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stringlengths 7
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stringlengths 1
94
| start
stringlengths 6
10
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stringlengths 6
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stringlengths 1
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2.09M
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rw [bottcherNear_zero] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ā fun x => bottcherNear f d x) 0 ā 0
ā¢ MDifferentiableAt I I i (bottcherNear f d 0) | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ā fun x => bottcherNear f d x) 0 ā 0
ā¢ MDifferentiableAt I I i 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ā fun x => bottcherNear f d x) 0 ā 0
ā¢ MDifferentiableAt I I i (bottcherNear f d 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | exact ia.mdifferentiableAt | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ā fun x => bottcherNear f d x) 0 ā 0
ā¢ MDifferentiableAt I I i 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
d0 : mfderiv I I (i ā fun x => bottcherNear f d x) 0 ā 0
ā¢ MDifferentiableAt I I i 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | apply HolomorphicAt.analyticAt I I | case intro.intro.intro.intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ AnalyticAt ā (fun z => i (a * bottcherNear f d z)) 0 | case intro.intro.intro.intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ HolomorphicAt I I (fun z => i (a * bottcherNear f d z)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ AnalyticAt ā (fun z => i (a * bottcherNear f d z)) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | refine ia.comp_of_eq (holomorphicAt_const.mul (ba.holomorphicAt I I)) ?_ | case intro.intro.intro.intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ HolomorphicAt I I (fun z => i (a * bottcherNear f d z)) 0 | case intro.intro.intro.intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ a * bottcherNear f d 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ HolomorphicAt I I (fun z => i (a * bottcherNear f d z)) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | simp only [bottcherNear_zero, s.f0, MulZeroClass.mul_zero] | case intro.intro.intro.intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ a * bottcherNear f d 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ a * bottcherNear f d 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | simp only [bottcherNear_zero, MulZeroClass.mul_zero, i0] | case intro.intro.intro.intro.intro.intro.refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ (fun z => i (a * bottcherNear f d z)) 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ (fun z => i (a * bottcherNear f d z)) 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | simp only [eventually_nhdsWithin_iff, mem_compl_singleton_iff] | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ āį¶ (z : ā) in š[ā ] 0, (fun z => i (a * bottcherNear f d z)) z ā z ā§ f ((fun z => i (a * bottcherNear f d z)) z) = f z | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ āį¶ (z : ā) in š[ā ] 0, (fun z => i (a * bottcherNear f d z)) z ā z ā§ f ((fun z => i (a * bottcherNear f d z)) z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have t0 : ContinuousAt (fun z ā¦ a * bottcherNear f d z) 0 :=
continuousAt_const.mul ba.continuousAt | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have t1 : ContinuousAt (fun z ā¦ f (i (a * bottcherNear f d z))) 0 := by
refine s.fa0.continuousAt.comp_of_eq (ia.continuousAt.comp_of_eq t0 ?_) ?_
repeat' simp only [bottcherNear_zero, MulZeroClass.mul_zero, i0] | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have t2 : ContinuousAt f 0 := s.fa0.continuousAt | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have m0 : āį¶ z in š 0, i (a * bottcherNear f d z) ā t := by
refine (ia.continuousAt.comp_of_eq t0 ?_).eventually_mem (s.o.mem_nhds ?_)
repeat' simp only [bottcherNear_zero, MulZeroClass.mul_zero, i0, s.t0, Function.comp] | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have m1 : āį¶ z in š 0, z ā t := s.o.eventually_mem s.t0 | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | simp only [ContinuousAt, bottcherNear_zero, MulZeroClass.mul_zero, i0, s.f0] at t0 t1 t2 | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have tp := t0.prod_mk ba.continuousAt | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š 0 ĆĖ¢ š (bottcherNear f d 0))
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | simp only [ā nhds_prod_eq, ContinuousAt, bottcherNear_zero] at tp | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š 0 ĆĖ¢ š (bottcherNear f d 0))
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š 0 ĆĖ¢ š (bottcherNear f d 0))
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | apply (tp.eventually inj).mp | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
ā¢ āį¶ (x : ā) in š 0,
(i (a * bottcherNear f d x, bottcherNear f d x).1 = i (a * bottcherNear f d x, bottcherNear f d x).2 ā
(a * bottcherNear f d x, bottcherNear f d x).1 = (a * bottcherNear f d x, bottcherNear f d x).2) ā
x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
ā¢ āį¶ (x : ā) in š 0, x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | refine ib.mp (bi.mp ((t1.eventually ib).mp
