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/AnalyticManifold/Nontrivial.lean | IsPreconnected.nontrivialAnalyticOn | [79, 1] | [88, 41] | have h' := (h.filter_mono (nhdsWithin_le_nhds (s := {z}αΆ))).frequently | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
β’ β x β s, β x_1 β s, f x = f x_1 | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
h' : βαΆ (x : β) in π[β ] z, f x = f z
β’ β x β s, β x_1 β s, f x = f x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
β’ β x β s, β x_1 β s, f x = f x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsPreconnected.nontrivialAnalyticOn | [79, 1] | [88, 41] | have e := fa.eqOn_of_preconnected_of_frequently_eq analyticOn_const p zs h' | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
h' : βαΆ (x : β) in π[β ] z, f x = f z
β’ β x β s, β x_1 β s, f x = f x_1 | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
h' : βαΆ (x : β) in π[β ] z, f x = f z
e : EqOn f (fun x => f z) s
β’ β x β s, β x_1 β s, f x = f x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
h' : βαΆ (x : β) in π[β ] z, f x = f z
β’ β x β s, β x_1 β s, f x = f x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsPreconnected.nontrivialAnalyticOn | [79, 1] | [88, 41] | intro x xs y ys | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
h' : βαΆ (x : β) in π[β ] z, f x = f z
e : EqOn f (fun x => f z) s
β’ β x β s, β x_1 β s, f x = f x_1 | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
h' : βαΆ (x : β) in π[β ] z, f x = f z
e : EqOn f (fun x => f z) s
x : β
xs : x β s
y : β
ys : y β s
β’ f x = f y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
h' : βαΆ (x : β) in π[β ] z, f x = f z
e : EqOn f (fun x => f z) s
β’ β x β s, β x_1 β s, f x = f x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsPreconnected.nontrivialAnalyticOn | [79, 1] | [88, 41] | rw [e xs, e ys] | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
h' : βαΆ (x : β) in π[β ] z, f x = f z
e : EqOn f (fun x => f z) s
x : β
xs : x β s
y : β
ys : y β s
β’ f x = f y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p : IsPreconnected s
fa : AnalyticOn β f s
z : β
zs : z β s
h : βαΆ (x : β) in π z, f x = f z
h' : βαΆ (x : β) in π[β ] z, f x = f z
e : EqOn f (fun x => f z) s
x : β
xs : x β s
y : β
ys : y β s
β’ f x = f y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | Entire.nontrivialAnalyticOn | [91, 1] | [93, 97] | refine isPreconnected_univ.nontrivialAnalyticOn fa ?_ | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
fa : AnalyticOn β f univ
ne : β a b, f a β f b
β’ NontrivialAnalyticOn f univ | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
fa : AnalyticOn β f univ
ne : β a b, f a β f b
β’ β a b, a β univ β§ b β univ β§ f a β f b | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
fa : AnalyticOn β f univ
ne : β a b, f a β f b
β’ NontrivialAnalyticOn f univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | Entire.nontrivialAnalyticOn | [91, 1] | [93, 97] | simpa only [Set.mem_univ, true_and_iff] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
fa : AnalyticOn β f univ
ne : β a b, f a β f b
β’ β a b, a β univ β§ b β univ β§ f a β f b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
fa : AnalyticOn β f univ
ne : β a b, f a β f b
β’ β a b, a β univ β§ b β univ β§ f a β f b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | rw [β singletons_open_iff_discrete] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a : β
β’ DiscreteTopology β(s β© f β»ΒΉ' {a}) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a : β
β’ β (a_1 : β(s β© f β»ΒΉ' {a})), IsOpen {a_1} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a : β
β’ DiscreteTopology β(s β© f β»ΒΉ' {a})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | intro β¨z, mβ© | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a : β
β’ β (a_1 : β(s β© f β»ΒΉ' {a})), IsOpen {a_1} | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
m : z β s β© f β»ΒΉ' {a}
β’ IsOpen {β¨z, mβ©} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a : β
β’ β (a_1 : β(s β© f β»ΒΉ' {a})), IsOpen {a_1}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff] at m | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
m : z β s β© f β»ΒΉ' {a}
β’ IsOpen {β¨z, mβ©} | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
β’ IsOpen {β¨z, mββ©} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
m : z β s β© f β»ΒΉ' {a}
β’ IsOpen {β¨z, mβ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | by_cases h : βαΆ z in π[{z}αΆ] z, f z = a | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
β’ IsOpen {β¨z, mββ©} | case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (z : β) in π[β ] z, f z = a
β’ IsOpen {β¨z, mββ©}
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : Β¬βαΆ (z : β) in π[β ] z, f z = a
β’ IsOpen {β¨z, mββ©} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
β’ IsOpen {β¨z, mββ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | have i := (n.isolated' m.1 a).and_frequently h | case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (z : β) in π[β ] z, f z = a
β’ IsOpen {β¨z, mββ©} | case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (z : β) in π[β ] z, f z = a
i : βαΆ (x : β) in π[β ] z, f x β a β§ f x = a
β’ IsOpen {β¨z, mββ©} | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (z : β) in π[β ] z, f z = a
β’ IsOpen {β¨z, mββ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | simp only [not_and_self_iff, Filter.frequently_const] at i | case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (z : β) in π[β ] z, f z = a
i : βαΆ (x : β) in π[β ] z, f x β a β§ f x = a
β’ IsOpen {β¨z, mββ©} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (z : β) in π[β ] z, f z = a
i : βαΆ (x : β) in π[β ] z, f x β a β§ f x = a
