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/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | intro e ep | case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
β’ β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < Ξ΅ | case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
β’ β Ξ΅ > 0, β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < Ξ΅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | generalize d4 : (1 - a) * (e / 4) = d | case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have dp : d > 0 := by rw [β d4]; bound | case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rcases Filter.eventually_atTop.mp (Metric.tendstoUniformlyOn_iff.mp u d dp) with β¨n, hn'β© | case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn' : β b β₯ n, β x β s, dist (g x) (f b x) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | set hn := hn' n | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn' : β b β₯ n, β x β s, dist (g x) (f b x) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn' : β b β₯ n, β x β s, dist (g x) (f b x) < d
hn : n β₯ n β β x β s, dist (g x) (f n x) < d := hn' n
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn' : β b β₯ n, β x β s, dist (g x) (f b x) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | simp at hn | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn' : β b β₯ n, β x β s, dist (g x) (f b x) < d
hn : n β₯ n β β x β s, dist (g x) (f n x) < d := hn' n
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn' : β b β₯ n, β x β s, dist (g x) (f b x) < d
hn : β x β s, dist (g x) (f n x) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn' : β b β₯ n, β x β s, dist (g x) (f b x) < d
hn : n β₯ n β β x β s, dist (g x) (f n x) < d := hn' n
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | clear hn' u | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn' : β b β₯ n, β x β s, dist (g x) (f b x) < d
hn : β x β s, dist (g x) (f n x) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn' : β b β₯ n, β x β s, dist (g x) (f b x) < d
hn : β x β s, dist (g x) (f n x) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have dfg : dist (f n (c + y)) (g (c + y)) β€ d := by
apply le_of_lt; rw [dist_comm]
refine hn (c + y) ?_
apply cb
simp; exact yr.le | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | set hs := (hpf n).hasSum yb | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
hs : HasSum (fun n_1 => (pr n n_1) fun x => y) (f n (c + y)) := (hpf n).hasSum yb
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [HasSum, Metric.tendsto_atTop] at hs | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
hs : HasSum (fun n_1 => (pr n n_1) fun x => y) (f n (c + y)) := (hpf n).hasSum yb
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
hs : β Ξ΅ > 0, β N, β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < Ξ΅
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
hs : HasSum (fun n_1 => (pr n n_1) fun x => y) (f n (c + y)) := (hpf n).hasSum yb
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rcases hs d dp with β¨N, NMβ© | case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
hs : β Ξ΅ > 0, β N, β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < Ξ΅
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
hs : β Ξ΅ > 0, β N, β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < Ξ΅
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
hs : β Ξ΅ > 0, β N, β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < Ξ΅
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | clear hs | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
hs : β Ξ΅ > 0, β N, β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < Ξ΅
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
hs : β Ξ΅ > 0, β N, β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < Ξ΅
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | exists N | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
β’ β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
β’ β N, β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | intro M NlM | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
β’ β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
M : Finset β
NlM : M β₯ N
β’ dist (M.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
β’ β n β₯ N, dist (n.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have dpf := (NM M NlM).le | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
M : Finset β
NlM : M β₯ N
β’ dist (M.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
M : Finset β
NlM : M β₯ N
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ dist (M.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
M : Finset β
NlM : M β₯ N
β’ dist (M.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | clear NM NlM N yb | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
M : Finset β
NlM : M β₯ N
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ dist (M.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ dist (M.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
N : Finset β
NM : β n_1 β₯ N, dist (n_1.sum fun b => (pr n b) fun x => y) (f n (c + y)) < d
M : Finset β
NlM : M β₯ N
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ dist (M.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | generalize hMp : M.sum (fun k : β β¦ p k fun _ β¦ y) = Mp | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
dppr : dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) β€ e / 4
β’ dist (M.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
dppr : dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) β€ e / 4
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
β’ dist Mp (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
dppr : dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) β€ e / 4
β’ dist (M.sum fun b => (cauchyPowerSeries g c (βr) b) fun x => y) (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [hMp] at dppr | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