((t0.eventually bi).mp ((t2.eventually ib).mp (m0.mp (m1.mp ?_)))))) | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
ā¢ āį¶ (x : ā) in š 0,
(i (a * bottcherNear f d x, bottcherNear f d x).1 = i (a * bottcherNear f d x, bottcherNear f d x).2 ā
(a * bottcherNear f d x, bottcherNear f d x).1 = (a * bottcherNear f d x, bottcherNear f d x).2) ā
x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
ā¢ āį¶ (x : ā) in š 0,
x ā t ā
i (a * bottcherNear f d x) ā t ā
i (bottcherNear f d (f x)) = f x ā
bottcherNear f d (i (a * bottcherNear f d x)) = a * bottcherNear f d x ā
i (bottcherNear f d (f (i (a * bottcherNear f d x)))) = f (i (a * bottcherNear f d x)) ā
bottcherNear f d (i x) = x ā
i (bottcherNear f d x) = x ā
(i (a * bottcherNear f d x, bottcherNear f d x).1 = i (a * bottcherNear f d x, bottcherNear f d x).2 ā
(a * bottcherNear f d x, bottcherNear f d x).1 = (a * bottcherNear f d x, bottcherNear f d x).2) ā
x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
ā¢ āį¶ (x : ā) in š 0,
(i (a * bottcherNear f d x, bottcherNear f d x).1 = i (a * bottcherNear f d x, bottcherNear f d x).2 ā
(a * bottcherNear f d x, bottcherNear f d x).1 = (a * bottcherNear f d x, bottcherNear f d x).2) ā
x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | refine eventually_of_forall fun z m1 m0 t2 t0 t1 _ ib tp z0 ā¦ āØ?_, ?_ā© | case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
ā¢ āį¶ (x : ā) in š 0,
x ā t ā
i (a * bottcherNear f d x) ā t ā
i (bottcherNear f d (f x)) = f x ā
bottcherNear f d (i (a * bottcherNear f d x)) = a * bottcherNear f d x ā
i (bottcherNear f d (f (i (a * bottcherNear f d x)))) = f (i (a * bottcherNear f d x)) ā
bottcherNear f d (i x) = x ā
i (bottcherNear f d x) = x ā
(i (a * bottcherNear f d x, bottcherNear f d x).1 = i (a * bottcherNear f d x, bottcherNear f d x).2 ā
(a * bottcherNear f d x, bottcherNear f d x).1 = (a * bottcherNear f d x, bottcherNear f d x).2) ā
x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x | case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
tp :
i (a * bottcherNear f d z, bottcherNear f d z).1 = i (a * bottcherNear f d z, bottcherNear f d z).2 ā
(a * bottcherNear f d z, bottcherNear f d z).1 = (a * bottcherNear f d z, bottcherNear f d z).2
z0 : z ā 0
ā¢ i (a * bottcherNear f d z) ā z
case intro.intro.intro.intro.intro.intro.refine_3.refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
tp :
i (a * bottcherNear f d z, bottcherNear f d z).1 = i (a * bottcherNear f d z, bottcherNear f d z).2 ā
(a * bottcherNear f d z, bottcherNear f d z).1 = (a * bottcherNear f d z, bottcherNear f d z).2
z0 : z ā 0
ā¢ f (i (a * bottcherNear f d z)) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0 : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1 : āį¶ (z : ā) in š 0, z ā t
t0 : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1 : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2 : Tendsto f (š 0) (š 0)
tp : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
ā¢ āį¶ (x : ā) in š 0,
x ā t ā
i (a * bottcherNear f d x) ā t ā
i (bottcherNear f d (f x)) = f x ā
bottcherNear f d (i (a * bottcherNear f d x)) = a * bottcherNear f d x ā
i (bottcherNear f d (f (i (a * bottcherNear f d x)))) = f (i (a * bottcherNear f d x)) ā
bottcherNear f d (i x) = x ā
i (bottcherNear f d x) = x ā
(i (a * bottcherNear f d x, bottcherNear f d x).1 = i (a * bottcherNear f d x, bottcherNear f d x).2 ā
(a * bottcherNear f d x, bottcherNear f d x).1 = (a * bottcherNear f d x, bottcherNear f d x).2) ā
x ā 0 ā i (a * bottcherNear f d x) ā x ā§ f (i (a * bottcherNear f d x)) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | refine s.fa0.continuousAt.comp_of_eq (ia.continuousAt.comp_of_eq t0 ?_) ?_ | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0 | case refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ a * bottcherNear f d 0 = 0
case refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ i (a * bottcherNear f d 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | repeat' simp only [bottcherNear_zero, MulZeroClass.mul_zero, i0] | case refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ a * bottcherNear f d 0 = 0
case refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ i (a * bottcherNear f d 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ a * bottcherNear f d 0 = 0
case refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ i (a * bottcherNear f d 0) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | simp only [bottcherNear_zero, MulZeroClass.mul_zero, i0] | case refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ i (a * bottcherNear f d 0) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
ā¢ i (a * bottcherNear f d 0) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | refine (ia.continuousAt.comp_of_eq t0 ?_).eventually_mem (s.o.mem_nhds ?_) | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t | case refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ a * bottcherNear f d 0 = 0
case refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ (i ā fun z => a * bottcherNear f d z) 0 ā t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | repeat' simp only [bottcherNear_zero, MulZeroClass.mul_zero, i0, s.t0, Function.comp] | case refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ a * bottcherNear f d 0 = 0
case refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ (i ā fun z => a * bottcherNear f d z) 0 ā t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ a * bottcherNear f d 0 = 0
case refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ (i ā fun z => a * bottcherNear f d z) 0 ā t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | simp only [bottcherNear_zero, MulZeroClass.mul_zero, i0, s.t0, Function.comp] | case refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ (i ā fun z => a * bottcherNear f d z) 0 ā t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ib : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
t0 : ContinuousAt (fun z => a * bottcherNear f d z) 0
t1 : ContinuousAt (fun z => f (i (a * bottcherNear f d z))) 0
t2 : ContinuousAt f 0
ā¢ (i ā fun z => a * bottcherNear f d z) 0 ā t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | contrapose tp | case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
tp :
i (a * bottcherNear f d z, bottcherNear f d z).1 = i (a * bottcherNear f d z, bottcherNear f d z).2 ā
(a * bottcherNear f d z, bottcherNear f d z).1 = (a * bottcherNear f d z, bottcherNear f d z).2
z0 : z ā 0
ā¢ i (a * bottcherNear f d z) ā z | case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : Ā¬i (a * bottcherNear f d z) ā z
ā¢ Ā¬(i (a * bottcherNear f d z, bottcherNear f d z).1 = i (a * bottcherNear f d z, bottcherNear f d z).2 ā
(a * bottcherNear f d z, bottcherNear f d z).1 = (a * bottcherNear f d z, bottcherNear f d z).2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
tp :
i (a * bottcherNear f d z, bottcherNear f d z).1 = i (a * bottcherNear f d z, bottcherNear f d z).2 ā
(a * bottcherNear f d z, bottcherNear f d z).1 = (a * bottcherNear f d z, bottcherNear f d z).2
z0 : z ā 0
ā¢ i (a * bottcherNear f d z) ā z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | simp only [ne_eq, Decidable.not_not, Classical.not_imp] at tp ā¢ | case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : Ā¬i (a * bottcherNear f d z) ā z
ā¢ Ā¬(i (a * bottcherNear f d z, bottcherNear f d z).1 = i (a * bottcherNear f d z, bottcherNear f d z).2 ā
(a * bottcherNear f d z, bottcherNear f d z).1 = (a * bottcherNear f d z, bottcherNear f d z).2) | case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
ā¢ i (a * bottcherNear f d z) = i (bottcherNear f d z) ā§ Ā¬a * bottcherNear f d z = bottcherNear f d z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : Ā¬i (a * bottcherNear f d z) ā z
ā¢ Ā¬(i (a * bottcherNear f d z, bottcherNear f d z).1 = i (a * bottcherNear f d z, bottcherNear f d z).2 ā
(a * bottcherNear f d z, bottcherNear f d z).1 = (a * bottcherNear f d z, bottcherNear f d z).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rw [ib] | case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
ā¢ i (a * bottcherNear f d z) = i (bottcherNear f d z) ā§ Ā¬a * bottcherNear f d z = bottcherNear f d z | case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
ā¢ i (a * bottcherNear f d z) = z ā§ Ā¬a * bottcherNear f d z = bottcherNear f d z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
ā¢ i (a * bottcherNear f d z) = i (bottcherNear f d z) ā§ Ā¬a * bottcherNear f d z = bottcherNear f d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | use tp | case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
ā¢ i (a * bottcherNear f d z) = z ā§ Ā¬a * bottcherNear f d z = bottcherNear f d z | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
ā¢ Ā¬a * bottcherNear f d z = bottcherNear f d z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
ā¢ i (a * bottcherNear f d z) = z ā§ Ā¬a * bottcherNear f d z = bottcherNear f d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | contrapose a1 | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
ā¢ Ā¬a * bottcherNear f d z = bottcherNear f d z | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : Ā¬Ā¬a * bottcherNear f d z = bottcherNear f d z
ā¢ Ā¬a ā 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
ā¢ Ā¬a * bottcherNear f d z = bottcherNear f d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | simp only [not_not] at a1 ā¢ | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : Ā¬Ā¬a * bottcherNear f d z = bottcherNear f d z
ā¢ Ā¬a ā 1 | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : a * bottcherNear f d z = bottcherNear f d z
ā¢ a = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : Ā¬Ā¬a * bottcherNear f d z = bottcherNear f d z
ā¢ Ā¬a ā 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | have b0 := bottcherNear_ne_zero s m1 z0 | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : a * bottcherNear f d z = bottcherNear f d z
ā¢ a = 1 | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : a * bottcherNear f d z = bottcherNear f d z
b0 : bottcherNear f d z ā 0
ā¢ a = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : a * bottcherNear f d z = bottcherNear f d z
ā¢ a = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | calc a
_ = a * bottcherNear f d z / bottcherNear f d z := by field_simp [b0]
_ = bottcherNear f d z / bottcherNear f d z := by rw [a1]
_ = 1 := div_self b0 | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : a * bottcherNear f d z = bottcherNear f d z
b0 : bottcherNear f d z ā 0
ā¢ a = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : a * bottcherNear f d z = bottcherNear f d z
b0 : bottcherNear f d z ā 0