β’ IsOpen {β¨z, mββ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | simp only [Filter.not_frequently, eventually_nhdsWithin_iff, Set.mem_compl_singleton_iff] at h | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : Β¬βαΆ (z : β) in π[β ] z, f z = a
β’ IsOpen {β¨z, mββ©} | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
β’ IsOpen {β¨z, mββ©} | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : Β¬βαΆ (z : β) in π[β ] z, f z = a
β’ IsOpen {β¨z, mββ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | rcases eventually_nhds_iff.mp h with β¨t, t0, o, tzβ© | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
β’ IsOpen {β¨z, mββ©} | case neg.intro.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ IsOpen {β¨z, mββ©} | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
β’ IsOpen {β¨z, mββ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | simp only [isOpen_induced_iff] | case neg.intro.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ IsOpen {β¨z, mββ©} | case neg.intro.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ β t, IsOpen t β§ Subtype.val β»ΒΉ' t = {β¨z, mββ©} | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ IsOpen {β¨z, mββ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | use t, o | case neg.intro.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ β t, IsOpen t β§ Subtype.val β»ΒΉ' t = {β¨z, mββ©} | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ Subtype.val β»ΒΉ' t = {β¨z, mββ©} | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ β t, IsOpen t β§ Subtype.val β»ΒΉ' t = {β¨z, mββ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | apply Set.ext | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ Subtype.val β»ΒΉ' t = {β¨z, mββ©} | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ β (x : β(s β© f β»ΒΉ' {a})), x β Subtype.val β»ΒΉ' t β x β {β¨z, mββ©} | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ Subtype.val β»ΒΉ' t = {β¨z, mββ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | intro β¨w, mβ© | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ β (x : β(s β© f β»ΒΉ' {a})), x β Subtype.val β»ΒΉ' t β x β {β¨z, mββ©} | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
w : β
m : w β s β© f β»ΒΉ' {a}
β’ β¨w, mβ© β Subtype.val β»ΒΉ' t β β¨w, mβ© β {β¨z, mβΒΉβ©} | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβ : z β s β© f β»ΒΉ' {a}
m : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
β’ β (x : β(s β© f β»ΒΉ' {a})), x β Subtype.val β»ΒΉ' t β x β {β¨z, mββ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | simp only [Set.mem_preimage, Subtype.coe_mk, Set.mem_singleton_iff, Subtype.mk_eq_mk] | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
w : β
m : w β s β© f β»ΒΉ' {a}
β’ β¨w, mβ© β Subtype.val β»ΒΉ' t β β¨w, mβ© β {β¨z, mβΒΉβ©} | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
w : β
m : w β s β© f β»ΒΉ' {a}
β’ w β t β w = z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
w : β
m : w β s β© f β»ΒΉ' {a}
β’ β¨w, mβ© β Subtype.val β»ΒΉ' t β β¨w, mβ© β {β¨z, mβΒΉβ©}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff] at m | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
w : β
m : w β s β© f β»ΒΉ' {a}
β’ w β t β w = z | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
β’ w β t β w = z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
w : β
m : w β s β© f β»ΒΉ' {a}
β’ w β t β w = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | specialize t0 w | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
β’ w β t β w = z | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w β z β Β¬f w = a
β’ w β t β w = z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
t0 : β x β t, x β z β Β¬f x = a
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
β’ w β t β w = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | simp only [m.2, imp_false, eq_self_iff_true, not_true, not_not] at t0 | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w β z β Β¬f w = a
β’ w β t β w = z | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
β’ w β t β w = z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w β z β Β¬f w = a
β’ w β t β w = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | use t0 | case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
β’ w β t β w = z | case mpr
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
β’ w = z β w β t | Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
β’ w β t β w = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | intro wz | case mpr
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
β’ w = z β w β t | case mpr
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
wz : w = z
β’ w β t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
β’ w = z β w β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | rw [wz] | case mpr
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
wz : w = z
β’ w β t | case mpr
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
wz : w = z
β’ z β t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
wz : w = z
β’ w β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.discreteTopology | [96, 1] | [111, 40] | exact tz | case mpr
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
wz : w = z
β’ z β t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
a z : β
mβΒΉ : z β s β© f β»ΒΉ' {a}
mβ : z β s β§ f z = a
h : βαΆ (x : β) in π z, x β z β Β¬f x = a
t : Set β
o : IsOpen t
tz : z β t
w : β
m : w β s β§ f w = a
t0 : w β t β w = z
wz : w = z