dppr : dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) β€ e / 4
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
β’ dist Mp (g (c + y)) < e | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
Mp : β
dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) β€ e / 4
hMp : (M.sum fun k => (p k) fun x => y) = Mp
β’ dist Mp (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
dppr : dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) β€ e / 4
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
β’ dist Mp (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | generalize hMpr : M.sum (fun k β¦ pr n k fun _ β¦ y) = Mpr | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
Mp : β
dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) β€ e / 4
hMp : (M.sum fun k => (p k) fun x => y) = Mp
β’ dist Mp (g (c + y)) < e | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
Mp : β
dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) β€ e / 4
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ dist Mp (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
Mp : β
dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) β€ e / 4
hMp : (M.sum fun k => (p k) fun x => y) = Mp
β’ dist Mp (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [hMpr] at dpf dppr | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
Mp : β
dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) β€ e / 4
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ dist Mp (g (c + y)) < e | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ dist Mp (g (c + y)) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
Mp : β
dppr : dist Mp (M.sum fun k => (pr n k) fun x => y) β€ e / 4
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ dist Mp (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | calc dist Mp (g (c + y))
_ β€ dist Mp (f n (c + y)) + dist (f n (c + y)) (g (c + y)) := dist_triangle _ _ _
_ β€ dist Mp Mpr + dist Mpr (f n (c + y)) + d := by bound
_ β€ e / 4 + d + d := by bound
_ = e / 4 + 2 * (1 - a) * (e / 4) := by rw [β d4]; ring
_ β€ e / 4 + 2 * (1 - 0) * (e / 4) := by bound
_ = 3 / 4 * e := by ring
_ < 1 * e := (mul_lt_mul_of_pos_right (by norm_num) ep)
_ = e := by simp | case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ dist Mp (g (c + y)) < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ dist Mp (g (c + y)) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | intro n | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
β’ β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
β’ β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have cs := cauchy_on_cball_radius rp (hb n) | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c βr) c βr
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have pn : pr n = cauchyPowerSeries (f n) c r := rfl | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c βr) c βr
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c βr) c βr
pn : pr n = cauchyPowerSeries (f n) c βr
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c βr) c βr
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [β pn] at cs | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c βr) c βr
pn : pr n = cauchyPowerSeries (f n) c βr
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
cs : HasFPowerSeriesOnBall (f n) (pr n) c βr
pn : pr n = cauchyPowerSeries (f n) c βr
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
cs : HasFPowerSeriesOnBall (f n) (cauchyPowerSeries (f n) c βr) c βr
pn : pr n = cauchyPowerSeries (f n) c βr
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | exact cs | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
cs : HasFPowerSeriesOnBall (f n) (pr n) c βr
pn : pr n = cauchyPowerSeries (f n) c βr
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
o : IsOpen s
h : β (n : I), AnalyticOn β (f n) s
u : TendstoUniformlyOn f g atTop s
c : β
hc : c β s
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
hb : β (n : I), AnalyticOn β (f n) (closedBall c βr)
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
n : I
cs : HasFPowerSeriesOnBall (f n) (pr n) c βr
pn : pr n = cauchyPowerSeries (f n) c βr
β’ HasFPowerSeriesOnBall (f n) (pr n) c βr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | bound | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
β’ a β₯ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
β’ a β₯ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | bound | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
β’ 1 - a > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
β’ 1 - a > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [β d4] | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
β’ d > 0 | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
β’ (1 - a) * (e / 4) > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
β’ d > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | bound | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
β’ (1 - a) * (e / 4) > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
u : TendstoUniformlyOn f g atTop s
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
β’ (1 - a) * (e / 4) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | apply le_of_lt | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ dist (f n (c + y)) (g (c + y)) β€ d | case a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ dist (f n (c + y)) (g (c + y)) < d | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ dist (f n (c + y)) (g (c + y)) β€ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [dist_comm] | case a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ dist (f n (c + y)) (g (c + y)) < d | case a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ dist (g (c + y)) (f n (c + y)) < d | Please generate a tactic in lean4 to solve the state.
