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/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | intros h₁ | Γ : Ctx
f g : Form
⊢ BProof (insert f Γ) g → (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
⊢ BProof (insert f Γ) g → (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | have ⟨s₁,h₂,prf₁⟩ := BProof.compactness h₁ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | let lst₁ := (Finset.erase s₁ f).toList | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | have l₁ : ↑s₁ ⊆ {h : Form | h = f ∨ h ∈ lst₁} := by
intros k h₂
cases decEq f k
case isTrue h₃ => rw [h₃]; exact Or.inl rfl
case isFalse h₃ =>
have l₂ : k ∈ lst₁ := by
apply Finset.mem_toList.mpr
exact Finset.mem_erase.mpr ⟨h₃ ∘ Eq.symm, h₂⟩
exact Or.inr l₂ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | have prf₂ := BProof.listCompression $ BProof.monotone l₁ prf₁ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.co... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | refine ⟨lst₁,?_,prf₂⟩ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
⊢ (l : List Form) × (_ : { x | x ∈ l } ⊆ Γ) ×' BProof {Form.conjoinList f l} g | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
⊢ { x | x ∈ lst₁ } ⊆ Γ | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
⊢ (l : List Form) × ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | intros k h₃ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
⊢ { x | x ∈ lst₁ } ⊆ Γ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { x | x ∈ lst₁ }
⊢ k ∈ Γ | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
⊢ { x | x ∈ lst₁ } ⊆... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | have : k ∈ Finset.erase s₁ f := Finset.mem_toList.mp h₃ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { x | x ∈ lst₁ }
⊢ k ∈ Γ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { x | x ∈ lst₁ }
this : k ∈ Finset.erase s₁ f
⊢ k ∈ Γ | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | have ⟨l₅,l₆⟩: k ≠ f ∧ k ∈ s₁ := Finset.mem_erase.mp this | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { x | x ∈ lst₁ }
this : k ∈ Finset.erase s₁ f
⊢ k ∈ Γ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { x | x ∈ lst₁ }
this : k ∈ Finset.erase s₁ f
l₅ : k ≠ f
l₆ : k... | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | have : k ∈ insert f Γ := h₂ $ Finset.mem_coe.mpr l₆ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { x | x ∈ lst₁ }
this : k ∈ Finset.erase s₁ f
l₅ : k ≠ f
l₆ : k... | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { x | x ∈ lst₁ }
this✝ : k ∈ Finset.erase s₁ f
l₅ : k ≠ f
l₆ : ... | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | exact Set.mem_of_mem_insert_of_ne this l₅ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { x | x ∈ lst₁ }
this✝ : k ∈ Finset.erase s₁ f
l₅ : k ≠ f
l₆ : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
l₁ : ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
prf₂ : BProof {Form.conjoinList f lst₁} g
k : Form
h₃ : k ∈ { ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | intros k h₂ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
⊢ ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ } | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ } | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
⊢ ↑s₁ ⊆ { h | h = f ∨ h ∈ lst₁ }
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | cases decEq f k | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ } | case isFalse
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h✝ : ¬f = k
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ }
case isTrue
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ ... | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ }
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | case isTrue h₃ => rw [h₃]; exact Or.inl rfl | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : f = k
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : f = k
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ }
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | case isFalse h₃ =>
have l₂ : k ∈ lst₁ := by
apply Finset.mem_toList.mpr
exact Finset.mem_erase.mpr ⟨h₃ ∘ Eq.symm, h₂⟩
exact Or.inr l₂ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ }
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | rw [h₃] | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : f = k