β’ z β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | powNontrivial | [114, 1] | [116, 86] | apply Entire.nontrivialAnalyticOn fun _ _ β¦ (analyticAt_id _ _).pow _ | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ NontrivialAnalyticOn (fun z => z ^ d) univ | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ β a b, id a ^ d β id b ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ NontrivialAnalyticOn (fun z => z ^ d) univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | powNontrivial | [114, 1] | [116, 86] | use 0, 1 | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ β a b, id a ^ d β id b ^ d | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ id 0 ^ d β id 1 ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ β a b, id a ^ d β id b ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | powNontrivial | [114, 1] | [116, 86] | simp only [id, one_pow, zero_pow (Nat.pos_iff_ne_zero.mp dp), Pi.pow_def] | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ id 0 ^ d β id 1 ^ d | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ 0 β 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ id 0 ^ d β id 1 ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | powNontrivial | [114, 1] | [116, 86] | norm_num | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ 0 β 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
d : β
dp : 0 < d
β’ 0 β 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | allRootsOfUnity.ne_zero | [123, 1] | [125, 49] | rcases m with β¨n, n0, z1β© | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
m : z β allRootsOfUnity
β’ z β 0 | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : z ^ n = 1
β’ z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
m : z β allRootsOfUnity
β’ z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | allRootsOfUnity.ne_zero | [123, 1] | [125, 49] | contrapose z1 | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : z ^ n = 1
β’ z β 0 | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : Β¬z β 0
β’ Β¬z ^ n = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : z ^ n = 1
β’ z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | allRootsOfUnity.ne_zero | [123, 1] | [125, 49] | simp only [not_not] at z1 | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : Β¬z β 0
β’ Β¬z ^ n = 1 | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : z = 0
β’ Β¬z ^ n = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : Β¬z β 0
β’ Β¬z ^ n = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | allRootsOfUnity.ne_zero | [123, 1] | [125, 49] | simp only [z1, zero_pow n0] | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : z = 0
β’ Β¬z ^ n = 1 | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : z = 0
β’ Β¬0 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : z = 0
β’ Β¬z ^ n = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | allRootsOfUnity.ne_zero | [123, 1] | [125, 49] | exact zero_ne_one | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : z = 0
β’ Β¬0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
z : β
n : β
n0 : n β 0
z1 : z = 0
β’ Β¬0 = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | apply IsCountable.isTotallyDisconnected | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ IsTotallyDisconnected _root_.allRootsOfUnity | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ _root_.allRootsOfUnity.Countable | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ IsTotallyDisconnected _root_.allRootsOfUnity
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | simp only [_root_.allRootsOfUnity, setOf_exists] | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ _root_.allRootsOfUnity.Countable | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ (β i, {x | i β 0 β§ x ^ i = 1}).Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ _root_.allRootsOfUnity.Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | apply countable_iUnion | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ (β i, {x | i β 0 β§ x ^ i = 1}).Countable | case h.ht
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ β (i : β), {x | i β 0 β§ x ^ i = 1}.Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ (β i, {x | i β 0 β§ x ^ i = 1}).Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | intro n | case h.ht
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ β (i : β), {x | i β 0 β§ x ^ i = 1}.Countable | case h.ht
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
β’ {x | n β 0 β§ x ^ n = 1}.Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case h.ht
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
β’ β (i : β), {x | i β 0 β§ x ^ i = 1}.Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | by_cases n0 : n = 0 | case h.ht
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
β’ {x | n β 0 β§ x ^ n = 1}.Countable | case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : n = 0
β’ {x | n β 0 β§ x ^ n = 1}.Countable
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
β’ {x | n β 0 β§ x ^ n = 1}.Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case h.ht
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
β’ {x | n β 0 β§ x ^ n = 1}.Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | simp only [n0, Ne, eq_self_iff_true, not_true, false_and_iff, setOf_false, countable_empty] | case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : n = 0
β’ {x | n β 0 β§ x ^ n = 1}.Countable
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
β’ {x | n β 0 β§ x ^ n = 1}.Countable | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
β’ {x | n β 0 β§ x ^ n = 1}.Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : n = 0
β’ {x | n β 0 β§ x ^ n = 1}.Countable
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
β’ {x | n β 0 β§ x ^ n = 1}.Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | simp only [Ne, n0, not_false_iff, true_and_iff] | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
