url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | forall_const_and_distrib | [208, 1] | [210, 80] | have d := @forall_and A (fun _ ↦ p) q | E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
inst✝² : CompleteSpace E
inst✝¹ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A : Type
inst✝ : Nonempty A
p : Prop
q : A → Prop
⊢ (∀ (x : A), p ∧ q x) ↔ p ∧ ∀ (x : A), q x | E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
inst✝² : CompleteSpace E
inst✝¹ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A : Type
inst✝ : Nonempty A
p : Prop
q : A → Prop
d : (∀ (x : A), p ∧ q x) ↔ (A → p) ∧ ∀ (x : A), q x
⊢ (∀ (x : A), p ∧ q x)... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
inst✝² : CompleteSpace E
inst✝¹ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A : Type
inst✝ : Nonempty A
p : Prop
q : A → Prop
⊢ (∀ (x : A),... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | forall_const_and_distrib | [208, 1] | [210, 80] | simp only [forall_const] at d | E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
inst✝² : CompleteSpace E
inst✝¹ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A : Type
inst✝ : Nonempty A
p : Prop
q : A → Prop
d : (∀ (x : A), p ∧ q x) ↔ (A → p) ∧ ∀ (x : A), q x
⊢ (∀ (x : A), p ∧ q x)... | E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
inst✝² : CompleteSpace E
inst✝¹ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A : Type
inst✝ : Nonempty A
p : Prop
q : A → Prop
d : (∀ (x : A), p ∧ q x) ↔ p ∧ ∀ (x : A), q x
⊢ (∀ (x : A), p ∧ q x) ↔ p ∧... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
inst✝² : CompleteSpace E
inst✝¹ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A : Type
inst✝ : Nonempty A
p : Prop
q : A → Prop
d : (∀ (x : A... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | forall_const_and_distrib | [208, 1] | [210, 80] | exact d | E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
inst✝² : CompleteSpace E
inst✝¹ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A : Type
inst✝ : Nonempty A
p : Prop
q : A → Prop
d : (∀ (x : A), p ∧ q x) ↔ p ∧ ∀ (x : A), q x
⊢ (∀ (x : A), p ∧ q x) ↔ p ∧... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
inst✝² : CompleteSpace E
inst✝¹ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A : Type
inst✝ : Nonempty A
p : Prop
q : A → Prop
d : (∀ (x : A... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | rw [Set.setOf_and] | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : S... | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : S... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | simp only [Set.setOf_mem_eq] | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : S... | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : S... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | set t := {p : B × B | p.fst ≤ p.snd} | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : S... | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : S... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | set fg := fun x ↦ (f x, g x) | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : S... | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : S... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | have e : {x | f x ≤ g x} = fg ⁻¹' t := by
apply Set.ext; intro x; simp only [Set.preimage_setOf_eq, t] | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : S... | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | rw [e] | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : ... | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalS... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | exact ContinuousOn.preimage_isClosed_of_isClosed (ContinuousOn.prod fc gc) sc
OrderClosedTopology.isClosed_le' | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e✝ : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalS... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | apply Set.ext | E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology B
s : S... | case h
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | intro x | case h
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology ... | case h
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : Topolo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | ContinuousOn.isClosed_le | [213, 1] | [223, 37] | simp only [Set.preimage_setOf_eq, t] | case h
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : Preorder B
inst✝ : OrderClosedTopology ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : SecondCountableTopology E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
A B : Type
inst✝³ : TopologicalSpace A
inst✝² : Topolo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | set re := min r e | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
⊢ ∃ c0', ∃ t > 0, c0' ∈ closedBall c0 e ∧ AnalyticOn ℂ f (ball c0' t ... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
⊢ ∃ c0', ∃ t > 0, c0' ∈ closedBall c0 e ∧ AnalyticO... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
⊢ ∃ c0',... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have esub : closedBall c0 re ⊆ closedBall c0 r :=
Metric.closedBall_subset_closedBall (min_le_left _ _) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
⊢ ∃ c0', ∃ t > 0, c0' ∈ closedBall c0 e ∧ AnalyticO... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
⊢ ∃ c0', ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | generalize hS : (fun b : ℕ ↦
{z0 | z0 ∈ closedBall c0 re ∧ ∀ z1, z1 ∈ closedBall c1 r → ‖f (z0, z1)‖ ≤ b}) = S | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
⊢ ∃ c0', ... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → S... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have hU : (interior (⋃ b, S b)).Nonempty := by
refine Set.nonempty_of_mem
(mem_interior.mpr ⟨ball c0 re, ?_, Metric.isOpen_ball, Metric.mem_ball_self (lt_min rp ep)⟩)
rw [Set.subset_def]; intro z0 z0s; rw [Set.mem_iUnion]
have z0s' := esub (mem_open_closed z0s)
