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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [subset_def, mem_Ioo, and_imp, mem_preimage, Function.comp] at xt ty h | case right.intro.mk.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
h : Ioo (x, y).1 (x, y).2 ⊆ f ∘ p.extend ⁻¹' interior s
xt : (x, y).1 < t
ty : t < (x, y).2
⊢ False | case right.intro.mk.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
h : Ioo (x, y).1 (x, y).2 ⊆ f ∘ p.extend ⁻¹' interior s
xt : (x, y).1 < t
ty : t < (x, y).2
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rcases exists_between (lt_min ty t1) with ⟨z, tz, zy1⟩ | case right.intro.mk.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
⊢ False | case right.intro.mk.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rcases lt_min_iff.mp zy1 with ⟨zy, z1⟩ | case right.intro.mk.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
⊢ False | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | suffices h : z ∈ u by linarith [le_csSup bdd h] | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ False | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ z ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rw [← hu] | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ z ∈ u | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ z ∈ Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ z ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | refine ⟨⟨_root_.trans m.1 tz.le, z1.le⟩, ?_⟩ | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ z ∈ Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ z ∈ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ z ∈ Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [mem_iInter₂, mem_Iic] | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ z ∈ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ ∀ (i : ℝ), f (p.extend i) ∉ s → z ≤ i | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ z ∈ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | intro w ws | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ ∀ (i : ℝ), f (p.extend i) ∉ s → z ≤ i | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : f (p.extend w) ∉ s
⊢ z ≤ w | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
⊢ ∀ (i : ℝ), f (p.extend i) ∉ s → z ≤ i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | contrapose ws | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : f (p.extend w) ∉ s
⊢ z ≤ w | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : ¬z ≤ w
⊢ ¬f (p.extend w) ∉ s | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : f (p.extend w) ∉ s
⊢ z ≤ w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [not_not, not_le] at ws ⊢ | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : ¬z ≤ w
⊢ ¬f (p.extend w) ∉ s | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
⊢ f (p.extend w) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : ¬z ≤ w
⊢ ¬f (p.extend w) ∉ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | by_cases xw : x < w | case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
⊢ f (p.extend w) ∈ s | case pos
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : x < w
⊢ f (p.extend w) ∈ s
case neg
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : ¬x < w
⊢ f (p.extend w) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
⊢ f (p.extend w) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | refine interior_subset (h _ xw (_root_.trans ws zy)) | case pos
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : x < w
⊢ f (p.extend w) ∈ s
case neg
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : ¬x < w
⊢ f (p.extend w) ∈ s | case neg
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : ¬x < w
⊢ f (p.extend w) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : x < w
⊢ f (p.extend w) ∈ s
case neg
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : ¬x < w
⊢ f (p.extend w) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [not_lt] at xw | case neg
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : ¬x < w
⊢ f (p.extend w) ∈ s | case neg
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : w ≤ x
⊢ f (p.extend w) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : ¬x < w
⊢ f (p.extend w) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | exact lo _ (_root_.trans xw xt.le) | case neg
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : w ≤ x
⊢ f (p.extend w) ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
w : ℝ
ws : w < z
xw : w ≤ x
⊢ f (p.extend w) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [Path.extend, IccExtend_apply, min_eq_right m.2, max_eq_right m.1] | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
⊢ p ⟨t, m⟩ = p.extend t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
⊢ p ⟨t, m⟩ = p.extend t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | linarith [le_csSup bdd h] | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝¹ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h✝ : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
