {z0 | β (x : β), z0 β closedBall c0 re β§ (x β closedBall c1 r β βf (z0, x)β β€ βb)} | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed (β i, {x | x β closedBall c0 re β§ (i β closedBall c1 r β βf (x, i)β β€ βb)}) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed {z0 | β (x : β), z0 β closedBall c0 re β§ (x β closedBall c1 r β βf (z0, x)β β€ βb)}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | apply isClosed_iInter | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed (β i, {x | x β closedBall c0 re β§ (i β closedBall c1 r β βf (x, i)β β€ βb)}) | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ β (i : β), IsClosed {x | x β closedBall c0 re β§ (i β closedBall c1 r β βf (x, i)β β€ βb)} | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ IsClosed (β i, {x | x β closedBall c0 re β§ (i β closedBall c1 r β βf (x, i)β β€ βb)})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro z1 | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ β (i : β), IsClosed {x | x β closedBall c0 re β§ (i β closedBall c1 r β βf (x, i)β β€ βb)} | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
β’ β (i : β), IsClosed {x | x β closedBall c0 re β§ (i β closedBall c1 r β βf (x, i)β β€ βb)}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | by_cases z1r : z1 β closedBall c1 r | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)} | case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)}
case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : Β¬z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [z1r, false_imp_iff, and_true_iff, Set.setOf_mem_eq, Metric.isClosed_ball] | case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.not_not_mem] at z1r | case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : Β¬z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)} | case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)} | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : Β¬z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [z1r, true_imp_iff] | case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)} | case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ βf (x, z1)β β€ βb} | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ (z1 β closedBall c1 r β βf (x, z1)β β€ βb)}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | refine ContinuousOn.isClosed_le Metric.isClosed_ball ?_ continuousOn_const | case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ βf (x, z1)β β€ βb} | case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ ContinuousOn (fun x => βf (x, z1)β) (closedBall c0 re) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ IsClosed {x | x β closedBall c0 re β§ βf (x, z1)β β€ βb}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | apply ContinuousOn.norm | case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ ContinuousOn (fun x => βf (x, z1)β) (closedBall c0 re) | case neg.h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ ContinuousOn (fun x => f (x, z1)) (closedBall c0 re) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ ContinuousOn (fun x => βf (x, z1)β) (closedBall c0 re)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | exact ContinuousOn.mono (h.on0 z1r).continuousOn esub | case neg.h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ ContinuousOn (fun x => f (x, z1)) (closedBall c0 re) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
b : β
z1 : β
z1r : z1 β closedBall c1 r
β’ ContinuousOn (fun x => f (x, z1)) (closedBall c0 re)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | refine Set.nonempty_of_mem
(mem_interior.mpr β¨ball c0 re, ?_, Metric.isOpen_ball, Metric.mem_ball_self (lt_min rp ep)β©) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
β’ (interior (β b, S b)).Nonempty | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
β’ ball c0 re β β b, S b | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
β’ (interior (β b, S b)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.subset_def] | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
β’ ball c0 re β β b, S b | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
β’ β x β ball c0 re, x β β b, S b | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
β’ ball c0 re β β b, S b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro z0 z0s | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
β’ β x β ball c0 re, x β β b, S b | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
β’ z0 β β b, S b | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0 z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
β’ β x β ball c0 re, x β β b, S b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.mem_iUnion] | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
β’ z0 β β b, S b | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
β’ β i, z0 β S i | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
β’ z0 β β b, S b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have z0s' := esub (mem_open_closed z0s) | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
β’ β i, z0 β S i | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
β’ β i, z0 β S i | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
