Split (1)

train · 219k rows

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.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]	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) × (_ : { 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]	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 } ⊆ Γ TACTIC:
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 ∈ { x \| x ∈ lst₁ } ⊢ k ∈ Γ TACTIC:
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 ∈ s₁ ⊢ 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 ∈ { x \| x ∈ lst₁ } this : k ∈ Finset.erase s₁ f ⊢ k ∈ Γ TACTIC:
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 ∈ s₁ ⊢ 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 ∈ s₁ this : k ∈ insert f Γ ⊢ 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 ∈ { x \| x ∈ lst₁ } this : k ∈ Finset.erase s₁ f l₅ : k ≠ f l₆ : k ∈ s₁ ⊢ k ∈ Γ TACTIC:
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 : k ∈ s₁ this : k ∈ insert f Γ ⊢ k ∈ Γ	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 ∈ { x \| x ∈ lst₁ } this✝ : k ∈ Finset.erase s₁ f l₅ : k ≠ f l₆ : k ∈ s₁ this : k ∈ insert f Γ ⊢ k ∈ Γ TACTIC:
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✝ : ↑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 }	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 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	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 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 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 ⊢ ∀ ⦃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) ⊆ { x \| ∃ n, x ∈ fam n } → ∃ n, ↑(insert a s) ⊆ fam n TACTIC:
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) ⊆ { x \| ∃ n, x ∈ fam n } → ∃ n, ↑(insert a s) ⊆ fam n TACTIC:
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 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 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 TACTIC:
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 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 : β l₂ : 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 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 x fs) ⊆ fam n TACTIC:
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 ∈ 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 : β l₂ : x ∈ fam n l₃ : ↑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 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 ∈ fam n ⊢ ∃ n, ↑(insert x fs) ⊆ fam n TACTIC:
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 ∈ fam n l₃ : ↑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 : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m ⊢ ∃ 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 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } ⊢ ∃ n, ↑(insert x fs) ⊆ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m ⊢ ∃ n, ↑(insert x fs) ⊆ fam n	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 l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m h✝ : n ≤ m ⊢ ∃ n, ↑(insert x fs) ⊆ fam n case inr 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m h✝ : m ≤ 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 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m ⊢ ∃ n, ↑(insert x fs) ⊆ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m ⊢ ∃ n, ↑(insert x fs) ⊆ 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 : 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m ⊢ ∃ n, ↑(insert x fs) ⊆ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n ⊢ ∃ n, ↑(insert x fs) ⊆ 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 : 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n ⊢ ∃ n, ↑(insert x fs) ⊆ fam n TACTIC:
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 ∈ fam n ⊢ ↑fs ⊆ { x \| ∃ n, x ∈ 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 : β l₂ : x ∈ fam n y : α h₅ : y ∈ ↑fs ⊢ y ∈ { x \| ∃ n, x ∈ 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 : 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 ∈ fam n ⊢ ↑fs ⊆ { x \| ∃ n, x ∈ fam n } TACTIC:
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 ∈ fam n y : α h₅ : y ∈ ↑fs ⊢ y ∈ { x \| ∃ n, x ∈ 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 : 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 ∈ fam n y : α h₅ : y ∈ ↑fs ⊢ y ∈ { x \| ∃ n, x ∈ fam n } TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m ⊢ ∃ 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 : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m ⊢ ∃ 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 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m ⊢ ∃ n, ↑(insert x fs) ⊆ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m ⊢ ∃ 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 : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m ⊢ ↑(insert x fs) ⊆ fam m	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, x ∈ fam n } l₁ : x ∈ { x \| ∃ n, x ∈ fam n } n : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m ⊢ ∃ n, ↑(insert x fs) ⊆ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m ⊢ ↑(insert x fs) ⊆ fam 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) ⊢ y ∈ fam m	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, x ∈ fam n } l₁ : x ∈ { x \| ∃ n, x ∈ fam n } n : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m ⊢ ↑(insert x fs) ⊆ fam m TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) ⊢ y ∈ fam m	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 l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h✝ : y = x ⊢ y ∈ fam m case inr 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h✝ : y ∈ fs ⊢ y ∈ fam m	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, x ∈ fam n } l₁ : x ∈ { x \| ∃ n, x ∈ fam n } n : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) ⊢ y ∈ fam m TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ y ∈ fam m	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, x ∈ fam n } l₁ : x ∈ { x \| ∃ n, x ∈ fam n } n : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ y ∈ fam m TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y ∈ fs ⊢ y ∈ fam m	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, x ∈ fam n } l₁ : x ∈ { x \| ∃ n, x ∈ fam n } n : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y ∈ fs ⊢ y ∈ fam m TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ y ∈ fam 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ x ∈ fam m	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, x ∈ fam n } l₁ : x ∈ { x \| ∃ n, x ∈ fam n } n : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ y ∈ fam m TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ x ∈ fam m	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, x ∈ fam n } l₁ : x ∈ { x \| ∃ n, x ∈ fam n } n : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ x ∈ fam m TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y ∈ fs ⊢ y ∈ fam m	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, x ∈ fam n } l₁ : x ∈ { x \| ∃ n, x ∈ fam n } n : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m leqthan : n ≤ m l₅ : x ∈ fam m y : α h₄ : y ∈ ↑(insert x fs) h₅ : y ∈ fs ⊢ y ∈ fam m TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ 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 : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ 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 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n ⊢ ∃ n, ↑(insert x fs) ⊆ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ 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 : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam 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 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n ⊢ ∃ n, ↑(insert x fs) ⊆ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam 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 : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) ⊢ y ∈ 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 : 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n ⊢ ↑(insert x fs) ⊆ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) ⊢ y ∈ fam n	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 l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h✝ : y = x ⊢ y ∈ fam n case inr 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h✝ : y ∈ fs ⊢ y ∈ 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 : 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) ⊢ y ∈ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ y ∈ 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 : 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ y ∈ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y ∈ fs ⊢ y ∈ 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 : 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y ∈ fs ⊢ y ∈ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ y ∈ 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 : β l₂ : x ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ x ∈ 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 : 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ y ∈ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ x ∈ 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 : 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y = x ⊢ x ∈ fam n TACTIC:
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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y ∈ fs ⊢ y ∈ 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 : 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 ∈ fam n l₃ : ↑fs ⊆ { x \| ∃ n, x ∈ fam n } m : β l₄ : ↑fs ⊆ fam m geqthan : m ≤ n l₅ : ↑fs ≤ fam n y : α h₄ : y ∈ ↑(insert x fs) h₅ : y ∈ fs ⊢ y ∈ fam n TACTIC:
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✝); cons (decode f) (decode g)) = nat a✝ TACTIC:
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 a✝) + 1); let_fun this := (_ : m < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝	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 a✝) + 1); let_fun this := (_ : m < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝	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.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 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝	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) => let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : m < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = 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]	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.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 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = 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.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 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝	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 : 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 := (_ : m✝ < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = 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]	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 := (_ : m✝ < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝	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.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : m✝ < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = 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 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 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝	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 a✝) + 1); let_fun this := (_ : m✝ < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝	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 (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : m✝ < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = 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]	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 a✝) + 1); let_fun this := (_ : m✝ < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝	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.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : Nat.mkpair (encode a✝¹) (encode a✝) < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝	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) => let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : m✝ < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = 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 [*]	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.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : Nat.mkpair (encode a✝¹) (encode a✝) < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝	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✝)) with \| (f, g) => let_fun hn' := (_ : 1 ≤ 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : Nat.mkpair (encode a✝¹) (encode a✝) < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this_1 := (_ : f < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); let_fun this := (_ : g < 2 * Nat.mkpair (encode a✝¹) (encode a✝) + 1); cons (decode f) (decode g)) = cons a✝¹ a✝ TACTIC:
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 f = 1 TACTIC:
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 ⊢ cindex c r f = 1 TACTIC:
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 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 g : ℂ → ℂ := dslope f z₀ ⊢ cindex c r f = 1 TACTIC:
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 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 h2 : ∀ z ∈ closedBall c r, g z ≠ 0 ⊢ 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U ⊢ cindex c r f = 1 TACTIC:
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 ∈ closedBall c r, g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 ⊢ 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 ⊢ cindex c r f = 1 TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 ⊢ cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g	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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 ⊢ cindex c r f = 1 TACTIC:
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_nhds hz' have h4 : deriv f z = deriv (λ w => (w - z₀) * g w) z := EventuallyEq.deriv_eq (eventually_of_mem e0 h3) have e1 : DifferentiableAt ℂ (λ y => y - z₀) z := differentiableAt_id.sub_const z₀ have e2 : DifferentiableAt ℂ g z := h1.differentiableAt e0 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] have e3 : g z ≠ 0 := h2 z (sphere_subset_closedBall hz) 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 ⊢ cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g	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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g	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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 ⊢ cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g	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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹ + deriv g z / g z) = (2 * ↑Real.pi * I)⁻¹ * ((∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + ∮ (z : ℂ) in C(c, r), deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹ + deriv g z / g z) = (2 * ↑Real.pi * I)⁻¹ * ((∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + ∮ (z : ℂ) in C(c, r), deriv g z / g 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 h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ (∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹ + deriv g z / g z) = (∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + ∮ (z : ℂ) in C(c, r), deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹ + deriv g z / g z) = (2 * ↑Real.pi * I)⁻¹ * ((∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + ∮ (z : ℂ) in C(c, r), deriv g z / g z) TACTIC:
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 h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ (∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹ + deriv g z / g z) = (∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + ∮ (z : ℂ) in C(c, r), deriv g z / g z	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 g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ CircleIntegrable (fun z => (z - z₀)⁻¹) c r case e_a.hg 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ CircleIntegrable (fun z => deriv g z / g z) c r	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 : 0 < r g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 h6 : ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z ⊢ (∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹ + deriv g z / g z) = (∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + ∮ (z : ℂ) in C(c, r), deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0	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 : z ∈ closedBall c r ⊢ g z ≠ 0	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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U ⊢ ∀ z ∈ closedBall c r, g z ≠ 0 TACTIC:
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 : z ∈ closedBall c r ⊢ g z ≠ 0	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 z : ℂ hz : z ∈ closedBall c r h : z = z₀ ⊢ g z ≠ 0 case neg 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 : z ∈ closedBall c r h : ¬z = z₀ ⊢ g z ≠ 0	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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U z : ℂ hz : z ∈ closedBall c r ⊢ g z ≠ 0 TACTIC:
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 : z ∈ closedBall c r h : z = z₀ ⊢ g z ≠ 0	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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U z : ℂ hz : z ∈ closedBall c r h : z = z₀ ⊢ g z ≠ 0 TACTIC:
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 : z ∈ closedBall c r h : ¬z = z₀ ⊢ g z ≠ 0	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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U z : ℂ hz : z ∈ closedBall c r h : ¬z = z₀ ⊢ g z ≠ 0 TACTIC:
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 : z ∈ closedBall c r h : z = z₀ ⊢ g z ≠ 0	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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U z : ℂ hz : z ∈ closedBall c r h : z = z₀ ⊢ g z ≠ 0 TACTIC:
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 : z ∈ closedBall c r h : ¬z = z₀ ⊢ g z ≠ 0	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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U z : ℂ hz : z ∈ closedBall c r h : ¬z = z₀ ⊢ g z ≠ 0 TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 this : cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g ⊢ 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 h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 this : cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g ⊢ (2 * ↑Real.pi * I)⁻¹ * (2 * ↑Real.pi * I) + 0 = 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 this : cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g ⊢ cindex c r f = 1 TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 this : cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g ⊢ (2 * ↑Real.pi * I)⁻¹ * (2 * ↑Real.pi * I) + 0 = 1	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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 this : cindex c r f = ((2 * ↑Real.pi * I)⁻¹ * ∮ (z : ℂ) in C(c, r), (z - z₀)⁻¹) + cindex c r g ⊢ (2 * ↑Real.pi * I)⁻¹ * (2 * ↑Real.pi * I) + 0 = 1 TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 ⊢ ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 ⊢ ∀ z ∈ sphere c r, deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z h5 : deriv f z = g z + (z - z₀) * deriv g z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z h5 : deriv f z = g z + (z - z₀) * deriv g z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z h5 : deriv f z = g z + (z - z₀) * deriv g z e3 : g z ≠ 0 ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z h5 : deriv f z = g z + (z - z₀) * deriv g z ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z h5 : deriv f z = g z + (z - z₀) * deriv g z e3 : g z ≠ 0 ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z h5 : deriv f z = g z + (z - z₀) * deriv g z e3 : g z ≠ 0 ⊢ deriv f z / f z = (z - z₀)⁻¹ + deriv g z / g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z✝ : ℂ hz : z✝ ∈ sphere c r z : ℂ x✝ : z ∈ U ⊢ f z = (z - z₀) * g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z✝ : ℂ hz : z✝ ∈ sphere c r z : ℂ x✝ : z ∈ U ⊢ f z = (z - z₀) * g z TACTIC:
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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z ⊢ deriv f z = g z + (z - z₀) * deriv g 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 ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z this : deriv (fun y => y - z₀) z = 1 ⊢ deriv f z = g z + (z - z₀) * deriv g 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 g : ℂ → ℂ := dslope f z₀ h1 : DifferentiableOn ℂ g U h2 : ∀ z ∈ closedBall c r, g z ≠ 0 h10 : ∀ z ∈ sphere c r, z - z₀ ≠ 0 z : ℂ hz : z ∈ sphere c r h3 : ∀ z ∈ U, f z = (z - z₀) * g z hz' : z ∈ U e0 : U ∈ 𝓝 z h4 : deriv f z = deriv (fun w => (w - z₀) * g w) z e1 : DifferentiableAt ℂ (fun y => y - z₀) z e2 : DifferentiableAt ℂ g z ⊢ deriv f z = g z + (z - z₀) * deriv g z TACTIC:

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