h : z ∈ u
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x✝ : X
fx : f x✝ ∈ s
p : Path x✝ b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
h✝¹ : f (p.extend t) ∈ interior s
o : IsOpen (f ∘ p.extend ⁻¹' interior s)
x y : ℝ
xt : x < t
ty : t < y
h✝ : ∀ (x_1 : ℝ), x < x_1 → x_1 < y → f (p.extend x_1) ∈ interior s
z : ℝ
tz : t < z
zy1 : z < min y 1
zy : z < y
z1 : z < 1
h : z ∈ u
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rw [← hq] | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
⊢ Continuous q | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
⊢ Continuous fun a => p.extend (min (↑a) t) | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
⊢ Continuous q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | exact p.continuous_extend.comp (continuous_subtype_val.min continuous_const) | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
⊢ Continuous fun a => p.extend (min (↑a) t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
⊢ Continuous fun a => p.extend (min (↑a) t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [← hq] | case neg.refine_1
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ { toFun := q, continuous_toFun := qc }.toFun 0 = x | case neg.refine_1
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ p.extend (min (↑0) t) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_1
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ { toFun := q, continuous_toFun := qc }.toFun 0 = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [Icc.coe_zero, min_eq_left m.1, p.extend_zero] | case neg.refine_1
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ p.extend (min (↑0) t) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_1
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ p.extend (min (↑0) t) = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [← hq] | case neg.refine_2
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ { toFun := q, continuous_toFun := qc }.toFun 1 = p ⟨t, m⟩ | case neg.refine_2
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ p.extend (min (↑1) t) = p ⟨t, m⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_2
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ { toFun := q, continuous_toFun := qc }.toFun 1 = p ⟨t, m⟩
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [Icc.coe_one, min_eq_right m.2, Path.extend, IccExtend_apply, max_eq_right m.1] | case neg.refine_2
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ p.extend (min (↑1) t) = p ⟨t, m⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_2
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ p.extend (min (↑1) t) = p ⟨t, m⟩
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | intro ⟨a, n⟩ | case neg.refine_3
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ ∀ (t_1 : ↑unitInterval), { toFun := q, continuous_toFun := qc, source' := ⋯, target' := ⋯ } t_1 ∈ f ⁻¹' s | case neg.refine_3
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
a : ℝ
n : a ∈ unitInterval
⊢ { toFun := q, continuous_toFun := qc, source' := ⋯, target' := ⋯ } ⟨a, n⟩ ∈ f ⁻¹' s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_3
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
⊢ ∀ (t_1 : ↑unitInterval), { toFun := q, continuous_toFun := qc, source' := ⋯, target' := ⋯ } t_1 ∈ f ⁻¹' s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [mem_preimage, Path.coe_mk_mk, ← hq, Subtype.coe_mk] | case neg.refine_3
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
a : ℝ
n : a ∈ unitInterval
⊢ { toFun := q, continuous_toFun := qc, source' := ⋯, target' := ⋯ } ⟨a, n⟩ ∈ f ⁻¹' s | case neg.refine_3
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
a : ℝ
n : a ∈ unitInterval
⊢ f (p.extend (min a t)) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_3
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
a : ℝ
n : a ∈ unitInterval
⊢ { toFun := q, continuous_toFun := qc, source' := ⋯, target' := ⋯ } ⟨a, n⟩ ∈ f ⁻¹' s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | exact lo _ (min_le_right _ _) | case neg.refine_3
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
a : ℝ
n : a ∈ unitInterval
⊢ f (p.extend (min a t)) ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_3
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' frontier s)
fc : Continuous f
sc : IsClosed s
b : X
fb : f b ∈ frontier s
j : ∀ {y : X}, f y ∈ frontier s → JoinedIn (f ⁻¹' frontier s) b y
bs : f b ∈ s
x : X
fx : f x ∈ s
p : Path x b
u : Set ℝ
hu : Icc 0 1 ∩ ⋂ a, ⋂ (_ : f (p.extend a) ∉ s), Iic a = u
bdd : BddAbove u
un : u.Nonempty
uc : IsClosed u
t : ℝ
ht : sSup u = t
tu : t ∈ u
m : t ∈ Icc 0 1
lo : ∀ a ≤ t, f (p.extend a) ∈ s
t1 : t < 1
ft : f (p ⟨t, m⟩) ∈ frontier s
q : ↑unitInterval → X
hq : (fun a => p.extend (min (↑a) t)) = q
qc : Continuous q
a : ℝ
n : a ∈ unitInterval