β’ {x | n β 0 β§ x ^ n = 1}.Countable | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
β’ {x | x ^ n = 1}.Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
β’ {x | n β 0 β§ x ^ n = 1}.Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | have np : 0 < n := Nat.pos_of_ne_zero n0 | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
β’ {x | x ^ n = 1}.Countable | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
β’ {x | x ^ n = 1}.Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
β’ {x | x ^ n = 1}.Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | generalize hn' : (β¨n, npβ© : β+) = n' | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
β’ {x | x ^ n = 1}.Countable | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ {x | x ^ n = 1}.Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
β’ {x | x ^ n = 1}.Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | apply Set.Countable.mono e | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
e : {z | z ^ n = 1} β (fun x => βx) '' β(rootsOfUnity n' β)
β’ {x | x ^ n = 1}.Countable | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
e : {z | z ^ n = 1} β (fun x => βx) '' β(rootsOfUnity n' β)
β’ ((fun x => βx) '' β(rootsOfUnity n' β)).Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
e : {z | z ^ n = 1} β (fun x => βx) '' β(rootsOfUnity n' β)
β’ {x | x ^ n = 1}.Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | clear e | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
e : {z | z ^ n = 1} β (fun x => βx) '' β(rootsOfUnity n' β)
β’ ((fun x => βx) '' β(rootsOfUnity n' β)).Countable | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ ((fun x => βx) '' β(rootsOfUnity n' β)).Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
e : {z | z ^ n = 1} β (fun x => βx) '' β(rootsOfUnity n' β)
β’ ((fun x => βx) '' β(rootsOfUnity n' β)).Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | apply Countable.image | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ ((fun x => βx) '' β(rootsOfUnity n' β)).Countable | case neg.hs
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ (β(rootsOfUnity n' β)).Countable | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ ((fun x => βx) '' β(rootsOfUnity n' β)).Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | apply Set.Finite.countable | case neg.hs
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ (β(rootsOfUnity n' β)).Countable | case neg.hs.hs
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ (β(rootsOfUnity n' β)).Finite | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hs
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ (β(rootsOfUnity n' β)).Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | rw [Set.finite_def] | case neg.hs.hs
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ (β(rootsOfUnity n' β)).Finite | case neg.hs.hs
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ Nonempty (Fintype ββ(rootsOfUnity n' β)) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hs.hs
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ (β(rootsOfUnity n' β)).Finite
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | exact β¨_root_.rootsOfUnity.fintype β n'β© | case neg.hs.hs
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ Nonempty (Fintype ββ(rootsOfUnity n' β)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hs.hs
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ Nonempty (Fintype ββ(rootsOfUnity n' β))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | intro z e | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ {z | z ^ n = 1} β (fun x => βx) '' β(rootsOfUnity n' β) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z β {z | z ^ n = 1}
β’ z β (fun x => βx) '' β(rootsOfUnity n' β) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
β’ {z | z ^ n = 1} β (fun x => βx) '' β(rootsOfUnity n' β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | simp only [mem_setOf] at e | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z β {z | z ^ n = 1}
β’ z β (fun x => βx) '' β(rootsOfUnity n' β) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
β’ z β (fun x => βx) '' β(rootsOfUnity n' β) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z β {z | z ^ n = 1}
β’ z β (fun x => βx) '' β(rootsOfUnity n' β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | simp only [mem_image, SetLike.mem_coe, mem_rootsOfUnity, PNat.mk_coe] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
β’ z β (fun x => βx) '' β(rootsOfUnity n' β) | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
β’ β x, x ^ βn' = 1 β§ βx = z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
β’ z β (fun x => βx) '' β(rootsOfUnity n' β)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | by_cases z0 : z = 0 | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
β’ β x, x ^ βn' = 1 β§ βx = z | case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
z0 : z = 0
β’ β x, x ^ βn' = 1 β§ βx = z
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
z0 : Β¬z = 0
β’ β x, x ^ βn' = 1 β§ βx = z | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
β’ β x, x ^ βn' = 1 β§ βx = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | simp only [z0, zero_pow n0, zero_ne_one] at e | case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
z0 : z = 0
β’ β x, x ^ βn' = 1 β§ βx = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
z0 : z = 0
β’ β x, x ^ βn' = 1 β§ βx = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | use Units.mk0 z z0 | case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
z0 : Β¬z = 0
β’ β x, x ^ βn' = 1 β§ βx = z | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
z0 : Β¬z = 0
β’ Units.mk0 z z0 ^ βn' = 1 β§ β(Units.mk0 z z0) = z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
z0 : Β¬z = 0