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ } | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : f = k
⊢ k ∈ { h | h = k ∨ h ∈ lst₁ } | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : f = k
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ }
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | exact Or.inl rfl | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : f = k
⊢ k ∈ { h | h = k ∨ h ∈ lst₁ } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : f = k
⊢ k ∈ { h | h = k ∨ h ∈ lst₁ }
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | have l₂ : k ∈ lst₁ := by
apply Finset.mem_toList.mpr
exact Finset.mem_erase.mpr ⟨h₃ ∘ Eq.symm, h₂⟩ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ } | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
l₂ : k ∈ lst₁
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ } | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ }
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | exact Or.inr l₂ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
l₂ : k ∈ lst₁
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
l₂ : k ∈ lst₁
⊢ k ∈ { h | h = f ∨ h ∈ lst₁ }
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | apply Finset.mem_toList.mpr | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
⊢ k ∈ lst₁ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
⊢ k ∈ Finset.erase s₁ f | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
⊢ k ∈ lst₁
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/SystemB/Hilbert.lean | BProof.sentenceCompactness | [209, 1] | [228, 44] | exact Finset.mem_erase.mpr ⟨h₃ ∘ Eq.symm, h₂⟩ | Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
⊢ k ∈ Finset.erase s₁ f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Γ : Ctx
f g : Form
h₁ : BProof (insert f Γ) g
s₁ : Finset Form
h₂✝ : ↑s₁ ⊆ insert f Γ
prf₁ : BProof (↑s₁) g
lst₁ : List Form := Finset.toList (Finset.erase s₁ f)
k : Form
h₂ : k ∈ ↑s₁
h₃ : ¬f = k
⊢ k ∈ Finset.erase s₁ f
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | intros h₁ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
⊢ Monotone fam → ↑fin ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fin ⊆ fam n | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ↑fin ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fin ⊆ fam n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
⊢ Monotone fam → ↑fin ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fin ⊆ fam n
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | apply @Finset.induction_on α (λfs => ↑fs ⊆ {x : α | ∃n : β, x ∈ fam n } → ∃n : β, ↑fs ⊆ fam n) instDec fin | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ↑fin ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fin ⊆ fam n | case empty
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ↑∅ ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑∅ ⊆ fam n
case insert
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ↑fin ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fin ⊆ fam n
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | case empty =>
intros _
refine ⟨default,?_⟩
intros h₂ h₃
contradiction | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ↑∅ ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑∅ ⊆ fam n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ↑∅ ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑∅ ⊆ fam n
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | case insert =>
intros x fs _ h₃ h₄
have l₁ : x ∈ {x : α | ∃n : β, x ∈ fam n } := h₄ $ Finset.mem_insert_self x fs
have ⟨n, l₂⟩ := l₁
have l₃ : ↑fs ⊆ {x : α | ∃n : β, x ∈ fam n } := by
intros y h₅
exact h₄ $ Finset.mem_insert_of_mem (Finset.mem_coe.mp h₅)
have ⟨m, l₄⟩ := h₃ l₃
cases le_total n m
c... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ∀ ⦃a : α⦄ {s : Finset α},
¬a ∈ s →
(↑s ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑s ⊆ fam n) →
↑(insert a s) ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑(insert a s) ⊆ fam n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ∀ ⦃a : α⦄ {s : Finset α},
¬a ∈ s →
(↑s ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑s ⊆ fam n) →
↑(insert a s... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | intros _ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ↑∅ ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑∅ ⊆ fam n | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
