url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | rcases h with ⟨r, ⟨rp, rxy⟩, rr⟩ | case hX
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
h : ∃ a, (0 < a ∧ a < dist x y) ∧ a ∉ R
⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U | case hX.intro.intro.intro
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U | Please generate a tactic in lean4 to solve the state.
STATE:
case hX
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
h : ∃ a, (0 < a ∧ a < dist x y) ∧ a ∉ R
⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | have e : ball x r = closedBall x r := by
apply Set.ext; intro z; simp only [mem_ball, mem_closedBall]
simp only [mem_setOf, not_exists, ← hR] at rr; simp only [Ne.le_iff_lt (rr z x)] | case hX.intro.intro.intro
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U | case hX.intro.intro.intro
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U | Please generate a tactic in lean4 to solve the state.
STATE:
case hX.intro.intro.intro
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | refine ⟨ball x r, ⟨?_, isOpen_ball⟩, ?_⟩ | case hX.intro.intro.intro
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U | case hX.intro.intro.intro.refine_1
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ IsClosed (ball x r)
case hX.intro.intro.intro.refine_2
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ x ∈ ball x r ∧ y ∉ ball x r | Please generate a tactic in lean4 to solve the state.
STATE:
case hX.intro.intro.intro
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | rw [e] | case hX.intro.intro.intro.refine_1
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ IsClosed (ball x r)
case hX.intro.intro.intro.refine_2
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ x ∈ ball x r ∧ y ∉ ball x r | case hX.intro.intro.intro.refine_1
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ IsClosed (closedBall x r)
case hX.intro.intro.intro.refine_2
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ x ∈ ball x r ∧ y ∉ ball x r | Please generate a tactic in lean4 to solve the state.
STATE:
case hX.intro.intro.intro.refine_1
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ IsClosed (ball x r)
case hX.intro.intro.intro.refine_2
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ x ∈ ball x r ∧ y ∉ ball x r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | exact isClosed_ball | case hX.intro.intro.intro.refine_1
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ IsClosed (closedBall x r)
case hX.intro.intro.intro.refine_2
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ x ∈ ball x r ∧ y ∉ ball x r | case hX.intro.intro.intro.refine_2
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ x ∈ ball x r ∧ y ∉ ball x r | Please generate a tactic in lean4 to solve the state.
STATE:
case hX.intro.intro.intro.refine_1
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ IsClosed (closedBall x r)
case hX.intro.intro.intro.refine_2
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ x ∈ ball x r ∧ y ∉ ball x r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | use mem_ball_self rp | case hX.intro.intro.intro.refine_2
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ x ∈ ball x r ∧ y ∉ ball x r | case right
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ y ∉ ball x r | Please generate a tactic in lean4 to solve the state.
STATE:
case hX.intro.intro.intro.refine_2
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ x ∈ ball x r ∧ y ∉ ball x r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | simp only [mem_ball, not_lt] | case right
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ y ∉ ball x r | case right
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ r ≤ dist y x | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ y ∉ ball x r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | rw [dist_comm] | case right
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ r ≤ dist y x | case right
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ r ≤ dist x y | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ r ≤ dist y x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | exact rxy.le | case right
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ r ≤ dist x y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
e : ball x r = closedBall x r
⊢ r ≤ dist x y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | have e : R = range (uncurry dist) := by
apply Set.ext; intro r; simp only [mem_setOf, mem_range, Prod.exists, uncurry, ← hR]; rfl | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
⊢ R.Countable | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
e : R = range (uncurry dist)
⊢ R.Countable | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
⊢ R.Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | rw [e] | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
e : R = range (uncurry dist)
⊢ R.Countable | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
e : R = range (uncurry dist)