rcases (isCompact_closedBall _ _).bddAbove_i... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → S... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → S... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rcases NonemptyInterior.nonempty_interior_of_iUnion_of_closed hc hU with ⟨b, bi⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → S... | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rcases bi with ⟨c0', c0's⟩ | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0... | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closed... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | use c0' | case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closed... | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rcases mem_interior.mp c0's with ⟨s', s's, so, c0s'⟩ | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | case h.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rcases Metric.isOpen_iff.mp so c0' c0s' with ⟨t, tp, ts⟩ | case h.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ... | case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : close... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have tr : ball c0' t ⊆ closedBall c0 re := by
rw [Set.subset_def]; intro z0 z0t
have z0b := _root_.trans ts s's z0t
rw [← hS] at z0b; simp only [Set.setOf_and, Set.setOf_mem_eq, Set.mem_inter_iff] at z0b
exact z0b.left | case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : close... | case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : close... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have c0e : c0' ∈ closedBall c0 e :=
_root_.trans tr (Metric.closedBall_subset_closedBall (min_le_right _ _))
(Metric.mem_ball_self tp) | case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : close... | case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : close... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have fb : ∀ z, z ∈ ball c0' t ×ˢ ball c1 r → ‖f z‖ ≤ b := by
intro z zs; rw [Set.mem_prod] at zs
have zb := _root_.trans ts s's zs.left
rw [← hS] at zb
simp only [Metric.mem_ball, Metric.mem_closedBall, le_min_iff, Set.mem_setOf_eq] at zb zs
have zb' := zb.right z.snd zs.right.le
simp only [Prod.mk.eta] at ... | case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : close... | case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : close... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | use t, tp, c0e | case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : close... | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | refine of_bounded (h.mono ?_) (IsOpen.prod isOpen_ball isOpen_ball) fb | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [← closedBall_prod_same] | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | exact Set.prod_mono (_root_.trans tr esub) Metric.ball_subset_closedBall | case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro b | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → S... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [← hS] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [← forall_const_and_distrib] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.setOf_forall] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | apply isClosed_iInter | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro z1 | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S... | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | by_cases z1r : z1 ∉ closedBall c1 r | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | case pos
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [z1r, false_imp_iff, and_true_iff, Set.setOf_mem_eq, Metric.isClosed_ball] | case pos
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e >... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.not_not_mem] at z1r | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 ... | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e >... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [z1r, true_imp_iff] | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 ... | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e >... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | refine ContinuousOn.isClosed_le Metric.isClosed_ball ?_ continuousOn_const | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 ... | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e >... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | apply ContinuousOn.norm | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 ... | case neg.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e >... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | exact ContinuousOn.mono (h.on0 z1r).continuousOn esub | case neg.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | refine Set.nonempty_of_mem
(mem_interior.mpr ⟨ball c0 re, ?_, Metric.isOpen_ball, Metric.mem_ball_self (lt_min rp ep)⟩) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → S... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → S... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.subset_def] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → S... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → S... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro z0 z0s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → S... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.mem_iUnion] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have z0s' := esub (mem_open_closed z0s) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rcases (isCompact_closedBall _ _).bddAbove_image (h.on1 z0s').continuousOn.norm with ⟨b, fb⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ → ... | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [mem_upperBounds, Set.forall_mem_image] at fb | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | use Nat.ceil b | case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c... | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [← hS] | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [Set.mem_setOf] | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | refine ⟨mem_open_closed z0s, ?_⟩ | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [Metric.mem_closedBall] at fb ⊢ | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro z1 z1r | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
... | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | exact _root_.trans (fb z1r) (Nat.le_ceil _) | case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1✝ w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.subset_def] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro z0 z0t | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have z0b := _root_.trans ts s's z0t | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ ... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [← hS] at z0b | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ ... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [Set.setOf_and, Set.setOf_mem_eq, Set.mem_inter_iff] at z0b | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ ... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | exact z0b.left | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | intro z zs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [Set.mem_prod] at zs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have zb := _root_.trans ts s's zs.left | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | rw [← hS] at zb | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [Metric.mem_ball, Metric.mem_closedBall, le_min_iff, Set.mem_setOf_eq] at zb zs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | have zb' := zb.right z.snd zs.right.le | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | simp only [Prod.mk.eta] at zb' | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | on_subdisk | [228, 1] | [278, 75] | exact zb' | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ := min r e
esub : closedBall c0 re ⊆ closedBall c0 r
S : ℕ →... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0'✝ c1 z0 z1 w0 w1 : ℂ
r r0 r1 b✝ e : ℝ
h : Har f (closedBall (c0, c1) r)
rp : r > 0
ep : e > 0
re : ℝ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | by_cases r0 : 0 = r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ Metric.diam (ball c r) = 2 * r | case pos
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0✝ r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
r0 : 0 = r
⊢ Metric.diam (ball c r) = 2 * r
case neg
E : Type
inst✝³ : NormedAddCommGroup ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
⊢ Metric.diam (ball c r) = 2 * r
TACTIC:... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | have rp' := Ne.lt_of_le r0 rp | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0✝ r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
r0 : ¬0 = r
⊢ Metric.diam (ball c r) = 2 * r | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0✝ r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
r0 : ¬0 = r
rp' : 0 < r
⊢ Metric.diam (ball c r) = 2 * r | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0✝ r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
r0 : ¬0 = r
⊢ Metric.diam (bal... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | clear r0 | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0✝ r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
r0 : ¬0 = r
rp' : 0 < r
⊢ Metric.diam (ball c r) = 2 * r | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
⊢ Metric.diam (ball c r) = 2 * r | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0✝ r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
r0 : ¬0 = r
rp' : 0 < r
⊢ Metr... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | apply le_antisymm (Metric.diam_ball rp) | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
⊢ Metric.diam (ball c r) = 2 * r | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
⊢ 2 * r ≤ Metric.diam (ball c r) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
⊢ Metric.diam (ball... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | apply le_of_forall_small_le_add rp' | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
⊢ 2 * r ≤ Metric.diam (ball c r) | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
⊢ ∀ (e : ℝ), 0 < e → e < r → 2 * r ≤ Metric.diam (ball c r) + e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
⊢ 2 * r ≤ Metric.di... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | intro e ep er | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
⊢ ∀ (e : ℝ), 0 < e → e < r → 2 * r ≤ Metric.diam (ball c r) + e | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
⊢ 2 * r ≤ Metric.diam (ball c r) + e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
⊢ ∀ (e : ℝ), 0 < e ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | have m : ∀ t : ℝ, |t| ≤ 1 → c + t * (r - e / 2) ∈ ball c r := by
intro t t1
simp only [Complex.dist_eq, Metric.mem_ball, add_sub_cancel_left, AbsoluteValue.map_mul,
Complex.abs_ofReal]
have re : r - e / 2 ≥ 0 := by linarith [_root_.trans (half_lt_self ep) er]
calc |t| * abs (↑r - ↑e / 2 : ℂ)
_ = |t| * a... | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
⊢ 2 * r ≤ Metric.diam (ball c r) + e | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ b... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | have lo :=
Metric.dist_le_diam_of_mem Metric.isBounded_ball (m 1 (by norm_num)) (m (-1) (by norm_num)) | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ b... | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ b... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | have e : abs (2 * ↑r - ↑e : ℂ) = 2 * r - e := by
have re : 2 * r - e ≥ 0 := by trans r - e; linarith; simp only [sub_nonneg, er.le]
calc abs (2 * ↑r - ↑e : ℂ)
_ = abs (↑(2 * r - e) : ℂ) := by
simp only [Complex.ofReal_sub, Complex.ofReal_mul, Complex.ofReal_one]
norm_num
_ = 2 * r - e := by rw [... | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ b... | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e✝