STATE:
case a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ dist (f n (c + y)) (g (c + y)) < d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | refine hn (c + y) ?_ | case a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ dist (g (c + y)) (f n (c + y)) < d | case a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ c + y β s | Please generate a tactic in lean4 to solve the state.
STATE:
case a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ dist (g (c + y)) (f n (c + y)) < d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | apply cb | case a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ c + y β s | case a.a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ c + y β closedBall c βr | Please generate a tactic in lean4 to solve the state.
STATE:
case a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ c + y β s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | simp | case a.a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ c + y β closedBall c βr | case a.a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ Complex.abs y β€ βr | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ c + y β closedBall c βr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | exact yr.le | case a.a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ Complex.abs y β€ βr | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yb : y β EMetric.ball 0 βr
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
β’ Complex.abs y β€ βr
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | trans M.sum fun k : β β¦ dist (p k fun _ β¦ y) (pr n k fun _ β¦ y) | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) β€ e / 4 | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) β€
M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) β€ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) β€ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | apply dist_sum_sum_le M (fun k : β β¦ p k fun _ β¦ y) fun k : β β¦ pr n k fun _ β¦ y | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) β€
M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) β€ e / 4 | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) β€ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ dist (M.sum fun k => (p k) fun x => y) (M.sum fun k => (pr n k) fun x => y) β€
M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) β€ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | trans M.sum fun k β¦ a ^ k * d | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) β€ e / 4 | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) β€ M.sum fun k => a ^ k * d
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => a ^ k * d) β€ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) β€ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | apply Finset.sum_le_sum | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) β€ M.sum fun k => a ^ k * d | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ β i β M, dist ((p i) fun x => y) ((pr n i) fun x => y) β€ a ^ i * d | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => dist ((p k) fun x => y) ((pr n k) fun x => y)) β€ M.sum fun k => a ^ k * d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | intro k _ | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ β i β M, dist ((p i) fun x => y) ((pr n i) fun x => y) β€ a ^ i * d | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ a ^ k * d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ β i β M, dist ((p i) fun x => y) ((pr n i) fun x => y) β€ a ^ i * d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have hak : a ^ k = abs y ^ k * rβ»ΒΉ ^ k := by
calc (abs y / r) ^ k
_ = (abs y * rβ»ΒΉ) ^ k := by rw [div_eq_mul_inv, NNReal.coe_inv]
_ = abs y ^ k * rβ»ΒΉ ^ k := mul_pow _ _ _ | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ a ^ k * d | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ a ^ k * d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ a ^ k * d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [hak] | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ a ^ k * d | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ a ^ k * d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | generalize hd' : d.toNNReal = d' | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have dd : (d' : β) = d := by rw [β hd']; exact Real.coe_toNNReal d dp.le | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have hcb : β z, z β closedBall c r β abs (g z - f n z) β€ d' := by
intro z zb; exact _root_.trans (hn z (cb zb)).le (le_of_eq dd.symm) | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
hcb : β z β closedBall c βr, Complex.abs (g z - f n z) β€ βd'
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | exact _root_.trans (cauchy_dist k y rp cg (cf n) hcb)
(mul_le_mul_of_nonneg_left (le_of_eq dd) (by bound)) | case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
hcb : β z β closedBall c βr, Complex.abs (g z - f n z) β€ βd'
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
hcb : β z β closedBall c βr, Complex.abs (g z - f n z) β€ βd'
β’ dist ((p k) fun x => y) ((pr n k) fun x => y) β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k * d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | calc (abs y / r) ^ k
_ = (abs y * rβ»ΒΉ) ^ k := by rw [div_eq_mul_inv, NNReal.coe_inv]