⊢ (range (uncurry dist)).Countable | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
e : R = range (uncurry dist)
⊢ R.Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | exact countable_range _ | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
e : R = range (uncurry dist)
⊢ (range (uncurry dist)).Countable | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
e : R = range (uncurry dist)
⊢ (range (uncurry dist)).Countable
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | apply Set.ext | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
⊢ R = range (uncurry dist) | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
⊢ ∀ (x : ℝ), x ∈ R ↔ x ∈ range (uncurry dist) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
⊢ R = range (uncurry dist)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | intro r | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
⊢ ∀ (x : ℝ), x ∈ R ↔ x ∈ range (uncurry dist) | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
r : ℝ
⊢ r ∈ R ↔ r ∈ range (uncurry dist) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
⊢ ∀ (x : ℝ), x ∈ R ↔ x ∈ range (uncurry dist)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | simp only [mem_setOf, mem_range, Prod.exists, uncurry, ← hR] | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
r : ℝ
⊢ r ∈ R ↔ r ∈ range (uncurry dist) | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
r : ℝ
⊢ (∃ x y, dist x y = r) ↔ ∃ a b, dist a b = r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
r : ℝ
⊢ r ∈ R ↔ r ∈ range (uncurry dist)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | rfl | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
r : ℝ
⊢ (∃ x y, dist x y = r) ↔ ∃ a b, dist a b = r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
r : ℝ
⊢ (∃ x y, dist x y = r) ↔ ∃ a b, dist a b = r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | by_contra h | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
⊢ ¬Ioo 0 (dist x y) ⊆ R | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
h : Ioo 0 (dist x y) ⊆ R
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
⊢ ¬Ioo 0 (dist x y) ⊆ R
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | exact not_countable_Ioo xy (rc.mono h) | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
h : Ioo 0 (dist x y) ⊆ R
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
h : Ioo 0 (dist x y) ⊆ R
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | apply Set.ext | X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
⊢ ball x r = closedBall x r | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
⊢ ∀ (x_1 : X), x_1 ∈ ball x r ↔ x_1 ∈ closedBall x r | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
⊢ ball x r = closedBall x r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | intro z | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
⊢ ∀ (x_1 : X), x_1 ∈ ball x r ↔ x_1 ∈ closedBall x r | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
z : X
⊢ z ∈ ball x r ↔ z ∈ closedBall x r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
⊢ ∀ (x_1 : X), x_1 ∈ ball x r ↔ x_1 ∈ closedBall x r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | simp only [mem_ball, mem_closedBall] | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
z : X
⊢ z ∈ ball x r ↔ z ∈ closedBall x r | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
z : X
⊢ dist z x < r ↔ dist z x ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
z : X
⊢ z ∈ ball x r ↔ z ∈ closedBall x r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | simp only [mem_setOf, not_exists, ← hR] at rr | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
z : X
⊢ dist z x < r ↔ dist z x ≤ r | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rp : 0 < r
rxy : r < dist x y
z : X
rr : ∀ (x x_1 : X), ¬dist x x_1 = r
⊢ dist z x < r ↔ dist z x ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rr : r ∉ R
rp : 0 < r
rxy : r < dist x y
z : X
⊢ dist z x < r ↔ dist z x ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | Countable.totallyDisconnectedSpace | [62, 1] | [78, 61] | simp only [Ne.le_iff_lt (rr z x)] | case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rp : 0 < r
rxy : r < dist x y
z : X
rr : ∀ (x x_1 : X), ¬dist x x_1 = r
⊢ dist z x < r ↔ dist z x ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝¹ : MetricSpace X
inst✝ : Countable X
R : Set ℝ
hR : {r | ∃ x y, dist x y = r} = R
rc : R.Countable
x y : X
xy : 0 < dist x y
r : ℝ
rp : 0 < r
rxy : r < dist x y
z : X
rr : ∀ (x x_1 : X), ¬dist x x_1 = r
⊢ dist z x < r ↔ dist z x ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | IsCountable.isTotallyDisconnected | [81, 1] | [84, 74] | rw [← isTotallyDisconnected_iff_totally_disconnected_subtype] | X : Type
inst✝ : MetricSpace X
s : Set X
h : s.Countable
⊢ IsTotallyDisconnected s | X : Type
inst✝ : MetricSpace X
s : Set X
h : s.Countable
⊢ TotallyDisconnectedSpace ↑s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MetricSpace X
s : Set X
h : s.Countable