a✝ : ↑∅ ⊆ { x | ∃ n, x ∈ fam n }
⊢ ∃ n, ↑∅ ⊆ fam n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ↑∅ ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑∅ ⊆ fam n
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | refine ⟨default,?_⟩ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
a✝ : ↑∅ ⊆ { x | ∃ n, x ∈ fam n }
⊢ ∃ n, ↑∅ ⊆ fam n | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
a✝ : ↑∅ ⊆ { x | ∃ n, x ∈ fam n }
⊢ ↑∅ ⊆ fam default | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
a✝ : ↑∅ ⊆ { x | ∃ n, x ∈ fam n }
⊢ ∃ n, ↑∅ ⊆ fam n
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | intros h₂ h₃ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
a✝ : ↑∅ ⊆ { x | ∃ n, x ∈ fam n }
⊢ ↑∅ ⊆ fam default | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
a✝ : ↑∅ ⊆ { x | ∃ n, x ∈ fam n }
h₂ : α
h₃ : h₂ ∈ ↑∅
⊢ h₂ ∈ fam default | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
a✝ : ↑∅ ⊆ { x | ∃ n, x ∈ fam n }
⊢ ↑∅ ⊆ fam default
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | contradiction | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
a✝ : ↑∅ ⊆ { x | ∃ n, x ∈ fam n }
h₂ : α
h₃ : h₂ ∈ ↑∅
⊢ h₂ ∈ fam default | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
a✝ : ↑∅ ⊆ { x | ∃ n, x ∈ fam n }
h₂ : α
h₃ : h₂ ∈ ↑∅
⊢ h₂ ∈ fam default
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | intros x fs _ h₃ h₄ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ∀ ⦃a : α⦄ {s : Finset α},
¬a ∈ s →
(↑s ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑s ⊆ fam n) →
↑(insert a s) ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑(insert a s) ⊆ fam n | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
⊢ ∃ n, ↑(insert x fs) ⊆ fam n | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
⊢ ∀ ⦃a : α⦄ {s : Finset α},
¬a ∈ s →
(↑s ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑s ⊆ fam n) →
↑(insert a s... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | have l₁ : x ∈ {x : α | ∃n : β, x ∈ fam n } := h₄ $ Finset.mem_insert_self x fs | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
⊢ ∃ n, ↑(insert x fs) ⊆ fam n | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
⊢ ∃ n, ↑(insert... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | have ⟨n, l₂⟩ := l₁ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
⊢ ∃ n, ↑(insert... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | have l₃ : ↑fs ⊆ {x : α | ∃n : β, x ∈ fam n } := by
intros y h₅
exact h₄ $ Finset.mem_insert_of_mem (Finset.mem_coe.mp h₅) | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | have ⟨m, l₄⟩ := h₃ l₃ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | cases le_total n m | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | case inl
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | case inl leqthan =>
have l₅ := (h₁ leqthan) l₂
refine ⟨m,?_⟩
intros y h₄
cases Finset.mem_insert.mp h₄
case inl h₅ => rw [h₅]; assumption
case inr h₅ => exact l₄ h₅ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | case inr geqthan =>
have l₅ := le_trans l₄ (h₁ geqthan)
refine ⟨n,?_⟩
intros y h₄
cases Finset.mem_insert.mp h₄
case inl h₅ => rw [h₅]; assumption
case inr h₅ => exact l₅ h₅ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | intros y h₅ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | exact h₄ $ Finset.mem_insert_of_mem (Finset.mem_coe.mp h₅) | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | have l₅ := (h₁ leqthan) l₂ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | refine ⟨m,?_⟩ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | intros y h₄ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | cases Finset.mem_insert.mp h₄ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | case inl
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | case inl h₅ => rw [h₅]; assumption | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | case inr h₅ => exact l₄ h₅ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | rw [h₅] | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | assumption | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | exact l₄ h₅ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | have l₅ := le_trans l₄ (h₁ geqthan) | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | refine ⟨n,?_⟩ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | intros y h₄ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈ ... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄ : ↑(insert x fs) ⊆ { x | ∃ n... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | cases Finset.mem_insert.mp h₄ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | case inl