⊢ f (p.extend (min a t)) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | have m : ∀ᶠ y : ℂ × ℂ in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near := by
refine ContinuousAt.eventually_mem ?_ (s.isOpen_near.mem_nhds mem)
exact continuousAt_fst.prod (s.continuousAt_iter continuousAt_fst holo.continuousAt) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | apply holo.eventually.mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x), HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), Eqn s n r y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | apply loc.mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x), HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n → HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x), HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | apply m.mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n → HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n → HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | apply eventually_of_forall | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ (x : ℂ × ℂ),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, x),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | intro _ m l h | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ (x : ℂ × ℂ),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m✝ : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
x✝ : ℂ × ℂ
m : (x✝.1, (f x✝.1)^[n] (r x✝.1 x✝.2)) ∈ s.near
l : s.bottcherNear x✝.1 ((f x✝.1)^[n] (r x✝.1 x✝.2)) = x✝.2 ^ d ^ n
h : HolomorphicAt (I.prod I) I (uncurry r) x✝
⊢ Eqn s n r x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
⊢ ∀ (x : ℂ × ℂ),
(x.1, (f x.1)^[n] (r x.1 x.2)) ∈ s.near →
s.bottcherNear x.1 ((f x.1)^[n] (r x.1 x.2)) = x.2 ^ d ^ n →
HolomorphicAt (I.prod I) I (uncurry r) x → Eqn s n r x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | exact ⟨h, m, l⟩ | case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m✝ : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
x✝ : ℂ × ℂ
m : (x✝.1, (f x✝.1)^[n] (r x✝.1 x✝.2)) ∈ s.near
l : s.bottcherNear x✝.1 ((f x✝.1)^[n] (r x✝.1 x✝.2)) = x✝.2 ^ d ^ n
h : HolomorphicAt (I.prod I) I (uncurry r) x✝
⊢ Eqn s n r x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
m✝ : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
x✝ : ℂ × ℂ
m : (x✝.1, (f x✝.1)^[n] (r x✝.1 x✝.2)) ∈ s.near
l : s.bottcherNear x✝.1 ((f x✝.1)^[n] (r x✝.1 x✝.2)) = x✝.2 ^ d ^ n
h : HolomorphicAt (I.prod I) I (uncurry r) x✝
⊢ Eqn s n r x✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | refine ContinuousAt.eventually_mem ?_ (s.isOpen_near.mem_nhds mem) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ContinuousAt (fun y => (y.1, (f y.1)^[n] (r y.1 y.2))) (c, x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), (y.1, (f y.1)^[n] (r y.1 y.2)) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | eqn_near | [71, 1] | [79, 61] | exact continuousAt_fst.prod (s.continuousAt_iter continuousAt_fst holo.continuousAt) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ContinuousAt (fun y => (y.1, (f y.1)^[n] (r y.1 y.2))) (c, x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
p : ℝ
s✝ : Super f d a
r✝ : ℂ → ℂ → S
s : Super f d a
n : ℕ
r : ℂ → ℂ → S
c x : ℂ
holo : HolomorphicAt (I.prod I) I (uncurry r) (c, x)
mem : (c, (f c)^[n] (r c x)) ∈ s.near
loc : ∀ᶠ (y : ℂ × ℂ) in 𝓝 (c, x), s.bottcherNear y.1 ((f y.1)^[n] (r y.1 y.2)) = y.2 ^ d ^ n
⊢ ContinuousAt (fun y => (y.1, (f y.1)^[n] (r y.1 y.2))) (c, x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.congr | [82, 1] | [88, 41] | have s := loc.self_of_nhds | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
⊢ Eqn s n r1 x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : uncurry r0 x = uncurry r1 x
⊢ Eqn s✝ n r1 x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
⊢ Eqn s n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.congr | [82, 1] | [88, 41] | simp only [uncurry] at s | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : uncurry r0 x = uncurry r1 x
⊢ Eqn s✝ n r1 x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ Eqn s✝ n r1 x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : uncurry r0 x = uncurry r1 x
⊢ Eqn s✝ n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.congr | [82, 1] | [88, 41] | exact
{ holo := e.holo.congr loc
near := by simp only [← s, e.near]
eqn := by simp only [← s, e.eqn] } | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ Eqn s✝ n r1 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ Eqn s✝ n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.congr | [82, 1] | [88, 41] | simp only [← s, e.near] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ (x.1, (f x.1)^[n] (r1 x.1 x.2)) ∈ s✝.near | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ (x.1, (f x.1)^[n] (r1 x.1 x.2)) ∈ s✝.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.congr | [82, 1] | [88, 41] | simp only [← s, e.eqn] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ s✝.bottcherNear x.1 ((f x.1)^[n] (r1 x.1 x.2)) = x.2 ^ d ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