_ = abs y ^ k * rβ»ΒΉ ^ k := mul_pow _ _ _ | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
β’ a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
β’ a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [div_eq_mul_inv, NNReal.coe_inv] | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
β’ (Complex.abs y / βr) ^ k = (Complex.abs y * βrβ»ΒΉ) ^ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
β’ (Complex.abs y / βr) ^ k = (Complex.abs y * βrβ»ΒΉ) ^ k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [β hd'] | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
β’ βd' = d | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
β’ βd.toNNReal = d | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
β’ βd' = d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | exact Real.coe_toNNReal d dp.le | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
β’ βd.toNNReal = d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
β’ βd.toNNReal = d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | intro z zb | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
β’ β z β closedBall c βr, Complex.abs (g z - f n z) β€ βd' | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
z : β
zb : z β closedBall c βr
β’ Complex.abs (g z - f n z) β€ βd' | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
β’ β z β closedBall c βr, Complex.abs (g z - f n z) β€ βd'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | exact _root_.trans (hn z (cb zb)).le (le_of_eq dd.symm) | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
z : β
zb : z β closedBall c βr
β’ Complex.abs (g z - f n z) β€ βd' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
z : β
zb : z β closedBall c βr
β’ Complex.abs (g z - f n z) β€ βd'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | bound | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
hcb : β z β closedBall c βr, Complex.abs (g z - f n z) β€ βd'
β’ 0 β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
k : β
aβ : k β M
hak : a ^ k = Complex.abs y ^ k * βrβ»ΒΉ ^ k
d' : ββ₯0
hd' : d.toNNReal = d'
dd : βd' = d
hcb : β z β closedBall c βr, Complex.abs (g z - f n z) β€ βd'
β’ 0 β€ Complex.abs y ^ k * βrβ»ΒΉ ^ k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | have pgb : (M.sum fun k β¦ a ^ k) β€ (1 - a)β»ΒΉ := partial_geometric_bound M a0 a1 | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => a ^ k * d) β€ e / 4 | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (M.sum fun k => a ^ k * d) β€ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
β’ (M.sum fun k => a ^ k * d) β€ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | calc
(M.sum fun k β¦ a ^ k * d) = (M.sum fun k β¦ a ^ k) * d := by rw [β Finset.sum_mul]
_ β€ (1 - a)β»ΒΉ * d := by bound
_ = (1 - a)β»ΒΉ * ((1 - a) * (e / 4)) := by rw [β d4]
_ = (1 - a) * (1 - a)β»ΒΉ * (e / 4) := by ring
_ = 1 * (e / 4) := by rw [mul_inv_cancel a1p.ne']
_ = e / 4 := by ring | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (M.sum fun k => a ^ k * d) β€ e / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (M.sum fun k => a ^ k * d) β€ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [β Finset.sum_mul] | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (M.sum fun k => a ^ k * d) = (M.sum fun k => a ^ k) * d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (M.sum fun k => a ^ k * d) = (M.sum fun k => a ^ k) * d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | bound | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (M.sum fun k => a ^ k) * d β€ (1 - a)β»ΒΉ * d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (M.sum fun k => a ^ k) * d β€ (1 - a)β»ΒΉ * d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [β d4] | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (1 - a)β»ΒΉ * d = (1 - a)β»ΒΉ * ((1 - a) * (e / 4)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (1 - a)β»ΒΉ * d = (1 - a)β»ΒΉ * ((1 - a) * (e / 4))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | ring | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (1 - a)β»ΒΉ * ((1 - a) * (e / 4)) = (1 - a) * (1 - a)β»ΒΉ * (e / 4) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (1 - a)β»ΒΉ * ((1 - a) * (e / 4)) = (1 - a) * (1 - a)β»ΒΉ * (e / 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [mul_inv_cancel a1p.ne'] | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (1 - a) * (1 - a)β»ΒΉ * (e / 4) = 1 * (e / 4) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ (1 - a) * (1 - a)β»ΒΉ * (e / 4) = 1 * (e / 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | ring | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ 1 * (e / 4) = e / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
dpf : dist (M.sum fun b => (pr n b) fun x => y) (f n (c + y)) β€ d
pgb : (M.sum fun k => a ^ k) β€ (1 - a)β»ΒΉ