β’ β x, x ^ βn' = 1 β§ βx = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | IsTotallyDisconnected.allRootsOfUnity | [128, 1] | [145, 64] | simp only [β hn', PNat.mk_coe, β Units.eq_iff, Units.val_pow_eq_pow_val, Units.val_mk0, e,
Units.val_one, and_self] | case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
z0 : Β¬z = 0
β’ Units.mk0 z z0 ^ βn' = 1 β§ β(Units.mk0 z z0) = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : β
n0 : Β¬n = 0
np : 0 < n
n' : { n // 0 < n }
hn' : β¨n, npβ© = n'
z : β
e : z ^ n = 1
z0 : Β¬z = 0
β’ Units.mk0 z z0 ^ βn' = 1 β§ β(Units.mk0 z z0) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.const | [148, 1] | [156, 21] | have disc : DiscreteTopology (β₯(s β© f β»ΒΉ' {b})) := n.discreteTopology b | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
p1 : β x β t, p x = a
fp : β x β t, f (p x) = b
β’ β x β t, p x = a | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
p1 : β x β t, p x = a
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
β’ β x β t, p x = a | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
p1 : β x β t, p x = a
fp : β x β t, f (p x) = b
β’ β x β t, p x = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.const | [148, 1] | [156, 21] | rcases p1 with β¨z, zt, z1β© | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
p1 : β x β t, p x = a
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
β’ β x β t, p x = a | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
β’ β x β t, p x = a | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
p1 : β x β t, p x = a
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
β’ β x β t, p x = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.const | [148, 1] | [156, 21] | simp only [β z1] | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
β’ β x β t, p x = a | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
β’ β x β t, p x = p z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
β’ β x β t, p x = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.const | [148, 1] | [156, 21] | intro x xt | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
β’ β x β t, p x = p z | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
β’ p x = p z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
β’ β x β t, p x = p z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.const | [148, 1] | [156, 21] | refine @IsPreconnected.constant_of_mapsTo _ _ _ _ _ tc _ disc _ pc ?_ _ _ xt zt | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
β’ p x = p z | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
β’ MapsTo p t (s β© f β»ΒΉ' {b}) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
β’ p x = p z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.const | [148, 1] | [156, 21] | intro y yt | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
β’ MapsTo p t (s β© f β»ΒΉ' {b}) | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
y : X
yt : y β t
β’ p y β s β© f β»ΒΉ' {b} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
β’ MapsTo p t (s β© f β»ΒΉ' {b})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.const | [148, 1] | [156, 21] | simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff] | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
y : X
yt : y β t
β’ p y β s β© f β»ΒΉ' {b} | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
y : X
yt : y β t
β’ p y β s β§ f (p y) = b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
y : X
yt : y β t
β’ p y β s β© f β»ΒΉ' {b}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | NontrivialAnalyticOn.const | [148, 1] | [156, 21] | use ps yt, fp _ yt | case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
y : X
yt : y β t
β’ p y β s β§ f (p y) = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
n : NontrivialAnalyticOn f s
p : X β β
t : Set X
tc : IsPreconnected t
pc : ContinuousOn p t
ps : MapsTo p t s
a b : β
fp : β x β t, f (p x) = b
disc : DiscreteTopology β(s β© f β»ΒΉ' {b})
z : X
zt : z β t
z1 : p z = a
x : X
xt : x β t
y : X
yt : y β t
β’ p y β s β§ f (p y) = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | generalize hr : (fun x β¦ q x / p x) = r | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
β’ β x β t, p x = q x | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
β’ β x β t, p x = q x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
β’ β x β t, p x = q x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | have rc : ContinuousOn r t := by rw [β hr]; exact qc.div pc p0 | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
β’ β x β t, p x = q x | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
β’ β x β t, p x = q x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
β’ β x β t, p x = q x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | have h := eq_one_of_pow_eq_one rc tc dp ?_ ?_ | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
β’ β x β t, p x = q x | case refine_3
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
h : β x β t, r x = 1
β’ β x β t, p x = q x
case refine_1
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
β’ β x β t, r x = 1
case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
β’ β x β t, r x ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
β’ β x β t, p x = q x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | rw [β hr] | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
β’ ContinuousOn r t | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
β’ ContinuousOn (fun x => q x / p x) t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
β’ ContinuousOn r t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | exact qc.div pc p0 | X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