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | case inl h₅ => rw [h₅]; assumption | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | case inr h₅ => exact l₅ h₅ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | rw [h₅] | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | assumption | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/util.lean | finiteExhaustion | [4, 1] | [35, 35] | exact l₅ h₅ | α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ n, x ∈ fam n }
l₁ : x ∈ { x | ∃ n, x ∈ fam n }
n : β
l₂ : x ∈... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
instDec : DecidableEq α
inst✝¹ : LinearOrder β
inst✝ : Inhabited β
fam : β → Set α
fin : Finset α
h₁ : Monotone fam
x : α
fs : Finset α
a✝ : ¬x ∈ fs
h₃ : ↑fs ⊆ { x | ∃ n, x ∈ fam n } → ∃ n, ↑fs ⊆ fam n
h₄✝ : ↑(insert x fs) ⊆ { x | ∃ ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | induction c | c : ConsExp
⊢ decode (encode c) = c | case nat
a✝ : ℕ
⊢ decode (encode (nat a✝)) = nat a✝
case cons
a✝¹ a✝ : ConsExp
a_ih✝¹ : decode (encode a✝¹) = a✝¹
a_ih✝ : decode (encode a✝) = a✝
⊢ decode (encode (cons a✝¹ a✝)) = cons a✝¹ a✝ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ConsExp
⊢ decode (encode c) = c
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | next hf hg =>
rw [encode, decode]
split
next h => simp at h
next h =>
simp [Nat.div2_val] at h
cases h
simp [*] | case cons
a✝¹ a✝ : ConsExp
a_ih✝¹ : decode (encode a✝¹) = a✝¹
a_ih✝ : decode (encode a✝) = a✝
⊢ decode (encode (cons a✝¹ a✝)) = cons a✝¹ a✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
a✝¹ a✝ : ConsExp
a_ih✝¹ : decode (encode a✝¹) = a✝¹
a_ih✝ : decode (encode a✝) = a✝
⊢ decode (encode (cons a✝¹ a✝)) = cons a✝¹ a✝
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | simp [Nat.div2_val] at h | a✝ m✝ : ℕ
h : Nat.boddDiv2 (2 * a✝) = (false, m✝)
⊢ nat m✝ = nat a✝ | a✝ m✝ : ℕ
h : a✝ = m✝
⊢ nat m✝ = nat a✝ | Please generate a tactic in lean4 to solve the state.
STATE:
a✝ m✝ : ℕ
h : Nat.boddDiv2 (2 * a✝) = (false, m✝)
⊢ nat m✝ = nat a✝
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | simp [*] | a✝ m✝ : ℕ
h : a✝ = m✝
⊢ nat m✝ = nat a✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a✝ m✝ : ℕ
h : a✝ = m✝
⊢ nat m✝ = nat a✝
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | simp at h | a✝ m✝ : ℕ
h : Nat.boddDiv2 (2 * a✝) = (true, m✝)
⊢ (match hm : Nat.unpair m✝ with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * a✝);
let_fun this := (_ : m✝ < 2 * a✝);
let_fun this_1 := (_ : f < 2 * a✝);
let_fun this := (_ : g < 2 * a✝);
cons (decode f) (decode g)) =
nat a✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a✝ m✝ : ℕ
h : Nat.boddDiv2 (2 * a✝) = (true, m✝)
⊢ (match hm : Nat.unpair m✝ with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * a✝);
let_fun this := (_ : m✝ < 2 * a✝);
let_fun this_1 := (_ : f < 2 * a✝);
let_fun this := (_ : g < 2 * a✝);... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | rw [encode, decode] | a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
⊢ decode (encode (cons a✝¹ a✝)) = cons a✝¹ a✝ | a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
⊢ (match hn : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) with
| (false, m) => nat m
| (true, m) =>
match hm : Nat.unpair m with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (encode a✝¹) (encode... | Please generate a tactic in lean4 to solve the state.
STATE:
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
⊢ decode (encode (cons a✝¹ a✝)) = cons a✝¹ a✝
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | split | a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
⊢ (match hn : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) with
| (false, m) => nat m
| (true, m) =>
match hm : Nat.unpair m with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (encode a✝¹) (encode... | case h_1
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
heq✝ : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (false, m✝)
⊢ nat m✝ = cons a✝¹ a✝
case h_2
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
heq✝ : Nat.boddDiv2 (2 * Nat.m... | Please generate a tactic in lean4 to solve the state.