⊢ IsTotallyDisconnected s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/TotallyDisconnected.lean | IsCountable.isTotallyDisconnected | [81, 1] | [84, 74] | exact @Countable.totallyDisconnectedSpace _ _ (countable_coe_iff.mpr h) | X : Type
inst✝ : MetricSpace X
s : Set X
h : s.Countable
⊢ TotallyDisconnectedSpace ↑s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : MetricSpace X
s : Set X
h : s.Countable
⊢ TotallyDisconnectedSpace ↑s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.zz | [49, 1] | [50, 85] | simp only [Prod.snd, Cinv.z', PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ↑(extChartAt I z).symm (c, i.z').2 = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ↑(extChartAt I z).symm (c, i.z').2 = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.fa' | [60, 1] | [64, 13] | have fa := i.fa | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ AnalyticAt ℂ i.f' (c, i.z') | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ AnalyticAt ℂ i.f' (c, i.z') | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ AnalyticAt ℂ i.f' (c, i.z')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.fa' | [60, 1] | [64, 13] | simp only [holomorphicAt_iff, uncurry, extChartAt_prod, Function.comp, PartialEquiv.prod_coe_symm,
PartialEquiv.prod_coe] at fa | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ AnalyticAt ℂ i.f' (c, i.z') | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
fa :
ContinuousAt (uncurry f) (c, z) ∧
AnalyticAt ℂ (fun x => ↑(extChartAt I (f c z)) (f (↑(extChartAt I c).symm x.1) (↑(extChartAt I z).symm x.2)))
(↑(extChartAt I c) c, ↑(extChartAt I z) z)
⊢ AnalyticAt ℂ i.f' (c, i.z') | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ AnalyticAt ℂ i.f' (c, i.z')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.fa' | [60, 1] | [64, 13] | exact fa.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
fa :
ContinuousAt (uncurry f) (c, z) ∧
AnalyticAt ℂ (fun x => ↑(extChartAt I (f c z)) (f (↑(extChartAt I c).symm x.1) (↑(extChartAt I z).symm x.2)))
(↑(extChartAt I c) c, ↑(extChartAt I z) z)
⊢ AnalyticAt ℂ i.f' (c, i.z') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
fa :
ContinuousAt (uncurry f) (c, z) ∧
AnalyticAt ℂ (fun x => ↑(extChartAt I (f c z)) (f (↑(extChartAt I c).symm x.1) (↑(extChartAt I z).symm x.2)))
(↑(extChartAt I c) c, ↑(extChartAt I z) z)
⊢ AnalyticAt ℂ i.f' (c, i.z')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | apply HasMFDerivAt.comp (I' := I) (c, i.z') | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I i.f' (c, i.z') i.df' | case hg
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (↑(extChartAt I (f c z))) (f (c, i.z').1 (↑(extChartAt I z).symm (c, i.z').2)) i.de'
case hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun x => f x.1 (↑(extChartAt I z).symm x.2)) (c, i.z') i.df | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I i.f' (c, i.z') i.df'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | rw [i.zz] | case hg
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (↑(extChartAt I (f c z))) (f (c, i.z').1 (↑(extChartAt I z).symm (c, i.z').2)) i.de' | case hg
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (↑(extChartAt I (f c z))) (f (c, i.z').1 z) i.de' | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (↑(extChartAt I (f c z))) (f (c, i.z').1 (↑(extChartAt I z).symm (c, i.z').2)) i.de'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | exact (HolomorphicAt.extChartAt (mem_extChartAt_source _ _)).mdifferentiableAt.hasMFDerivAt | case hg
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (↑(extChartAt I (f c z))) (f (c, i.z').1 z) i.de' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (↑(extChartAt I (f c z))) (f (c, i.z').1 z) i.de'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | simp only [Cinv.df] | case hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun x => f x.1 (↑(extChartAt I z).symm x.2)) (c, i.z') i.df | case hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun x => f x.1 (↑(extChartAt I z).symm x.2)) (c, i.z')
(i.dfc.comp dc + i.dfz.comp (i.de.comp dz)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun x => f x.1 (↑(extChartAt I z).symm x.2)) (c, i.z') i.df
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | apply MDifferentiableAt.hasMFDerivAt_comp2 (J := I) (co := cms) | case hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun x => f x.1 (↑(extChartAt I z).symm x.2)) (c, i.z')
(i.dfc.comp dc + i.dfz.comp (i.de.comp dz)) | case hf.fd
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ MDifferentiableAt (I.prod I) I (uncurry f) ((c, i.z').1, ↑(extChartAt I z).symm (c, i.z').2)
case hf.gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun x => f x.1 (↑(extChartAt I z).symm x.2)) (c, i.z')
(i.dfc.comp dc + i.dfz.comp (i.de.comp dz))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | rw [i.zz] | case hf.fd
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ MDifferentiableAt (I.prod I) I (uncurry f) ((c, i.z').1, ↑(extChartAt I z).symm (c, i.z').2)
case hf.gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | case hf.fd