ā¢ a = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | field_simp [b0] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : a * bottcherNear f d z = bottcherNear f d z
b0 : bottcherNear f d z ā 0
ā¢ a = a * bottcherNear f d z / bottcherNear f d z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : a * bottcherNear f d z = bottcherNear f d z
b0 : bottcherNear f d z ā 0
ā¢ a = a * bottcherNear f d z / bottcherNear f d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rw [a1] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : a * bottcherNear f d z = bottcherNear f d z
b0 : bottcherNear f d z ā 0
ā¢ a * bottcherNear f d z / bottcherNear f d z = bottcherNear f d z / bottcherNear f d z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
z0 : z ā 0
tp : i (a * bottcherNear f d z) = z
a1 : a * bottcherNear f d z = bottcherNear f d z
b0 : bottcherNear f d z ā 0
ā¢ a * bottcherNear f d z / bottcherNear f d z = bottcherNear f d z / bottcherNear f d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | SuperAt.not_local_inj | [54, 1] | [104, 94] | rw [ā t1, bottcherNear_eqn s m0, t0, mul_pow, ad, one_mul, ā bottcherNear_eqn s m1, t2] | case intro.intro.intro.intro.intro.intro.refine_3.refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
tp :
i (a * bottcherNear f d z, bottcherNear f d z).1 = i (a * bottcherNear f d z, bottcherNear f d z).2 ā
(a * bottcherNear f d z, bottcherNear f d z).1 = (a * bottcherNear f d z, bottcherNear f d z).2
z0 : z ā 0
ā¢ f (i (a * bottcherNear f d z)) = f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.refine_3.refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
d : ā
sā : SuperAt f d
t : Set ā
s : SuperNear f d t
ba : AnalyticAt ā (bottcherNear f d) 0
nc : mfderiv I I (bottcherNear f d) 0 ā 0
i : ā ā ā
ia : HolomorphicAt I I i 0
ibā : āį¶ (x : ā) in š 0, i (bottcherNear f d x) = x
bi : āį¶ (x : ā) in š 0, bottcherNear f d (i x) = x
i0 : i 0 = 0
inj : āį¶ (p : ā Ć ā) in š (0, 0), i p.1 = i p.2 ā p.1 = p.2
a : ā
a1 : a ā 1
ad : a ^ d = 1
m0ā : āį¶ (z : ā) in š 0, i (a * bottcherNear f d z) ā t
m1ā : āį¶ (z : ā) in š 0, z ā t
t0ā : Tendsto (fun z => a * bottcherNear f d z) (š 0) (š 0)
t1ā : Tendsto (fun z => f (i (a * bottcherNear f d z))) (š 0) (š 0)
t2ā : Tendsto f (š 0) (š 0)
tpā : Tendsto (fun x => (a * bottcherNear f d x, bottcherNear f d x)) (š 0) (š (0, 0))
z : ā
m1 : z ā t
m0 : i (a * bottcherNear f d z) ā t
t2 : i (bottcherNear f d (f z)) = f z
t0 : bottcherNear f d (i (a * bottcherNear f d z)) = a * bottcherNear f d z
t1 : i (bottcherNear f d (f (i (a * bottcherNear f d z)))) = f (i (a * bottcherNear f d z))
xā : bottcherNear f d (i z) = z
ib : i (bottcherNear f d z) = z
tp :
i (a * bottcherNear f d z, bottcherNear f d z).1 = i (a * bottcherNear f d z, bottcherNear f d z).2 ā
(a * bottcherNear f d z, bottcherNear f d z).1 = (a * bottcherNear f d z, bottcherNear f d z).2
z0 : z ā 0
ā¢ f (i (a * bottcherNear f d z)) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | by_cases o0 : orderAt f 0 = 0 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case pos
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : orderAt f 0 = 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | have o1 : orderAt f 0 ā 1 := by
have d := df.deriv; contrapose d; simp only [not_not] at d
exact deriv_ne_zero_of_orderAt_eq_one d | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | have d2 : 2 ā¤ orderAt f 0 := by rw [Nat.two_le_iff]; use o0, o1 | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
d2 : 2 ā¤ orderAt f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | clear o1 df f0 | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
d2 : 2 ā¤ orderAt f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
d2 : 2 ā¤ orderAt f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | set a := leadingCoeff f 0 | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | have a0 : a ā 0 := leadingCoeff_ne_zero fa o0 | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | set g := fun z ā¦ aā»Ā¹ ā¢ f z | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | have s : SuperAt g (orderAt f 0) :=
{ d2
fa0 := analyticAt_const.mul fa
fd := by rw [orderAt_const_smul (inv_ne_zero a0)]
fc := by rw [leadingCoeff_const_smul]; simp only [smul_eq_mul, inv_mul_cancel a0] } | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | rcases s.not_local_inj with āØh, ha, h0, eā© | case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case neg.intro.intro.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0 : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | use h, ha, h0 | case neg.intro.intro.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0 : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0 : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
ā¢ āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ f (h z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0 : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | refine e.mp (eventually_of_forall ?_) | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0 : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
ā¢ āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ f (h z) = f z | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0 : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
ā¢ ā (x : ā), h x ā x ā§ g (h x) = g x ā h x ā x ā§ f (h x) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0 : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