STATE:
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
⊢ (match hn : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) with
| (false, m) => nat m
| (true, m) =>
match hm : Nat.unpair m with
| (f, g) =>
... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | next h => simp at h | case h_1
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
heq✝ : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (false, m✝)
⊢ nat m✝ = cons a✝¹ a✝
case h_2
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
heq✝ : Nat.boddDiv2 (2 * Nat.m... | case h_2
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
heq✝ : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, m✝)
⊢ (match hm : Nat.unpair m✝ with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1);
let_fun this := (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_1
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
heq✝ : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (false, m✝)
⊢ nat m✝ = cons a✝¹ a✝
case h_2
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : ... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | next h =>
simp [Nat.div2_val] at h
cases h
simp [*] | case h_2
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
heq✝ : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, m✝)
⊢ (match hm : Nat.unpair m✝ with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1);
let_fun this := (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h_2
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
heq✝ : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, m✝)
⊢ (match hm : Nat.unpair m✝ with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | simp at h | a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
h : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (false, m✝)
⊢ nat m✝ = cons a✝¹ a✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
h : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (false, m✝)
⊢ nat m✝ = cons a✝¹ a✝
TACTIC:
|
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | simp [Nat.div2_val] at h | a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
h : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, m✝)
⊢ (match hm : Nat.unpair m✝ with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1);
let_fun this := (_ : m✝ < 2 *... | a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
h✝ : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, m✝)
h : Nat.mkpair (encode a✝¹) (encode a✝) = m✝
⊢ (match hm : Nat.unpair m✝ with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (encode a✝¹) (encode... | Please generate a tactic in lean4 to solve the state.
STATE:
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
h : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, m✝)
⊢ (match hm : Nat.unpair m✝ with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (enco... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | cases h | a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
h✝ : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, m✝)
h : Nat.mkpair (encode a✝¹) (encode a✝) = m✝
⊢ (match hm : Nat.unpair m✝ with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (encode a✝¹) (encode... | case refl
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
h : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, Nat.mkpair (encode a✝¹) (encode a✝))
⊢ (match hm : Nat.unpair (Nat.mkpair (encode a✝¹) (encode a✝)) with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.mk... | Please generate a tactic in lean4 to solve the state.
STATE:
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
m✝ : ℕ
h✝ : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, m✝)
h : Nat.mkpair (encode a✝¹) (encode a✝) = m✝
⊢ (match hm : Nat.unpair m✝ with
| (f, g) =>
... |
https://github.com/gleachkr/Completeness-For-Fine-Semantics.git | 0d8cc9a4c9c53181a2bf1541d2ed5a39c2593f0f | Fine/Util/ConsExp.lean | ConsExp.decode_encode | [49, 1] | [64, 15] | simp [*] | case refl
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
h : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, Nat.mkpair (encode a✝¹) (encode a✝))
⊢ (match hm : Nat.unpair (Nat.mkpair (encode a✝¹) (encode a✝)) with
| (f, g) =>
let_fun hn' := (_ : 1 ≤ 2 * Nat.mk... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refl
a✝¹ a✝ : ConsExp
hf : decode (encode a✝¹) = a✝¹
hg : decode (encode a✝) = a✝
h : Nat.boddDiv2 (2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1) = (true, Nat.mkpair (encode a✝¹) (encode a✝))
⊢ (match hm : Nat.unpair (Nat.mkpair (encode a✝¹) (encode a✝)) ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have hr : 0 < r := dist_nonneg.trans_lt hz₀ | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
⊢ cindex c r f = 1 | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
⊢ cindex c r f = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
⊢ cindex c r... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | set g := dslope f z₀ | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
⊢ cindex c r f = 1 | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
⊢ cindex c r f = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
⊢... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have h1 : DifferentiableOn ℂ g U :=