r0 r1 : ℂ → ℂ → S
e : Eqn s✝ n r0 x
loc : (𝓝 x).EventuallyEq (uncurry r0) (uncurry r1)
s : r0 x.1 x.2 = r1 x.1 x.2
⊢ s✝.bottcherNear x.1 ((f x.1)^[n] (r1 x.1 x.2)) = x.2 ^ d ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.mono | [91, 1] | [97, 80] | refine Nat.le_induction e.eqn ?_ m nm | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
⊢ s.bottcherNear x.1 ((f x.1)^[m] (r x.1 x.2)) = x.2 ^ d ^ m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
⊢ ∀ (n_1 : ℕ),
n ≤ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1] (r x.1 x.2)) = x.2 ^ d ^ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1 + 1] (r x.1 x.2)) = x.2 ^ d ^ (n_1 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
⊢ s.bottcherNear x.1 ((f x.1)^[m] (r x.1 x.2)) = x.2 ^ d ^ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.mono | [91, 1] | [97, 80] | intro k nk h | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
⊢ ∀ (n_1 : ℕ),
n ≤ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1] (r x.1 x.2)) = x.2 ^ d ^ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1 + 1] (r x.1 x.2)) = x.2 ^ d ^ (n_1 + 1) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
k : ℕ
nk : n ≤ k
h : s.bottcherNear x.1 ((f x.1)^[k] (r x.1 x.2)) = x.2 ^ d ^ k
⊢ s.bottcherNear x.1 ((f x.1)^[k + 1] (r x.1 x.2)) = x.2 ^ d ^ (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
⊢ ∀ (n_1 : ℕ),
n ≤ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1] (r x.1 x.2)) = x.2 ^ d ^ n_1 →
s.bottcherNear x.1 ((f x.1)^[n_1 + 1] (r x.1 x.2)) = x.2 ^ d ^ (n_1 + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.mono | [91, 1] | [97, 80] | simp only [h, Function.iterate_succ_apply',
s.bottcherNear_eqn (s.iter_stays_near' e.near nk), pow_succ, pow_mul] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
k : ℕ
nk : n ≤ k
h : s.bottcherNear x.1 ((f x.1)^[k] (r x.1 x.2)) = x.2 ^ d ^ k
⊢ s.bottcherNear x.1 ((f x.1)^[k + 1] (r x.1 x.2)) = x.2 ^ d ^ (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
m : ℕ
nm : n ≤ m
k : ℕ
nk : n ≤ k
h : s.bottcherNear x.1 ((f x.1)^[k] (r x.1 x.2)) = x.2 ^ d ^ k
⊢ s.bottcherNear x.1 ((f x.1)^[k + 1] (r x.1 x.2)) = x.2 ^ d ^ (k + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | mem_domain_self | [112, 1] | [115, 37] | simp only [mem_prod_eq, mem_singleton_iff, eq_self_iff_true, mem_closedBall, Complex.dist_eq,
sub_zero, true_and_iff, le_refl] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
⊢ (c, x) ∈ {c} ×ˢ closedBall 0 (Complex.abs x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
c x : ℂ
⊢ (c, x) ∈ {c} ×ˢ closedBall 0 (Complex.abs x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | set u := Complex.abs '' (closedBall 0 (p + 1) \ t) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | by_cases ne : u = ∅ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : ¬u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | replace ne := nonempty_iff_ne_empty.mpr ne | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : ¬u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : ¬u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | have uc : IsClosed u :=
(((isCompact_closedBall _ _).diff ot).image Complex.continuous_abs).isClosed | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | have up : ∀ x : ℝ, x ∈ u → p < x := by
intro x m; rcases m with ⟨z, ⟨_, mt⟩, e⟩; rw [← e]; contrapose mt
simp only [not_not, not_lt] at mt ⊢
apply sub; simp only [mem_closedBall, Complex.dist_eq, sub_zero, mt] | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | have ub : BddBelow u := ⟨p, fun _ m ↦ (up _ m).le⟩ | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | have iu : sInf u ∈ u := IsClosed.csInf_mem uc ne ub | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | rcases exists_between (up _ iu) with ⟨q, pq, qi⟩ | case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case neg.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | use min q (p + 1), lt_min pq (by linarith) | case neg.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ closedBall 0 (min q (p + 1)) ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | intro z m | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ closedBall 0 (min q (p + 1)) ⊆ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : z ∈ closedBall 0 (min q (p + 1))
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ closedBall 0 (min q (p + 1)) ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | simp only [mem_closedBall, Complex.dist_eq, sub_zero, le_min_iff] at m | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : z ∈ closedBall 0 (min q (p + 1))
⊢ z ∈ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : Complex.abs z ≤ q ∧ Complex.abs z ≤ p + 1
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : z ∈ closedBall 0 (min q (p + 1))