β’ β i, z0 β S i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rcases (isCompact_closedBall _ _).bddAbove_image (h.on1 z0s').continuousOn.norm with β¨b, fbβ© | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
β’ β i, z0 β S i | case intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : b β upperBounds ((fun x => βf (z0, x)β) '' closedBall c1 r)
β’ β i, z0 β S i | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 b e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
β’ β i, z0 β S i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [mem_upperBounds, Set.forall_mem_image] at fb | case intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : b β upperBounds ((fun x => βf (z0, x)β) '' closedBall c1 r)
β’ β i, z0 β S i | case intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ β i, z0 β S i | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : b β upperBounds ((fun x => βf (z0, x)β) '' closedBall c1 r)
β’ β i, z0 β S i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | use Nat.ceil b | case intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ β i, z0 β S i | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ z0 β S βbββ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ β i, z0 β S i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [β hS] | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ z0 β S βbββ | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ z0 β (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) βbββ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ z0 β S βbββ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [Set.mem_setOf] | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ z0 β (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) βbββ | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ ββbββ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ z0 β (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) βbββ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | refine β¨mem_open_closed z0s, ?_β© | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ ββbββ | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ β z1 β closedBall c1 r, βf (z0, z1)β β€ ββbββ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ ββbββ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [Metric.mem_closedBall] at fb β’ | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ β z1 β closedBall c1 r, βf (z0, z1)β β€ ββbββ | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, dist x c1 β€ r β βf (z0, x)β β€ b
β’ β (z1 : β), dist z1 c1 β€ r β βf (z0, z1)β β€ ββbββ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, x β closedBall c1 r β βf (z0, x)β β€ b
β’ β z1 β closedBall c1 r, βf (z0, z1)β β€ ββbββ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro z1 z1r | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, dist x c1 β€ r β βf (z0, x)β β€ b
β’ β (z1 : β), dist z1 c1 β€ r β βf (z0, z1)β β€ ββbββ | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, dist x c1 β€ r β βf (z0, x)β β€ b
z1 : β
z1r : dist z1 c1 β€ r
β’ βf (z0, z1)β β€ ββbββ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, dist x c1 β€ r β βf (z0, x)β β€ b
β’ β (z1 : β), dist z1 c1 β€ r β βf (z0, z1)β β€ ββbββ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | exact _root_.trans (fb z1r) (Nat.le_ceil _) | case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, dist x c1 β€ r β βf (z0, x)β β€ b
z1 : β
z1r : dist z1 c1 β€ r
β’ βf (z0, z1)β β€ ββbββ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0' c1 z0β z1β w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
z0 : β
z0s : z0 β ball c0 re
z0s' : z0 β closedBall c0 r
b : β
fb : β β¦x : ββ¦, dist x c1 β€ r β βf (z0, x)β β€ b
z1 : β
z1r : dist z1 c1 β€ r
β’ βf (z0, z1)β β€ ββbββ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.subset_def] | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
β’ ball c0' t β closedBall c0 re | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
β’ β x β ball c0' t, x β closedBall c0 re | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
β’ ball c0' t β closedBall c0 re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro z0 z0t | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
β’ β x β ball c0' t, x β closedBall c0 re | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
β’ z0 β closedBall c0 re | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
β’ β x β ball c0' t, x β closedBall c0 re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have z0b := _root_.trans ts s's z0t | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
β’ z0 β closedBall c0 re | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
z0b : z0 β S b
β’ z0 β closedBall c0 re | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
β’ z0 β closedBall c0 re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [β hS] at z0b | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
z0b : z0 β S b
β’ z0 β closedBall c0 re | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
z0b : z0 β (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) b
β’ z0 β closedBall c0 re | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
z0b : z0 β S b
β’ z0 β closedBall c0 re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [Set.setOf_and, Set.setOf_mem_eq, Set.mem_inter_iff] at z0b | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