er : e✝ < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e✝ / 2... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | simp only [Complex.dist_eq, Complex.ofReal_one, one_mul, Complex.ofReal_neg, neg_mul, neg_sub,
add_sub_add_left_eq_sub] at lo | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e✝
er : e✝ < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e✝ / 2... | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e✝
er : e✝ < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e✝ / 2... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | ring_nf at lo | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e✝
er : e✝ < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e✝ / 2... | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e✝
er : e✝ < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e✝ / 2... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | rw [mul_comm, e] at lo | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e✝
er : e✝ < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e✝ / 2... | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e✝
er : e✝ < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e✝ / 2... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | linarith | case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e✝
er : e✝ < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e✝ / 2... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝¹ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e✝ : ℝ
ep : 0 < e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | simp only [← r0, Metric.ball_zero, Metric.diam_empty, MulZeroClass.mul_zero] | case pos
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0✝ r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
r0 : 0 = r
⊢ Metric.diam (ball c r) = 2 * r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0✝ r1 b e : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
r0 : 0 = r
⊢ Metric.diam (ball... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | intro t t1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
⊢ ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
⊢ c + ↑t * (↑r - ↑e / 2) ∈ ball c r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | simp only [Complex.dist_eq, Metric.mem_ball, add_sub_cancel_left, AbsoluteValue.map_mul,
Complex.abs_ofReal] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
⊢ c + ↑t * (↑r - ↑e / 2) ∈ ball c r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
⊢ |t| * Complex.abs (↑r - ↑e / 2) < r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | have re : r - e / 2 ≥ 0 := by linarith [_root_.trans (half_lt_self ep) er] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
⊢ |t| * Complex.abs (↑r - ↑e / 2) < r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
re : r - e / 2 ≥ 0
⊢ |t| * Complex.abs (↑... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | calc |t| * abs (↑r - ↑e / 2 : ℂ)
_ = |t| * abs (↑(r - e / 2) : ℂ) := by
simp only [Complex.ofReal_sub, Complex.ofReal_div, Complex.ofReal_one]
norm_num
_ = |t| * (r - e / 2) := by rw [Complex.abs_ofReal, abs_of_nonneg re]
_ ≤ 1 * (r - e / 2) := mul_le_mul_of_nonneg_right t1 re
_ = r - e / 2 := by ring
... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
re : r - e / 2 ≥ 0
⊢ |t| * Complex.abs (↑... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | linarith [_root_.trans (half_lt_self ep) er] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
⊢ r - e / 2 ≥ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | simp only [Complex.ofReal_sub, Complex.ofReal_div, Complex.ofReal_one] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
re : r - e / 2 ≥ 0
⊢ |t| * Complex.abs (↑... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
re : r - e / 2 ≥ 0
⊢ |t| * Complex.abs (↑... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | norm_num | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
re : r - e / 2 ≥ 0
⊢ |t| * Complex.abs (↑... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | rw [Complex.abs_ofReal, abs_of_nonneg re] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
re : r - e / 2 ≥ 0
⊢ |t| * Complex.abs ↑(... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | ring | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
re : r - e / 2 ≥ 0
⊢ 1 * (r - e / 2) = r ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | linarith | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
re : r - e / 2 ≥ 0
⊢ 0 < e / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | ring | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
t : ℝ
t1 : |t| ≤ 1
re : r - e / 2 ≥ 0
⊢ r - 0 = r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | norm_num | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
⊢... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | norm_num | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
⊢... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | have re : 2 * r - e ≥ 0 := by trans r - e; linarith; simp only [sub_nonneg, er.le] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
l... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
l... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | calc abs (2 * ↑r - ↑e : ℂ)
_ = abs (↑(2 * r - e) : ℂ) := by
simp only [Complex.ofReal_sub, Complex.ofReal_mul, Complex.ofReal_one]
norm_num
_ = 2 * r - e := by rw [Complex.abs_ofReal, abs_of_nonneg re] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
l... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | diam_ball_eq | [295, 1] | [326, 50] | trans r - e | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
l... | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r
m : ∀ (t : ℝ), |t| ≤ 1 → c + ↑t * (↑r - ↑e / 2) ∈ ball c r
l... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b e✝ : ℝ
c : ℂ
r : ℝ
rp : r ≥ 0
rp' : 0 < r
e : ℝ
ep : 0 < e
er : e < r... |
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