β’ ContinuousOn (fun x => q x / p x) t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
β’ ContinuousOn (fun x => q x / p x) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | intro x m | case refine_3
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
h : β x β t, r x = 1
β’ β x β t, p x = q x | case refine_3
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
h : β x β t, r x = 1
x : X
m : x β t
β’ p x = q x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
h : β x β t, r x = 1
β’ β x β t, p x = q x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | rw [β hr] at h | case refine_3
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
h : β x β t, r x = 1
x : X
m : x β t
β’ p x = q x | case refine_3
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
h : β x β t, (fun x => q x / p x) x = 1
x : X
m : x β t
β’ p x = q x | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
h : β x β t, r x = 1
x : X
m : x β t
β’ p x = q x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | exact ((div_eq_one_iff_eq (p0 _ m)).mp (h _ m)).symm | case refine_3
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
h : β x β t, (fun x => q x / p x) x = 1
x : X
m : x β t
β’ p x = q x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_3
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
h : β x β t, (fun x => q x / p x) x = 1
x : X
m : x β t
β’ p x = q x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | rcases pq with β¨x, m, eβ© | case refine_1
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
β’ β x β t, r x = 1 | case refine_1.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
e : p x = q x
β’ β x β t, r x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
β’ β x β t, r x = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | use x, m | case refine_1.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
e : p x = q x
β’ β x β t, r x = 1 | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
e : p x = q x
β’ r x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro.intro
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
e : p x = q x
β’ β x β t, r x = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | rw [β hr] | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
e : p x = q x
β’ r x = 1 | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
e : p x = q x
β’ (fun x => q x / p x) x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
e : p x = q x
β’ r x = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | exact (div_eq_one_iff_eq (p0 _ m)).mpr e.symm | case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
e : p x = q x
β’ (fun x => q x / p x) x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
e : p x = q x
β’ (fun x => q x / p x) x = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | intro x m | case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
β’ β x β t, r x ^ d = 1 | case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ r x ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
β’ β x β t, r x ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | simp only [div_pow, β hr] | case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ r x ^ d = 1 | case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ q x ^ d / p x ^ d = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ r x ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | rw [div_eq_one_iff_eq] | case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ q x ^ d / p x ^ d = 1 | case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ q x ^ d = p x ^ d
case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ p x ^ d β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ q x ^ d / p x ^ d = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | exact (pqd _ m).symm | case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ q x ^ d = p x ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ q x ^ d = p x ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | eq_of_pow_eq | [166, 1] | [183, 35] | exact pow_ne_zero _ (p0 _ m) | case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ p x ^ d β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
X : Type
instββΆ : TopologicalSpace X
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instβΒ³ : TopologicalSpace T
instβΒ² : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒΉ : TopologicalSpace U
instβ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
f : β β β
s : Set β
p q : X β β
t : Set X
d : β
pc : ContinuousOn p t
qc : ContinuousOn q t
tc : IsPreconnected t
dp : d > 0
pq : β x β t, p x = q x
p0 : β x β t, p x β 0
pqd : β x β t, p x ^ d = q x ^ d
r : X β β
hr : (fun x => q x / p x) = r
rc : ContinuousOn r t
x : X
m : x β t
β’ p x ^ d β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | simp only [holomorphicAt_iff, Function.comp] at fa ga | X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fa : HolomorphicAt π(β, β) π(β, β) f z
ga : HolomorphicAt π(β, β) π(β, β) g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fa :
ContinuousAt f z β§
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
ga :
ContinuousAt g z β§
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fa : HolomorphicAt π(β, β) π(β, β) f z
ga : HolomorphicAt π(β, β) π(β, β) g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | rcases fa with β¨fc, faβ© | X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fa :
ContinuousAt f z β§
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
ga :
ContinuousAt g z β§
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | case intro
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
ga :
ContinuousAt g z β§
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fa :
ContinuousAt f z β§
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
ga :