ā¢ āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ f (h z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | intro z āØh0, hzā© | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0 : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
ā¢ ā (x : ā), h x ā x ā§ g (h x) = g x ā h x ā x ā§ f (h x) = f x | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0ā : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
z : ā
h0 : h z ā z
hz : g (h z) = g z
ā¢ h z ā z ā§ f (h z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0 : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
ā¢ ā (x : ā), h x ā x ā§ g (h x) = g x ā h x ā x ā§ f (h x) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | use h0 | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0ā : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
z : ā
h0 : h z ā z
hz : g (h z) = g z
ā¢ h z ā z ā§ f (h z) = f z | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0ā : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
z : ā
h0 : h z ā z
hz : g (h z) = g z
ā¢ f (h z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0ā : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
z : ā
h0 : h z ā z
hz : g (h z) = g z
ā¢ h z ā z ā§ f (h z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | exact (IsUnit.smul_left_cancel (Ne.isUnit (inv_ne_zero a0))).mp hz | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0ā : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
z : ā
h0 : h z ā z
hz : g (h z) = g z
ā¢ f (h z) = f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
s : SuperAt g (orderAt f 0)
h : ā ā ā
ha : AnalyticAt ā h 0
h0ā : h 0 = 0
e : āį¶ (z : ā) in š[ā ] 0, h z ā z ā§ g (h z) = g z
z : ā
h0 : h z ā z
hz : g (h z) = g z
ā¢ f (h z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | simp only [orderAt_eq_zero_iff fa, f0, Ne, eq_self_iff_true, not_true, or_false_iff] at o0 | case pos
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : orderAt f 0 = 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case pos
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : orderAt f 0 = 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | use fun z ā¦ -z, (analyticAt_id _ _).neg, neg_zero | case pos
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ āį¶ (z : ā) in š[ā ] 0, -z ā z ā§ f (-z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ ā g, AnalyticAt ā g 0 ā§ g 0 = 0 ā§ āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | rw [eventually_nhdsWithin_iff] | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ āį¶ (z : ā) in š[ā ] 0, -z ā z ā§ f (-z) = f z | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ āį¶ (x : ā) in š 0, x ā {0}į¶ ā -x ā x ā§ f (-x) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ āį¶ (z : ā) in š[ā ] 0, -z ā z ā§ f (-z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | have e0 : āį¶ z in š 0, f (-z) = 0 := by
nth_rw 1 [ā neg_zero] at o0; exact continuousAt_neg.eventually o0 | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ āį¶ (x : ā) in š 0, x ā {0}į¶ ā -x ā x ā§ f (-x) = f x | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
ā¢ āį¶ (x : ā) in š 0, x ā {0}į¶ ā -x ā x ā§ f (-x) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ āį¶ (x : ā) in š 0, x ā {0}į¶ ā -x ā x ā§ f (-x) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | refine o0.mp (e0.mp (eventually_of_forall fun z f0' f0 z0 ā¦ ?_)) | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
ā¢ āį¶ (x : ā) in š 0, x ā {0}į¶ ā -x ā x ā§ f (-x) = f x | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0 z
z0 : z ā {0}į¶
ā¢ -z ā z ā§ f (-z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
ā¢ āį¶ (x : ā) in š 0, x ā {0}į¶ ā -x ā x ā§ f (-x) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | simp only [mem_compl_singleton_iff] at z0 | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0 z
z0 : z ā {0}į¶
ā¢ -z ā z ā§ f (-z) = f z | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0 z
z0 : z ā 0
ā¢ -z ā z ā§ f (-z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0 z
z0 : z ā {0}į¶
ā¢ -z ā z ā§ f (-z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | rw [Pi.zero_apply] at f0 | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0 z
z0 : z ā 0
ā¢ -z ā z ā§ f (-z) = f z | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0
z0 : z ā 0
ā¢ -z ā z ā§ f (-z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0 z
z0 : z ā 0
ā¢ -z ā z ā§ f (-z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | rw [f0, f0', eq_self_iff_true, and_true_iff, Ne, neg_eq_self_iff] | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0
z0 : z ā 0
ā¢ -z ā z ā§ f (-z) = f z | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0
z0 : z ā 0
ā¢ Ā¬z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0
z0 : z ā 0
ā¢ -z ā z ā§ f (-z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | exact z0 | case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0
z0 : z ā 0
ā¢ Ā¬z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0ā : f 0 = 0
o0 : (š 0).EventuallyEq f 0
e0 : āį¶ (z : ā) in š 0, f (-z) = 0
z : ā
f0' : f (-z) = 0
f0 : f z = 0
z0 : z ā 0
ā¢ Ā¬z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | nth_rw 1 [ā neg_zero] at o0 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ āį¶ (z : ā) in š 0, f (-z) = 0 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š (-0)).EventuallyEq f 0