β’ 1 * (e / 4) = e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | bound | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ dist Mp (f n (c + y)) + dist (f n (c + y)) (g (c + y)) β€ dist Mp Mpr + dist Mpr (f n (c + y)) + d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ dist Mp (f n (c + y)) + dist (f n (c + y)) (g (c + y)) β€ dist Mp Mpr + dist Mpr (f n (c + y)) + d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | bound | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ dist Mp Mpr + dist Mpr (f n (c + y)) + d β€ e / 4 + d + d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ dist Mp Mpr + dist Mpr (f n (c + y)) + d β€ e / 4 + d + d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | rw [β d4] | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ e / 4 + d + d = e / 4 + 2 * (1 - a) * (e / 4) | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ e / 4 + (1 - a) * (e / 4) + (1 - a) * (e / 4) = e / 4 + 2 * (1 - a) * (e / 4) | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ e / 4 + d + d = e / 4 + 2 * (1 - a) * (e / 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | ring | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ e / 4 + (1 - a) * (e / 4) + (1 - a) * (e / 4) = e / 4 + 2 * (1 - a) * (e / 4) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ e / 4 + (1 - a) * (e / 4) + (1 - a) * (e / 4) = e / 4 + 2 * (1 - a) * (e / 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | bound | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ e / 4 + 2 * (1 - a) * (e / 4) β€ e / 4 + 2 * (1 - 0) * (e / 4) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ e / 4 + 2 * (1 - a) * (e / 4) β€ e / 4 + 2 * (1 - 0) * (e / 4)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | ring | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ e / 4 + 2 * (1 - 0) * (e / 4) = 3 / 4 * e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ e / 4 + 2 * (1 - 0) * (e / 4) = 3 / 4 * e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | norm_num | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ 3 / 4 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ 3 / 4 < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Uniform.lean | uniform_analytic_lim | [189, 1] | [270, 21] | simp | I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ 1 * e = e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
I : Type
instβΒΉ : Lattice I
instβ : Nonempty I
f : I β β β β
g : β β β
s : Set β
c : β
r : ββ₯0
cb : closedBall c βr β s
rp : 0 < r
pr : I β FormalMultilinearSeries β β β := fun n => cauchyPowerSeries (f n) c βr
hpf : β (n : I), HasFPowerSeriesOnBall (f n) (pr n) c βr
cf : β (n : I), ContinuousOn (f n) (closedBall c βr)
cg : ContinuousOn g (closedBall c βr)
p : FormalMultilinearSeries β β β := cauchyPowerSeries g c βr
y : β
yr : Complex.abs y < βr
a : β := Complex.abs y / βr
a0 : a β₯ 0
a1 : a < 1
a1p : 1 - a > 0
e : β
ep : e > 0
d : β
d4 : (1 - a) * (e / 4) = d
dp : d > 0
n : I
hn : β x β s, dist (g x) (f n x) < d
dfg : dist (f n (c + y)) (g (c + y)) β€ d
M : Finset β
Mp : β
hMp : (M.sum fun k => (p k) fun x => y) = Mp
Mpr : β
dppr : dist Mp Mpr β€ e / 4
dpf : dist Mpr (f n (c + y)) β€ d
hMpr : (M.sum fun k => (pr n k) fun x => y) = Mpr
β’ 1 * e = e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | generalize hc : (fun n β¦ Classical.choose ((sc.bddAbove_image (fc n).norm).exists_ge 0)) = c | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | have cs : β n, 0 β€ c n β§ β x, x β s β βf n xβ β€ c n := fun n β¦ by
simpa only [β hc, mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iffβ] using
Classical.choose_spec ((sc.bddAbove_image (fc n).norm).exists_ge 0) | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | rw [Metric.uniformCauchySeqOn_iff] at u | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | rcases u 1 (by norm_num) with β¨N, Hβ© | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | clear u | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | generalize hbs : Finset.image c (Finset.range (N + 1)) = bs | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | have c0 : c 0 β bs := by simp [β hbs]; exists 0; simp | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | generalize hb : 1 + bs.max' β¨_, c0β© = b | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | exists b | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β b, 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | constructor | case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b | case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b
case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β (n : β), β x β s, βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b β§ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | simpa only [β hc, mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iffβ] using
Classical.choose_spec ((sc.bddAbove_image (fc n).norm).exists_ge 0) | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
n : β
β’ 0 β€ c n β§ β x β s, βf n xβ β€ c n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : UniformCauchySeqOn f atTop s
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
n : β
β’ 0 β€ c n β§ β x β s, βf n xβ β€ c n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | norm_num | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ 1 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
u : β Ξ΅ > 0, β N, β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < Ξ΅
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
β’ 1 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | simp [β hbs] | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ c 0 β bs | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β a < N + 1, c a = c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ c 0 β bs