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ MDifferentiableAt (I.prod I) I (uncurry f) ((c, i.z').1, z)
case hf.gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fd
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ MDifferentiableAt (I.prod I) I (uncurry f) ((c, i.z').1, ↑(extChartAt I z).symm (c, i.z').2)
case hf.gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | exact i.fa.mdifferentiableAt | case hf.fd
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ MDifferentiableAt (I.prod I) I (uncurry f) ((c, i.z').1, z)
case hf.gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | case hf.gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fd
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ MDifferentiableAt (I.prod I) I (uncurry f) ((c, i.z').1, z)
case hf.gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | apply hasMFDerivAt_fst | case hf.gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | refine HasMFDerivAt.comp _ ?_ (hasMFDerivAt_snd _ _ _) | case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I ↑(extChartAt I z).symm (c, i.z').2 i.de
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => ↑(extChartAt I z).symm y.2) (c, i.z') (i.de.comp dz)
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | exact (HolomorphicAt.extChartAt_symm (mem_extChartAt_target _ _)).mdifferentiableAt.hasMFDerivAt | case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I ↑(extChartAt I z).symm (c, i.z').2 i.de
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I ↑(extChartAt I z).symm (c, i.z').2 i.de
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | rw [i.zz] | case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y z) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y (↑(extChartAt I z).symm (c, i.z').2)) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | exact i.fa.along_fst.mdifferentiableAt.hasMFDerivAt | case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y z) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fh0
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f y z) (c, i.z').1 i.dfc
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | rw [i.zz] | case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz | case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) z i.dfz | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) (↑(extChartAt I z).symm (c, i.z').2) i.dfz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_df' | [106, 1] | [117, 67] | exact i.fa.along_snd.mdifferentiableAt.hasMFDerivAt | case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) z i.dfz | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.fh1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt I I (fun y => f (c, i.z').1 y) z i.dfz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_dh | [119, 1] | [120, 64] | refine HasMFDerivAt.prod ?_ i.has_df' | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) (I.prod I) i.h (c, i.z') i.dh | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) (I.prod I) i.h (c, i.z') i.dh
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_dh | [119, 1] | [120, 64] | apply hasMFDerivAt_fst | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de | [142, 1] | [146, 89] | intro t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ), i.dei (i.de t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.dei (i.de t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ), i.dei (i.de t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de | [142, 1] | [146, 89] | have h := ContinuousLinearMap.ext_iff.mp
(extChartAt_mderiv_right_inverse' (mem_extChartAt_source I z)) t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.dei (i.de t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I z)) z).comp (mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z))) t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I z) z))) t
⊢ i.dei (i.de t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.dei (i.de t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de | [142, 1] | [146, 89] | simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply] at h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I z)) z).comp (mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z))) t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I z) z))) t
⊢ i.dei (i.de t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h : (mfderiv I I (↑(extChartAt I z)) z) ((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) t) = t
⊢ i.dei (i.de t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I z)) z).comp (mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z))) t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I z) z))) t
⊢ i.dei (i.de t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de | [142, 1] | [146, 89] | exact h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h : (mfderiv I I (↑(extChartAt I z)) z) ((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) t) = t
⊢ i.dei (i.de t) = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h : (mfderiv I I (↑(extChartAt I z)) z) ((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) t) = t