ā¢ āį¶ (z : ā) in š 0, f (-z) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š 0).EventuallyEq f 0
ā¢ āį¶ (z : ā) in š 0, f (-z) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | exact continuousAt_neg.eventually o0 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š (-0)).EventuallyEq f 0
ā¢ āį¶ (z : ā) in š 0, f (-z) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : (š (-0)).EventuallyEq f 0
ā¢ āį¶ (z : ā) in š 0, f (-z) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | have d := df.deriv | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
ā¢ orderAt f 0 ā 1 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
d : deriv f 0 = 0
ā¢ orderAt f 0 ā 1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
ā¢ orderAt f 0 ā 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | contrapose d | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
d : deriv f 0 = 0
ā¢ orderAt f 0 ā 1 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
d : Ā¬orderAt f 0 ā 1
ā¢ Ā¬deriv f 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
d : deriv f 0 = 0
ā¢ orderAt f 0 ā 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | simp only [not_not] at d | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
d : Ā¬orderAt f 0 ā 1
ā¢ Ā¬deriv f 0 = 0 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
d : orderAt f 0 = 1
ā¢ Ā¬deriv f 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
d : Ā¬orderAt f 0 ā 1
ā¢ Ā¬deriv f 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | exact deriv_ne_zero_of_orderAt_eq_one d | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
d : orderAt f 0 = 1
ā¢ Ā¬deriv f 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
d : orderAt f 0 = 1
ā¢ Ā¬deriv f 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | rw [Nat.two_le_iff] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
ā¢ 2 ā¤ orderAt f 0 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
ā¢ orderAt f 0 ā 0 ā§ orderAt f 0 ā 1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
ā¢ 2 ā¤ orderAt f 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | use o0, o1 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
ā¢ orderAt f 0 ā 0 ā§ orderAt f 0 ā 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
df : HasDerivAt f 0 0
f0 : f 0 = 0
o0 : Ā¬orderAt f 0 = 0
o1 : orderAt f 0 ā 1
ā¢ orderAt f 0 ā 0 ā§ orderAt f 0 ā 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | rw [orderAt_const_smul (inv_ne_zero a0)] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
ā¢ orderAt g 0 = orderAt f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
ā¢ orderAt g 0 = orderAt f 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | rw [leadingCoeff_const_smul] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
ā¢ leadingCoeff g 0 = 1 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
ā¢ aā»Ā¹ ā¢ leadingCoeff (fun z => f z) 0 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
ā¢ leadingCoeff g 0 = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero' | [108, 1] | [135, 69] | simp only [smul_eq_mul, inv_mul_cancel a0] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
ā¢ aā»Ā¹ ā¢ leadingCoeff (fun z => f z) 0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
fa : AnalyticAt ā f 0
o0 : Ā¬orderAt f 0 = 0
d2 : 2 ā¤ orderAt f 0
a : ā := leadingCoeff f 0
a0 : a ā 0
g : ā ā ā := fun z => aā»Ā¹ ā¢ f z
ā¢ aā»Ā¹ ā¢ leadingCoeff (fun z => f z) 0 = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | set f' := fun z ā¦ f (z + c) - f c | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | have fa' : AnalyticAt ā f' 0 :=
AnalyticAt.sub
(AnalyticAt.comp (by simp only [zero_add, fa]) ((analyticAt_id _ _).add analyticAt_const))
analyticAt_const | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | have df' : HasDerivAt f' (0 * 1) 0 := by
refine HasDerivAt.sub_const ?_ _
have e : (fun z ā¦ f (z + c)) = f ā fun z ā¦ z + c := rfl
rw [e]; apply HasDerivAt.comp; simp only [zero_add, df]
exact HasDerivAt.add_const (hasDerivAt_id _) _ | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' (0 * 1) 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | simp only [MulZeroClass.zero_mul] at df' | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' (0 * 1) 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' (0 * 1) 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | have f0' : (fun z ā¦ f (z + c) - f c) 0 = 0 := by simp only [zero_add, sub_self] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
f0' : (fun z => f (z + c) - f c) 0 = 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | rcases not_local_inj_of_deriv_zero' fa' df' f0' with āØg, ga, e, hā© | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
f0' : (fun z => f (z + c) - f c) 0 = 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | case intro.intro.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
f0' : (fun z => f (z + c) - f c) 0 = 0
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | clear fa df fa' df' | case intro.intro.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | case intro.intro.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | refine āØfun z ā¦ g (z - c) + c, ?_, ?_, ?_ā© | case intro.intro.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z | case intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ AnalyticAt ā (fun z => g (z - c) + c) c