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | exists 0 | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β a < N + 1, c a = c 0 | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ 0 < N + 1 β§ c 0 = c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ β a < N + 1, c a = c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | simp | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ 0 < N + 1 β§ c 0 = c 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
β’ 0 < N + 1 β§ c 0 = c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | rw [β hb] | case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b | case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ 1 + bs.max' β― | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | exact add_nonneg (by norm_num) (_root_.trans (cs 0).1 (Finset.le_max' _ _ c0)) | case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ 1 + bs.max' β― | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ 1 + bs.max' β―
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | norm_num | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ 0 β€ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | intro n x xs | case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β (n : β), β x β s, βf n xβ β€ b | case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
β’ βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
β’ β (n : β), β x β s, βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | by_cases nN : n β€ N | case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
β’ βf n xβ β€ b | case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ βf n xβ β€ b
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : Β¬n β€ N
β’ βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | have cn : c n β bs := by simp [β hbs]; exists n; simp [Nat.lt_add_one_iff.mpr nN] | case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ βf n xβ β€ b | case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
cn : c n β bs
β’ βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | exact _root_.trans ((cs n).2 x xs) (_root_.trans (Finset.le_max' _ _ cn)
(by simp only [le_add_iff_nonneg_left, zero_le_one, β hb])) | case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
cn : c n β bs
β’ βf n xβ β€ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
cn : c n β bs
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | simp [β hbs] | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ c n β bs | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ β a < N + 1, c a = c n | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ c n β bs
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | exists n | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ β a < N + 1, c a = c n | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ n < N + 1 β§ c n = c n | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ β a < N + 1, c a = c n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | simp [Nat.lt_add_one_iff.mpr nN] | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ n < N + 1 β§ c n = c n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
β’ n < N + 1 β§ c n = c n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | simp only [le_add_iff_nonneg_left, zero_le_one, β hb] | X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
cn : c n β bs
β’ bs.max' β― β€ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : n β€ N
cn : c n β bs
β’ bs.max' β― β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | simp at nN | case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : Β¬n β€ N
β’ βf n xβ β€ b | case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
β’ βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : Β¬n β€ N
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | specialize H N le_rfl n nN.le x xs | case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
β’ βf n xβ β€ b | case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
H : β m β₯ N, β n β₯ N, β x β s, dist (f m x) (f n x) < 1
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | have cN : c N β bs := by simp [β hbs]; exists N; simp | case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ βf n xβ β€ b | case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
cN : c N β bs
β’ βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
β’ βf n xβ β€ b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | have bN := _root_.trans ((cs N).2 x xs) (Finset.le_max' _ _ cN) | case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
cN : c N β bs
β’ βf n xβ β€ b | case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
cN : c N β bs
bN : βf N xβ β€ bs.max' β―
β’ βf n xβ β€ b | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X Y : Type
instβΒΉ : TopologicalSpace X
instβ : NormedAddCommGroup Y
f : β β X β Y
s : Set X
fc : β (n : β), ContinuousOn (f n) s
sc : IsCompact s
c : β β β
hc : (fun n => Classical.choose β―) = c
cs : β (n : β), 0 β€ c n β§ β x β s, βf n xβ β€ c n
N : β
bs : Finset β
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 β bs
b : β
hb : 1 + bs.max' β― = b
n : β
x : X
xs : x β s
nN : N < n
H : dist (f N x) (f n x) < 1
cN : c N β bs
β’ βf n xβ β€ b
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.