(differentiableOn_dslope (hU.mem_nhds (hcr (ball_subset_closedBall hz₀)))).2 hf | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
⊢ cindex c r f = 1 | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
⊢ cindex... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have h2 : ∀ z ∈ closedBall c r, g z ≠ 0 := by
rintro z hz
by_cases h : z = z₀
case pos => simp [g, dslope, h, Function.update, hf'z₀]
case neg => simp [g, dslope, h, Function.update, slope, sub_ne_zero.2 h, hfz₀, hfz z hz h] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
⊢ cindex... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 :=
λ z hz => sub_ne_zero.2 (sphere_disjoint_ball.ne_of_mem hz hz₀) | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | suffices this : cindex c r f = ((2 * Real.pi * I)⁻¹ * ∮ z in C(c, r), (z - z₀)⁻¹) + cindex c r g
by rw [this, integral_sub_inv_of_mem_ball hz₀, cindex_eq_zero hU hr hcr h1 h2]
field_simp [two_ne_zero, Real.pi_ne_zero, I_ne_zero] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z := by
rintro z hz
have h3 : ∀ z ∈ U, f z = (z - z₀) * g z :=
λ z _ => by simpa only [smul_eq_mul, hfz₀, sub_zero] using (sub_smul_dslope f z₀ z).symm
have hz' : z ∈ U := hcr (sphere_subset_closedBall hz)
have e0 : U ∈ 𝓝 z := hU.mem_... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | simp only [cindex, integral_congr hr.le h6, ← mul_add] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | congr | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | case e_a
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | apply circleIntegral.integral_add | case e_a
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U... | case e_a.hf
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | rintro z hz | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
⊢ ∀ z ∈ ... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
z : ℂ
hz... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | by_cases h : z = z₀ | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
z : ℂ
hz... | case pos
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | case pos => simp [g, dslope, h, Function.update, hf'z₀] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
z : ℂ
hz... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | case neg => simp [g, dslope, h, Function.update, slope, sub_ne_zero.2 h, hfz₀, hfz z hz h] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
z : ℂ
hz... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | simp [g, dslope, h, Function.update, hf'z₀] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
z : ℂ
hz... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | simp [g, dslope, h, Function.update, slope, sub_ne_zero.2 h, hfz₀, hfz z hz h] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
z : ℂ
hz... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | rw [this, integral_sub_inv_of_mem_ball hz₀, cindex_eq_zero hU hr hcr h1 h2] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | field_simp [two_ne_zero, Real.pi_ne_zero, I_ne_zero] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | rintro z hz | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have h3 : ∀ z ∈ U, f z = (z - z₀) * g z :=
λ z _ => by simpa only [smul_eq_mul, hfz₀, sub_zero] using (sub_smul_dslope f z₀ z).symm | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have hz' : z ∈ U := hcr (sphere_subset_closedBall hz) | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have e0 : U ∈ 𝓝 z := hU.mem_nhds hz' | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have h4 : deriv f z = deriv (λ w => (w - z₀) * g w) z :=
EventuallyEq.deriv_eq (eventually_of_mem e0 h3) | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have e1 : DifferentiableAt ℂ (λ y => y - z₀) z := differentiableAt_id.sub_const z₀ | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have e2 : DifferentiableAt ℂ g z := h1.differentiableAt e0 | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have h5 : deriv f z = g z + (z - z₀) * deriv g z := by
have : deriv (fun y => y - z₀) z = 1 := by
change deriv (fun y => id y - z₀) z = 1
simp [deriv_sub_const]
simp [h4, deriv_mul e1 e2, this] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have e3 : g z ≠ 0 := h2 z (sphere_subset_closedBall hz) | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | field_simp [h3 z hz', h5, mul_comm, h10 z hz] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | simpa only [smul_eq_mul, hfz₀, sub_zero] using (sub_smul_dslope f z₀ z).symm | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/deriv_inj.lean | crucial | [7, 1] | [49, 57] | have : deriv (fun y => y - z₀) z = 1 := by
change deriv (fun y => id y - z₀) z = 1
simp [deriv_sub_const] | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
g : ℂ → ℂ := dslope f z₀
h1 : DifferentiableOn ℂ g U
h2 : ∀ z... | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
α : Type u_2
β : Type u_3
U : Set ℂ
c z₀ : ℂ
r : ℝ
f g✝ : ℂ → ℂ
hU : IsOpen U
hcr : closedBall c r ⊆ U
hz₀ : z₀ ∈ ball c r
hf : DifferentiableOn ℂ f U
hfz₀ : f z₀ = 0
hf'z₀ : deriv f z₀ ≠ 0
hfz : ∀ z ∈ closedBall c r, z ≠ z₀ → f z ≠ 0
hr : 0 < r
... |
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