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | rcases m with ⟨zq, zp⟩ | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : Complex.abs z ≤ q ∧ Complex.abs z ≤ p + 1
⊢ z ∈ t | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
m : Complex.abs z ≤ q ∧ Complex.abs z ≤ p + 1
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | have zi := lt_of_le_of_lt zq qi | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
⊢ z ∈ t | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : Complex.abs z < sInf u
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | contrapose zi | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : Complex.abs z < sInf u
⊢ z ∈ t | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ ¬Complex.abs z < sInf u | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : Complex.abs z < sInf u
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | simp only [not_lt] | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ ¬Complex.abs z < sInf u | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ sInf u ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ ¬Complex.abs z < sInf u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | refine csInf_le ub (mem_image_of_mem _ ?_) | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ sInf u ≤ Complex.abs z | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ z ∈ closedBall 0 (p + 1) \ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ sInf u ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | simp only [mem_diff, mem_closedBall, Complex.dist_eq, sub_zero] | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ z ∈ closedBall 0 (p + 1) \ t | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ Complex.abs z ≤ p + 1 ∧ z ∉ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ z ∈ closedBall 0 (p + 1) \ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | use zp, zi | case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ Complex.abs z ≤ p + 1 ∧ z ∉ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
z : ℂ
zq : Complex.abs z ≤ q
zp : Complex.abs z ≤ p + 1
zi : ¬z ∈ t
⊢ Complex.abs z ≤ p + 1 ∧ z ∉ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | refine ⟨p + 1, by bound, ?_⟩ | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ closedBall 0 (p + 1) ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ ∃ q, p < q ∧ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | rw [image_eq_empty, diff_eq_empty] at ne | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ closedBall 0 (p + 1) ⊆ t | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : closedBall 0 (p + 1) ⊆ t
⊢ closedBall 0 (p + 1) ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ closedBall 0 (p + 1) ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | exact ne | case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : closedBall 0 (p + 1) ⊆ t
⊢ closedBall 0 (p + 1) ⊆ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : closedBall 0 (p + 1) ⊆ t
⊢ closedBall 0 (p + 1) ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | bound | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ p < p + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u = ∅
⊢ p < p + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | intro x m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
⊢ ∀ x ∈ u, p < x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
m : x ∈ u
⊢ p < x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
⊢ ∀ x ∈ u, p < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | rcases m with ⟨z, ⟨_, mt⟩, e⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
m : x ∈ u
⊢ p < x | case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
m : x ∈ u
⊢ p < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | rw [← e] | case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < x | case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | contrapose mt | case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < Complex.abs z | case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : ¬p < Complex.abs z
⊢ ¬z ∉ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : z ∉ t
⊢ p < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | simp only [not_not, not_lt] at mt ⊢ | case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : ¬p < Complex.abs z
⊢ ¬z ∉ t | case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : ¬p < Complex.abs z
⊢ ¬z ∉ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | apply sub | case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ t | case intro.intro.intro.a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ closedBall 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | simp only [mem_closedBall, Complex.dist_eq, sub_zero, mt] | case intro.intro.intro.a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ closedBall 0 p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
x : ℝ
z : ℂ
e : Complex.abs z = x
left✝ : z ∈ closedBall 0 (p + 1)
mt : Complex.abs z ≤ p