case intro.intro.intro.refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ (fun z => g (z - c) + c) c = c
case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ āį¶ (z : ā) in š[ā ] c, (fun z => g (z - c) + c) z ā z ā§ f ((fun z => g (z - c) + c) z) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ ā g, AnalyticAt ā g c ā§ g c = c ā§ āį¶ (z : ā) in š[ā ] c, g z ā z ā§ f (g z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | simp only [zero_add, fa] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
ā¢ AnalyticAt ā f (0 + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
ā¢ AnalyticAt ā f (0 + c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | refine HasDerivAt.sub_const ?_ _ | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
ā¢ HasDerivAt f' (0 * 1) 0 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
ā¢ HasDerivAt (fun z => f (z + c)) (0 * 1) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
ā¢ HasDerivAt f' (0 * 1) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | have e : (fun z ā¦ f (z + c)) = f ā fun z ā¦ z + c := rfl | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
ā¢ HasDerivAt (fun z => f (z + c)) (0 * 1) 0 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (fun z => f (z + c)) (0 * 1) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
ā¢ HasDerivAt (fun z => f (z + c)) (0 * 1) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | rw [e] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (fun z => f (z + c)) (0 * 1) 0 | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (f ā fun z => z + c) (0 * 1) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (fun z => f (z + c)) (0 * 1) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | apply HasDerivAt.comp | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (f ā fun z => z + c) (0 * 1) 0 | case hhā
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt f 0 (0 + c)
case hh
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (fun z => z + c) 1 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (f ā fun z => z + c) (0 * 1) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | simp only [zero_add, df] | case hhā
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt f 0 (0 + c)
case hh
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (fun z => z + c) 1 0 | case hh
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (fun z => z + c) 1 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hhā
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt f 0 (0 + c)
case hh
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (fun z => z + c) 1 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | exact HasDerivAt.add_const (hasDerivAt_id _) _ | case hh
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (fun z => z + c) 1 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hh
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
e : (fun z => f (z + c)) = f ā fun z => z + c
ā¢ HasDerivAt (fun z => z + c) 1 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | simp only [zero_add, sub_self] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
ā¢ (fun z => f (z + c) - f c) 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
fa : AnalyticAt ā f c
df : HasDerivAt f 0 c
f' : ā ā ā := fun z => f (z + c) - f c
fa' : AnalyticAt ā f' 0
df' : HasDerivAt f' 0 0
ā¢ (fun z => f (z + c) - f c) 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | exact AnalyticAt.add (AnalyticAt.comp (by simp only [sub_self, ga])
((analyticAt_id _ _).sub analyticAt_const)) analyticAt_const | case intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ AnalyticAt ā (fun z => g (z - c) + c) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_1
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ AnalyticAt ā (fun z => g (z - c) + c) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | simp only [sub_self, ga] | S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ AnalyticAt ā g (c - c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ AnalyticAt ā g (c - c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | simp only [sub_self, e, zero_add] | case intro.intro.intro.refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ (fun z => g (z - c) + c) c = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_2
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ (fun z => g (z - c) + c) c = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | simp only [eventually_nhdsWithin_iff] at h ā¢ | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ āį¶ (z : ā) in š[ā ] c, (fun z => g (z - c) + c) z ā z ā§ f ((fun z => g (z - c) + c) z) = f z | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
ā¢ āį¶ (x : ā) in š c, x ā {c}į¶ ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (z : ā) in š[ā ] 0, g z ā z ā§ f' (g z) = f' z
ā¢ āį¶ (z : ā) in š[ā ] c, (fun z => g (z - c) + c) z ā z ā§ f ((fun z => g (z - c) + c) z) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | have sc : Tendsto (fun z ā¦ z - c) (š c) (š 0) := by
rw [ā sub_self c]; exact continuousAt_id.sub continuousAt_const | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
ā¢ āį¶ (x : ā) in š c, x ā {c}į¶ ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
ā¢ āį¶ (x : ā) in š c, x ā {c}į¶ ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