z0b : z0 β (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) b
β’ z0 β closedBall c0 re | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
z0b : z0 β closedBall c0 re β§ z0 β {a | β z1 β closedBall c1 r, βf (a, z1)β β€ βb}
β’ z0 β closedBall c0 re | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
z0b : z0 β (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) b
β’ z0 β closedBall c0 re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | exact z0b.left | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
z0b : z0 β closedBall c0 re β§ z0 β {a | β z1 β closedBall c1 r, βf (a, z1)β β€ βb}
β’ z0 β closedBall c0 re | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0β z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
z0 : β
z0t : z0 β ball c0' t
z0b : z0 β closedBall c0 re β§ z0 β {a | β z1 β closedBall c1 r, βf (a, z1)β β€ βb}
β’ z0 β closedBall c0 re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro z zs | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
β’ β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z β ball c0' t ΓΛ’ ball c1 r
β’ βf zβ β€ βb | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
β’ β z β ball c0' t ΓΛ’ ball c1 r, βf zβ β€ βb
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.mem_prod] at zs | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z β ball c0' t ΓΛ’ ball c1 r
β’ βf zβ β€ βb | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z.1 β ball c0' t β§ z.2 β ball c1 r
β’ βf zβ β€ βb | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z β ball c0' t ΓΛ’ ball c1 r
β’ βf zβ β€ βb
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have zb := _root_.trans ts s's zs.left | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z.1 β ball c0' t β§ z.2 β ball c1 r
β’ βf zβ β€ βb | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z.1 β ball c0' t β§ z.2 β ball c1 r
zb : z.1 β S b
β’ βf zβ β€ βb | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z.1 β ball c0' t β§ z.2 β ball c1 r
β’ βf zβ β€ βb
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [β hS] at zb | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z.1 β ball c0' t β§ z.2 β ball c1 r
zb : z.1 β S b
β’ βf zβ β€ βb | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z.1 β ball c0' t β§ z.2 β ball c1 r
zb : z.1 β (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) b
β’ βf zβ β€ βb | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z.1 β ball c0' t β§ z.2 β ball c1 r
zb : z.1 β S b
β’ βf zβ β€ βb
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [Metric.mem_ball, Metric.mem_closedBall, le_min_iff, Set.mem_setOf_eq] at zb zs | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z.1 β ball c0' t β§ z.2 β ball c1 r
zb : z.1 β (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) b
β’ βf zβ β€ βb | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zb : dist z.1 c0 β€ re β§ β (z1 : β), dist z1 c1 β€ r β βf (z.1, z1)β β€ βb
zs : dist z.1 c0' < t β§ dist z.2 c1 < r
β’ βf zβ β€ βb | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zs : z.1 β ball c0' t β§ z.2 β ball c1 r
zb : z.1 β (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) b
β’ βf zβ β€ βb
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have zb' := zb.right z.snd zs.right.le | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zb : dist z.1 c0 β€ re β§ β (z1 : β), dist z1 c1 β€ r β βf (z.1, z1)β β€ βb
zs : dist z.1 c0' < t β§ dist z.2 c1 < r
β’ βf zβ β€ βb | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zb : dist z.1 c0 β€ re β§ β (z1 : β), dist z1 c1 β€ r β βf (z.1, z1)β β€ βb
zs : dist z.1 c0' < t β§ dist z.2 c1 < r
zb' : βf (z.1, z.2)β β€ βb
β’ βf zβ β€ βb | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zb : dist z.1 c0 β€ re β§ β (z1 : β), dist z1 c1 β€ r β βf (z.1, z1)β β€ βb
zs : dist z.1 c0' < t β§ dist z.2 c1 < r
β’ βf zβ β€ βb
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [Prod.mk.eta] at zb' | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zb : dist z.1 c0 β€ re β§ β (z1 : β), dist z1 c1 β€ r β βf (z.1, z1)β β€ βb
zs : dist z.1 c0' < t β§ dist z.2 c1 < r
zb' : βf (z.1, z.2)β β€ βb
β’ βf zβ β€ βb | E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zb : dist z.1 c0 β€ re β§ β (z1 : β), dist z1 c1 β€ r β βf (z.1, z1)β β€ βb
zs : dist z.1 c0' < t β§ dist z.2 c1 < r
zb' : βf zβ β€ βb
β’ βf zβ β€ βb | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ³ : NormedAddCommGroup E
instβΒ² : NormedSpace β E
instβΒΉ : CompleteSpace E
instβ : SecondCountableTopology E
f : β Γ β β E
s : Set (β Γ β)
c0 c0'β c1 z0 z1 w0 w1 : β
r r0 r1 bβ e : β
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : β := min r e
esub : closedBall c0 re β closedBall c0 r
S : β β Set β
hS : (fun b => {z0 | z0 β closedBall c0 re β§ β z1 β closedBall c1 r, βf (z0, z1)β β€ βb}) = S
hc : β (b : β), IsClosed (S b)
hU : (interior (β b, S b)).Nonempty
b : β
c0' : β
c0's : c0' β interior (S b)
s' : Set β
s's : s' β S b
so : IsOpen s'
c0s' : c0' β s'
t : β
tp : t > 0
ts : ball c0' t β s'
tr : ball c0' t β closedBall c0 re
c0e : c0' β closedBall c0 e
z : β Γ β
zb : dist z.1 c0 β€ re β§ β (z1 : β), dist z1 c1 β€ r β βf (z.1, z1)β β€ βb
zs : dist z.1 c0' < t β§ dist z.2 c1 < r
zb' : βf (z.1, z.2)β β€ βb
β’ βf zβ β€ βb
TACTIC:
|
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