⊢ i.dei (i.de t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de' | [148, 1] | [152, 89] | intro t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : TangentSpace I (f c z)), i.dei' (i.de' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
⊢ i.dei' (i.de' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : TangentSpace I (f c z)), i.dei' (i.de' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de' | [148, 1] | [152, 89] | have h := ContinuousLinearMap.ext_iff.mp (extChartAt_mderiv_left_inverse
(mem_extChartAt_source I (f c z))) t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
⊢ i.dei' (i.de' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))).comp
(mfderiv I I (↑(extChartAt I (f c z))) (f c z)))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
⊢ i.dei' (i.de' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de' | [148, 1] | [152, 89] | simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply] at h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))).comp
(mfderiv I I (↑(extChartAt I (f c z))) (f c z)))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z)))
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))).comp
(mfderiv I I (↑(extChartAt I (f c z))) (f c z)))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de' | [148, 1] | [152, 89] | exact h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z)))
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z)))
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei | [154, 1] | [158, 89] | intro t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : TangentSpace I z), i.de (i.dei t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
⊢ i.de (i.dei t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : TangentSpace I z), i.de (i.dei t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei | [154, 1] | [158, 89] | have h := ContinuousLinearMap.ext_iff.mp (extChartAt_mderiv_left_inverse
(mem_extChartAt_source I z)) t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
⊢ i.de (i.dei t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)).comp (mfderiv I I (↑(extChartAt I z)) z)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
⊢ i.de (i.dei t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei | [154, 1] | [158, 89] | simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply] at h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)).comp (mfderiv I I (↑(extChartAt I z)) z)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) ((mfderiv I I (↑(extChartAt I z)) z) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)).comp (mfderiv I I (↑(extChartAt I z)) z)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei | [154, 1] | [158, 89] | exact h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) ((mfderiv I I (↑(extChartAt I z)) z) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) ((mfderiv I I (↑(extChartAt I z)) z) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei' | [160, 1] | [164, 89] | intro t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ), i.de' (i.dei' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.de' (i.dei' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ), i.de' (i.dei' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei' | [160, 1] | [164, 89] | have h := ContinuousLinearMap.ext_iff.mp (extChartAt_mderiv_right_inverse'
(mem_extChartAt_source I (f c z))) t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.de' (i.dei' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)).comp
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I (f c z)) (f c z)))) t
⊢ i.de' (i.dei' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.de' (i.dei' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei' | [160, 1] | [164, 89] | simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply] at h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)).comp
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I (f c z)) (f c z)))) t
⊢ i.de' (i.dei' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
(mfderiv I I (↑(extChartAt I (f c z))) (f c z))
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))) t) =
t
⊢ i.de' (i.dei' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)).comp
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I (f c z)) (f c z)))) t
⊢ i.de' (i.dei' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei' | [160, 1] | [164, 89] | exact h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
(mfderiv I I (↑(extChartAt I (f c z))) (f c z))
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))) t) =
t
⊢ i.de' (i.dei' t) = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
(mfderiv I I (↑(extChartAt I (f c z))) (f c z))
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))) t) =
t