⊢ z ∈ closedBall 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open' | [128, 1] | [147, 78] | linarith | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ p < p + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set ℂ
sub : closedBall 0 p ⊆ t
ot : IsOpen t
u : Set ℝ := ⇑Complex.abs '' (closedBall 0 (p + 1) \ t)
ne : u.Nonempty
uc : IsClosed u
up : ∀ x ∈ u, p < x
ub : BddBelow u
iu : sInf u ∈ u
q : ℝ
pq : p < q
qi : q < sInf u
⊢ p < p + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | have sub : closedBall 0 p ⊆ {b | (c, b) ∈ t} := by
intro z m; simp only [mem_setOf]; apply sub; exact ⟨mem_singleton _, m⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub : closedBall 0 p ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | rcases domain_open' sub (o.snd_preimage c) with ⟨q, pq, sub⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub : closedBall 0 p ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t | case intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub : closedBall 0 p ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | use q, pq | case intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ {c} ×ˢ closedBall 0 q ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ ∃ q, p < q ∧ {c} ×ˢ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | intro ⟨e, z⟩ ⟨ec, m⟩ | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ {c} ×ˢ closedBall 0 q ⊆ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : (e, z).1 ∈ {c}
m : (e, z).2 ∈ closedBall 0 q
⊢ (e, z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
⊢ {c} ×ˢ closedBall 0 q ⊆ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | simp only [mem_singleton_iff] at ec | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : (e, z).1 ∈ {c}
m : (e, z).2 ∈ closedBall 0 q
⊢ (e, z) ∈ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
m : (e, z).2 ∈ closedBall 0 q
ec : e = c
⊢ (e, z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : (e, z).1 ∈ {c}
m : (e, z).2 ∈ closedBall 0 q
⊢ (e, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | replace m := sub m | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
m : (e, z).2 ∈ closedBall 0 q
ec : e = c
⊢ (e, z) ∈ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z).2 ∈ {b | (c, b) ∈ t}
⊢ (e, z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
m : (e, z).2 ∈ closedBall 0 q
ec : e = c
⊢ (e, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | simp only [← ec, mem_setOf] at m | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z).2 ∈ {b | (c, b) ∈ t}
⊢ (e, z) ∈ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z) ∈ t
⊢ (e, z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z).2 ∈ {b | (c, b) ∈ t}
⊢ (e, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | exact m | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z) ∈ t
⊢ (e, z) ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub✝¹ : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
sub✝ : closedBall 0 p ⊆ {b | (c, b) ∈ t}
q : ℝ
pq : p < q
sub : closedBall 0 q ⊆ {b | (c, b) ∈ t}
e z : ℂ
ec : e = c
m : (e, z) ∈ t
⊢ (e, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | intro z m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
⊢ closedBall 0 p ⊆ {b | (c, b) ∈ t} | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ z ∈ {b | (c, b) ∈ t} | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
⊢ closedBall 0 p ⊆ {b | (c, b) ∈ t}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | simp only [mem_setOf] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ z ∈ {b | (c, b) ∈ t} | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ z ∈ {b | (c, b) ∈ t}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | apply sub | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ t | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ {c} ×ˢ closedBall 0 p | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | domain_open | [150, 1] | [156, 64] | exact ⟨mem_singleton _, m⟩ | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ {c} ×ˢ closedBall 0 p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n : ℕ
p✝ : ℝ
s : Super f d a
r : ℂ → ℂ → S
p : ℝ
t : Set (ℂ × ℂ)
sub : {c} ×ˢ closedBall 0 p ⊆ t
o : IsOpen t
z : ℂ
m : z ∈ closedBall 0 p
⊢ (c, z) ∈ {c} ×ˢ closedBall 0 p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | have e := e.self_of_nhdsSet (mem_domain c g.nonneg) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ r1 c 0 = a | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : uncurry r0 (c, 0) = uncurry r1 (c, 0)
⊢ r1 c 0 = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ r1 c 0 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | simp only [uncurry] at e | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : uncurry r0 (c, 0) = uncurry r1 (c, 0)
⊢ r1 c 0 = a | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r1 c 0 = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : uncurry r0 (c, 0) = uncurry r1 (c, 0)