ContinuousAt g z β§
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | rcases ga with β¨gc, gaβ© | case intro
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
ga :
ContinuousAt g z β§
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | case intro.intro
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
ga :
ContinuousAt g z β§
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | by_cases fg : f z β g z | case intro.intro
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | case pos
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z β g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w
case neg
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : Β¬f z β g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | simp only [not_not] at fg | case neg
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : Β¬f z β g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | case neg
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z = g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : Β¬f z β g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | cases' fa.eventually_eq_or_eventually_ne ga with e e | case neg
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z = g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | case neg.inl
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z = g z
e :
βαΆ (z_1 : β) in π (β(_root_.extChartAt π(β, β) z) z),
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm z_1)) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm z_1))
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w
case neg.inr
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z = g z
e :
βαΆ (z_1 : β) in π[β ] β(_root_.extChartAt π(β, β) z) z,
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm z_1)) β
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm z_1))
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z = g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | right | case pos
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z β g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z β g z
β’ βαΆ (w : S) in π[β ] z, f w β g w | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z β g z
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | contrapose fg | case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z β g z
β’ βαΆ (w : S) in π[β ] z, f w β g w | case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : Β¬βαΆ (w : S) in π[β ] z, f w β g w
β’ Β¬f z β g z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z β g z
β’ βαΆ (w : S) in π[β ] z, f w β g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | simp only [not_not] | case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : Β¬βαΆ (w : S) in π[β ] z, f w β g w
β’ Β¬f z β g z | case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : Β¬βαΆ (w : S) in π[β ] z, f w β g w
β’ f z = g z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : Β¬βαΆ (w : S) in π[β ] z, f w β g w
β’ Β¬f z β g z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | simp only [Filter.not_eventually, not_not] at fg | case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : Β¬βαΆ (w : S) in π[β ] z, f w β g w
β’ f z = g z | case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : βαΆ (x : S) in π[β ] z, f x = g x
β’ f z = g z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : Β¬βαΆ (w : S) in π[β ] z, f w β g w
β’ f z = g z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | exact tendsto_nhds_unique_of_frequently_eq fc gc (fg.filter_mono nhdsWithin_le_nhds) | case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : βαΆ (x : S) in π[β ] z, f x = g x
β’ f z = g z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : βαΆ (x : S) in π[β ] z, f x = g x
β’ f z = g z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | left | case neg.inl
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z = g z
e :
βαΆ (z_1 : β) in π (β(_root_.extChartAt π(β, β) z) z),
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm z_1)) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm z_1))
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z = g z
e :
βαΆ (z_1 : β) in π (β(_root_.extChartAt π(β, β) z) z),
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm z_1)) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm z_1))
β’ βαΆ (w : S) in π z, f w = g w | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z = g z
e :
βαΆ (z_1 : β) in π (β(_root_.extChartAt π(β, β) z) z),
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm z_1)) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm z_1))
β’ (βαΆ (w : S) in π z, f w = g w) β¨ βαΆ (w : S) in π[β ] z, f w β g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | clear fa ga | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z = g z
e :
βαΆ (z_1 : β) in π (β(_root_.extChartAt π(β, β) z) z),
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm z_1)) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm z_1))
β’ βαΆ (w : S) in π z, f w = g w | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
βαΆ (z_1 : β) in π (β(_root_.extChartAt π(β, β) z) z),
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm z_1)) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm z_1))
β’ βαΆ (w : S) in π z, f w = g w | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
fa :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
gc : ContinuousAt g z
ga :
AnalyticAt β (fun x => β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm x)))
(β(_root_.extChartAt π(β, β) z) z)
fg : f z = g z
e :
βαΆ (z_1 : β) in π (β(_root_.extChartAt π(β, β) z) z),
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm z_1)) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm z_1))