⊢ i.de' (i.dei' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dhi_dh | [166, 1] | [172, 54] | intro ⟨u, v⟩ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ × ℂ), i.dhi (i.dh t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dhi (i.dh (u, v)) = (u, v) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ × ℂ), i.dhi (i.dh t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dhi_dh | [166, 1] | [172, 54] | simp only [Cinv.dh, Cinv.dhi, dc, dz, Cinv.dfi', Cinv.df', Cinv.df, i.dei_de', i.dei_de,
i.dfzi_dfz, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
ContinuousLinearMap.sub_apply, ContinuousLinearMap.coe_fst', ContinuousLinearMap.coe_snd',
ContinuousLinearMap.add_apply, ContinuousLinearMap.map_add, ContinuousLinearMap.map_sub,
add_sub_cancel_left, ContinuousLinearMap.coe_snd] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dhi (i.dh (u, v)) = (u, v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dhi (i.dh (u, v)) = (u, v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dh_dhi | [174, 1] | [180, 100] | intro ⟨u, v⟩ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ × ℂ), i.dh (i.dhi t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dh (i.dhi (u, v)) = (u, v) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ × ℂ), i.dh (i.dhi t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dh_dhi | [174, 1] | [180, 100] | simp only [Cinv.dh, Cinv.dhi, dc, dz, Cinv.dfi', Cinv.df', Cinv.df, i.de_dei', i.de_dei,
i.dfz_dfzi, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
ContinuousLinearMap.sub_apply, ContinuousLinearMap.coe_fst', ContinuousLinearMap.coe_snd',
ContinuousLinearMap.add_apply, ContinuousLinearMap.map_add, ContinuousLinearMap.map_sub,
add_sub_cancel_left, ← add_sub_assoc, ContinuousLinearMap.coe_snd, ContinuousLinearMap.coe_fst] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dh (i.dhi (u, v)) = (u, v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dh (i.dhi (u, v)) = (u, v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | have a := ContDiffAt.localInverse_apply_image i.ha.contDiffAt i.has_dhe le_top | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | have e : ContDiffAt.localInverse i.ha.contDiffAt i.has_dhe le_top = i.he.symm := rfl | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | rw [e] at a | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | clear e | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | simp only [Cinv.z', Cinv.h, Cinv.f', PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] at a | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z)) = (c, ↑(extChartAt I z) z)
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | rw [a] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z)) = (c, ↑(extChartAt I z) z)
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z)) = (c, ↑(extChartAt I z) z)
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | generalize ht :
((extChartAt II (c, z)).source ∩ extChartAt II (c, z) ⁻¹' i.he.source : Set (ℂ × S)) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | have o : IsOpen t := by
rw [← ht]
exact (continuousOn_extChartAt _ _).isOpen_inter_preimage (isOpen_extChartAt_source _ _)
i.he.open_source | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | have m : (c, z) ∈ t := by
simp only [mem_inter_iff, mem_preimage, mem_extChartAt_source, true_and_iff, ← ht]
exact ContDiffAt.mem_toPartialHomeomorph_source i.ha.contDiffAt i.has_dhe le_top | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | apply Filter.eventuallyEq_of_mem (o.mem_nhds m) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ EqOn (fun x => i.g x.1 (f x.1 x.2)) (fun x => x.2) t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | intro x m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ EqOn (fun x => i.g x.1 (f x.1 x.2)) (fun x => x.2) t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x ∈ t
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ EqOn (fun x => i.g x.1 (f x.1 x.2)) (fun x => x.2) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | simp only [mem_inter_iff, mem_preimage, extChartAt_prod, extChartAt_eq_refl, ← ht,
PartialEquiv.prod_source, PartialEquiv.refl_source, mem_prod_eq, mem_univ, true_and_iff,
PartialEquiv.prod_coe, PartialEquiv.refl_coe, id] at m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x ∈ t
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x ∈ t
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | have inv := i.he.left_inv m.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | simp only [Cinv.g] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | generalize hq : i.he.symm = q | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | rw [hq] at inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | rw [Cinv.he, ContDiffAt.toPartialHomeomorph_coe i.ha.contDiffAt i.has_dhe le_top] at inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (i.h (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | simp only [Cinv.h, Cinv.f', PartialEquiv.left_inv _ m.1] at inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (i.h (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
inv : ↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (i.h (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | simp only [inv, PartialEquiv.left_inv _ m.1] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