⊢ r1 c 0 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | rw [← e] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r1 c 0 = a | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r0 c 0 = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r1 c 0 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | exact g.zero | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r0 c 0 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
e : r0 c 0 = r1 c 0
⊢ r0 c 0 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | refine g.start.mp ((e.filter_mono (nhds_le_nhdsSet (mem_domain c g.nonneg))).mp ?_) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r1 x.1 x.2) = x.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0),
uncurry r0 x = uncurry r1 x → s.bottcherNear x.1 (r0 x.1 x.2) = x.2 → s.bottcherNear x.1 (r1 x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (r1 x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | refine eventually_of_forall fun x e s ↦ ?_ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0),
uncurry r0 x = uncurry r1 x → s.bottcherNear x.1 (r0 x.1 x.2) = x.2 → s.bottcherNear x.1 (r1 x.1 x.2) = x.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : uncurry r0 x = uncurry r1 x
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0),
uncurry r0 x = uncurry r1 x → s.bottcherNear x.1 (r0 x.1 x.2) = x.2 → s.bottcherNear x.1 (r1 x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | simp only [uncurry] at e | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : uncurry r0 x = uncurry r1 x
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : uncurry r0 x = uncurry r1 x
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | rw [← e] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.2 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r1 x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | exact s | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s✝ : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s✝ c p n r0
e✝ : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
x : ℂ × ℂ
e : r0 x.1 x.2 = r1 x.1 x.2
s : s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
⊢ s✝.bottcherNear x.1 (r0 x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | have eqn := g.eqn | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r1 x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
eqn : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r0 x
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r1 x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | simp only [Filter.EventuallyEq, eventually_nhdsSet_iff_forall] at eqn e ⊢ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
eqn : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r0 x
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r1 x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
⊢ ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
e : (𝓝ˢ ({c} ×ˢ closedBall 0 p)).EventuallyEq (uncurry r0) (uncurry r1)
eqn : ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r0 x
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝ˢ ({c} ×ˢ closedBall 0 p), Eqn s n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | intro x m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
⊢ ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
⊢ ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | refine (eqn x m).mp ((e x m).eventually_nhds.mp (eventually_of_forall fun y e eqn ↦ ?_)) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
y : ℂ × ℂ
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝 y, uncurry r0 x = uncurry r1 x
eqn : Eqn s n r0 y
⊢ Eqn s n r1 y | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r1 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Grow.congr | [159, 1] | [173, 26] | exact eqn.congr e | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
y : ℂ × ℂ
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝 y, uncurry r0 x = uncurry r1 x
eqn : Eqn s n r0 y
⊢ Eqn s n r1 y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r r0 r1 : ℂ → ℂ → S
g : Grow s c p n r0
eqn✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, Eqn s n r0 x
e✝ : ∀ x ∈ {c} ×ˢ closedBall 0 p, ∀ᶠ (x : ℂ × ℂ) in 𝓝 x, uncurry r0 x = uncurry r1 x
x : ℂ × ℂ
m : x ∈ {c} ×ˢ closedBall 0 p
y : ℂ × ℂ
e : ∀ᶠ (x : ℂ × ℂ) in 𝓝 y, uncurry r0 x = uncurry r1 x
eqn : Eqn s n r0 y
⊢ Eqn s n r1 y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Grow.lean | Eqn.potential | [176, 1] | [178, 75] | simp only [s.potential_eq e.near, Super.potential', e.eqn, Complex.abs.map_pow, ← Nat.cast_pow,
Real.pow_rpow_inv_natCast (Complex.abs.nonneg _) (pow_ne_zero _ s.d0)] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
⊢ s.potential x.1 (r x.1 x.2) = Complex.abs x.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
p : ℝ
s : Super f d a
r : ℂ → ℂ → S
x : ℂ × ℂ
e : Eqn s n r x
⊢ s.potential x.1 (r x.1 x.2) = Complex.abs x.2
TACTIC:
|
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