ā¢ āį¶ (x : ā) in š c, x ā {c}į¶ ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | refine (sc.eventually h).mp (eventually_of_forall ?_) | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
ā¢ āį¶ (x : ā) in š c, x ā {c}į¶ ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
ā¢ ā (x : ā),
(x - c ā {0}į¶ ā g (x - c) ā x - c ā§ f' (g (x - c)) = f' (x - c)) ā
x ā {c}į¶ ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
ā¢ āį¶ (x : ā) in š c, x ā {c}į¶ ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | simp only [mem_compl_singleton_iff, sub_ne_zero] | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
ā¢ ā (x : ā),
(x - c ā {0}į¶ ā g (x - c) ā x - c ā§ f' (g (x - c)) = f' (x - c)) ā
x ā {c}į¶ ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
ā¢ ā (x : ā),
(x ā c ā g (x - c) ā x - c ā§ f' (g (x - c)) = f' (x - c)) ā x ā c ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
ā¢ ā (x : ā),
(x - c ā {0}į¶ ā g (x - c) ā x - c ā§ f' (g (x - c)) = f' (x - c)) ā
x ā {c}į¶ ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | intro z h zc | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
ā¢ ā (x : ā),
(x ā c ā g (x - c) ā x - c ā§ f' (g (x - c)) = f' (x - c)) ā x ā c ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ā¢ g (z - c) + c ā z ā§ f (g (z - c) + c) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
h : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
ā¢ ā (x : ā),
(x ā c ā g (x - c) ā x - c ā§ f' (g (x - c)) = f' (x - c)) ā x ā c ā g (x - c) + c ā x ā§ f (g (x - c) + c) = f x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | rcases h zc with āØgz, ffā© | case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ā¢ g (z - c) + c ā z ā§ f (g (z - c) + c) = f z | case intro.intro.intro.refine_3.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ g (z - c) + c ā z ā§ f (g (z - c) + c) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ā¢ g (z - c) + c ā z ā§ f (g (z - c) + c) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | constructor | case intro.intro.intro.refine_3.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ g (z - c) + c ā z ā§ f (g (z - c) + c) = f z | case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ g (z - c) + c ā z
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ g (z - c) + c ā z ā§ f (g (z - c) + c) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | contrapose gz | case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ g (z - c) + c ā z
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z | case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ff : f' (g (z - c)) = f' (z - c)
gz : Ā¬g (z - c) + c ā z
ā¢ Ā¬g (z - c) ā z - c
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ g (z - c) + c ā z
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | simp only [not_not] at gz ā¢ | case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ff : f' (g (z - c)) = f' (z - c)
gz : Ā¬g (z - c) + c ā z
ā¢ Ā¬g (z - c) ā z - c
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z | case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ff : f' (g (z - c)) = f' (z - c)
gz : g (z - c) + c = z
ā¢ g (z - c) = z - c
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ff : f' (g (z - c)) = f' (z - c)
gz : Ā¬g (z - c) + c ā z
ā¢ Ā¬g (z - c) ā z - c
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | nth_rw 2 [ā gz] | case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ff : f' (g (z - c)) = f' (z - c)
gz : g (z - c) + c = z
ā¢ g (z - c) = z - c
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z | case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ff : f' (g (z - c)) = f' (z - c)
gz : g (z - c) + c = z
ā¢ g (z - c) = g (z - c) + c - c
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ff : f' (g (z - c)) = f' (z - c)
gz : g (z - c) + c = z
ā¢ g (z - c) = z - c
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | ring | case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ff : f' (g (z - c)) = f' (z - c)
gz : g (z - c) + c = z
ā¢ g (z - c) = g (z - c) + c - c
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z | case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.left
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
ff : f' (g (z - c)) = f' (z - c)
gz : g (z - c) + c = z
ā¢ g (z - c) = g (z - c) + c - c
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multiple.lean | not_local_inj_of_deriv_zero | [140, 1] | [167, 65] | simp only [sub_left_inj, sub_add_cancel, f'] at ff | case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z | case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f (g (z - c) + c) = f z
ā¢ f (g (z - c) + c) = f z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3.intro.right
S : Type
instāāµ : TopologicalSpace S
instāā“ : ChartedSpace ā S
instāĀ³ : AnalyticManifold I S
T : Type
instāĀ² : TopologicalSpace T
instāĀ¹ : ChartedSpace ā T
instā : AnalyticManifold I T
f : ā ā ā
c : ā
f' : ā ā ā := fun z => f (z + c) - f c
f0' : (fun z => f (z + c) - f c) 0 = 0
g : ā ā ā
ga : AnalyticAt ā g 0
e : g 0 = 0
hā : āį¶ (x : ā) in š 0, x ā {0}į¶ ā g x ā x ā§ f' (g x) = f' x
sc : Tendsto (fun z => z - c) (š c) (š 0)
z : ā
h : z ā c ā g (z - c) ā z - c ā§ f' (g (z - c)) = f' (z - c)
zc : z ā c
gz : g (z - c) ā z - c
ff : f' (g (z - c)) = f' (z - c)
ā¢ f (g (z - c) + c) = f z
TACTIC:
|
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