β’ βαΆ (w : S) in π z, f w = g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | replace e := (continuousAt_extChartAt I z).eventually e | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
βαΆ (z_1 : β) in π (β(_root_.extChartAt π(β, β) z) z),
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm z_1)) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm z_1))
β’ βαΆ (w : S) in π z, f w = g w | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
βαΆ (x : S) in π z,
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))
β’ βαΆ (w : S) in π z, f w = g w | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
βαΆ (z_1 : β) in π (β(_root_.extChartAt π(β, β) z) z),
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm z_1)) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm z_1))
β’ βαΆ (w : S) in π z, f w = g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | replace e := Filter.EventuallyEq.fun_comp e (_root_.extChartAt I (f z)).symm | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
βαΆ (x : S) in π z,
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))
β’ βαΆ (w : S) in π z, f w = g w | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
(π z).EventuallyEq
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
β’ βαΆ (w : S) in π z, f w = g w | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
βαΆ (x : S) in π z,
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))) =
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))
β’ βαΆ (w : S) in π z, f w = g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | apply e.congr | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
(π z).EventuallyEq
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
β’ βαΆ (w : S) in π z, f w = g w | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
(π z).EventuallyEq
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
β’ βαΆ (x : S) in π z,
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z))
(f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
x =
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z))
(g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
x β
f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
(π z).EventuallyEq
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
β’ βαΆ (w : S) in π z, f w = g w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | simp only [Function.comp] | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
(π z).EventuallyEq
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
β’ βαΆ (x : S) in π z,
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z))
(f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
x =
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z))
(g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
x β
f x = g x | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
(π z).EventuallyEq
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
β’ βαΆ (x : S) in π z,
β(_root_.extChartAt π(β, β) (f z)).symm
(β(_root_.extChartAt π(β, β) (f z))
(f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))) =
β(_root_.extChartAt π(β, β) (f z)).symm
(β(_root_.extChartAt π(β, β) (g z))
(g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))) β
f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
(π z).EventuallyEq
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
β’ βαΆ (x : S) in π z,
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z))
(f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
x =
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z))
(g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
x β
f x = g x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nontrivial.lean | HolomorphicAt.eventually_eq_or_eventually_ne | [188, 1] | [220, 48] | clear e | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
(π z).EventuallyEq
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
β’ βαΆ (x : S) in π z,
β(_root_.extChartAt π(β, β) (f z)).symm
(β(_root_.extChartAt π(β, β) (f z))
(f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))) =
β(_root_.extChartAt π(β, β) (f z)).symm
(β(_root_.extChartAt π(β, β) (g z))
(g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))) β
f x = g x | case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
β’ βαΆ (x : S) in π z,
β(_root_.extChartAt π(β, β) (f z)).symm
(β(_root_.extChartAt π(β, β) (f z))
(f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))) =
β(_root_.extChartAt π(β, β) (f z)).symm
(β(_root_.extChartAt π(β, β) (g z))
(g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))) β
f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.h
X : Type
instββ· : TopologicalSpace X
S : Type
instββΆ : TopologicalSpace S
instββ΅ : ChartedSpace β S
cms : AnalyticManifold π(β, β) S
T : Type
instββ΄ : TopologicalSpace T
instβΒ³ : ChartedSpace β T
cmt : AnalyticManifold π(β, β) T
U : Type
instβΒ² : TopologicalSpace U
instβΒΉ : ChartedSpace β U
cmu : AnalyticManifold π(β, β) U
fβ : β β β
s : Set β
instβ : T2Space T
f g : S β T
z : S
fc : ContinuousAt f z
gc : ContinuousAt g z
fg : f z = g z
e :
(π z).EventuallyEq
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (f z)) (f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
(β(_root_.extChartAt π(β, β) (f z)).symm β fun x =>
β(_root_.extChartAt π(β, β) (g z)) (g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x))))
β’ βαΆ (x : S) in π z,
β(_root_.extChartAt π(β, β) (f z)).symm
(β(_root_.extChartAt π(β, β) (f z))
(f (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))) =
β(_root_.extChartAt π(β, β) (f z)).symm
(β(_root_.extChartAt π(β, β) (g z))
(g (β(_root_.extChartAt π(β, β) z).symm (β(_root_.extChartAt π(β, β) z) x)))) β
f x = g x
TACTIC:
|
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