inv : ↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
inv : ↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | rw [← ht] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ IsOpen t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ IsOpen ((extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ IsOpen t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | exact (continuousOn_extChartAt _ _).isOpen_inter_preimage (isOpen_extChartAt_source _ _)
i.he.open_source | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ IsOpen ((extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ IsOpen ((extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | simp only [mem_inter_iff, mem_preimage, mem_extChartAt_source, true_and_iff, ← ht] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ (c, z) ∈ t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ↑(extChartAt (I.prod I) (c, z)) (c, z) ∈ i.he.source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ (c, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | exact ContDiffAt.mem_toPartialHomeomorph_source i.ha.contDiffAt i.has_dhe le_top | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ↑(extChartAt (I.prod I) (c, z)) (c, z) ∈ i.he.source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ↑(extChartAt (I.prod I) (c, z)) (c, z) ∈ i.he.source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | intro x m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ x ∈ i.he.target, (↑i.he.symm x).1 = x.1 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
⊢ (↑i.he.symm x).1 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ x ∈ i.he.target, (↑i.he.symm x).1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | have e : i.he (i.he.symm x) = x := i.he.right_inv m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
⊢ (↑i.he.symm x).1 = x.1 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
⊢ (↑i.he.symm x).1 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
⊢ (↑i.he.symm x).1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | generalize hq : i.he.symm x = q | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
⊢ (↑i.he.symm x).1 = x.1 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
q : ℂ × ℂ
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
⊢ (↑i.he.symm x).1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | rw [hq] at e | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
q : ℂ × ℂ
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : ↑i.he q = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
q : ℂ × ℂ
hq : ↑i.he.symm x = q
⊢ q.1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | rw [Cinv.he, ContDiffAt.toPartialHomeomorph_coe i.ha.contDiffAt i.has_dhe le_top, Cinv.h] at e | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : ↑i.he q = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : (q.1, i.f' q) = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : ↑i.he q = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | rw [← e] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : (q.1, i.f' q) = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : (q.1, i.f' q) = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | generalize ht : ((extChartAt II (c, f c z)).source ∩ extChartAt II (c, f c z) ⁻¹' i.he.target
: Set (ℂ × T)) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have o : IsOpen t := by
rw [← ht]
exact (continuousOn_extChartAt _ _).isOpen_inter_preimage (isOpen_extChartAt_source _ _)
i.he.open_target | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have m' : (c, extChartAt I (f c z) (f c z)) ∈ i.he.toPartialEquiv.target := by
have m := ContDiffAt.image_mem_toPartialHomeomorph_target i.ha.contDiffAt i.has_dhe le_top
have e : i.h (c, i.z') = (c, extChartAt I (f c z) (f c z)) := by
simp only [Cinv.h, Cinv.z', Cinv.f', PartialEquiv.left_inv _ (mem_extChartAt_source _ _)]
rw [e] at m; exact m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have m : (c, f c z) ∈ t := by
simp only [m', mem_inter_iff, mem_preimage, mem_extChartAt_source, true_and_iff, ← ht,
extChartAt_prod, PartialEquiv.prod_coe, extChartAt_eq_refl, PartialEquiv.refl_coe, id,
PartialEquiv.prod_source, prod_mk_mem_set_prod_eq, PartialEquiv.refl_source, mem_univ] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | refine fm.mp (Filter.eventually_of_mem (o.mem_nhds m) ?_) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ ∀ x ∈ t,
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source →
f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | intro x m mf | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ ∀ x ∈ t,
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source →
f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x ∈ t
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ ∀ x ∈ t,
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source →
f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
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