[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.Modular).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.NumberTheory.Modular.exists_one_half_le_im_smul`:
State 0: ⊢ lemma exists_one_half_le_im_smul (τ : ℍ) : ∃ γ : SL(2, ℤ), 1 / 2 ≤ im (γ • τ)
Tactic: obtain ⟨γ, hγ⟩ := exists_smul_mem_fd τ
State 1: ⊢ <goal after obtain ⟨γ, hγ⟩ := exists_smul_mem_fd τ>
Tactic: use γ  nlinarith [three_le_four_mul_im_sq_of_mem_fd hγ, im_pos (γ • τ)]  /-- For every `τ : ℍ` there is some `γ ∈ SL(2, ℤ)` that sends it to an element whose  imaginary part is at least `1/2` and such that `denom γ τ` has norm at most 1. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.Modular).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.NumberTheory.Modular.exists_one_half_le_im_smul`:
State 0: ⊢ lemma exists_one_half_le_im_smul (τ : ℍ) : ∃ γ : SL(2, ℤ), 1 / 2 ≤ im (γ • τ)
Tactic: obtain ⟨γ, hγ⟩ := exists_smul_mem_fd τ
State 1: ⊢ <goal after obtain ⟨γ, hγ⟩ := exists_smul_mem_fd τ>
Tactic: use γ  nlinarith [three_le_four_mul_im_sq_of_mem_fd hγ, im_pos (γ • τ)]  /-- For every `τ : ℍ` there is some `γ ∈ SL(2, ℤ)` that sends it to an element whose  imaginary part is at least `1/2` and such that `denom γ τ` has norm at most 1. -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.ConditionallyCompleteLattice.Finset).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Order.ConditionallyCompleteLattice.Finset.Finset`:
State 0: ⊢ theorem Finset.ciSup_eq_max'_image {s : Finset ι} (h : ∃ x ∈ s, sSup ∅ ≤ f x)
    (h' : (s.image f).Nonempty
Tactic: rw [iSup, ← h'.csSup_eq_max', coe_image]
State 1: ⊢ <goal after rw [iSup, ← h'.csSup_eq_max', coe_image]>
Tactic: refine csSup_eq_csSup_of_forall_exists_le ?_ ?_  · simp only [ciSup_eq_ite, dite_eq_ite, Set.mem_range, Set.mem_image, mem_coe,  exists_exists_and_eq_and, forall_exists_index, forall_apply_eq_imp_iff]
State 2: ⊢ <goal after refine csSup_eq_csSup_of_forall_exists_le ?_ ?_>
Tactic: intro i  split_ifs  · exact ⟨_, by assumption, le_rfl⟩  · assumption  · simp only [Set.mem_image, mem_coe, ciSup_eq_ite, dite_eq_ite, Set.mem_range,  exists_exists_eq_and, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
State 3: ⊢ <goal after intro i>
Tactic: intro i hi
State 4: ⊢ <goal after intro i hi>
Tactic: refine ⟨i, ?_⟩
State 5: ⊢ <goal after refine ⟨i, ?_⟩>
Tactic: simp [hi]
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.ConditionallyCompleteLattice.Finset).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Order.ConditionallyCompleteLattice.Finset.Finset`:
State 0: ⊢ theorem Finset.ciSup_eq_max'_image {s : Finset ι} (h : ∃ x ∈ s, sSup ∅ ≤ f x)
    (h' : (s.image f).Nonempty
Tactic: rw [iSup, ← h'.csSup_eq_max', coe_image]
State 1: ⊢ <goal after rw [iSup, ← h'.csSup_eq_max', coe_image]>
Tactic: refine csSup_eq_csSup_of_forall_exists_le ?_ ?_  · simp only [ciSup_eq_ite, dite_eq_ite, Set.mem_range, Set.mem_image, mem_coe,  exists_exists_and_eq_and, forall_exists_index, forall_apply_eq_imp_iff]
State 2: ⊢ <goal after refine csSup_eq_csSup_of_forall_exists_le ?_ ?_>
Tactic: intro i  split_ifs  · exact ⟨_, by assumption, le_rfl⟩  · assumption  · simp only [Set.mem_image, mem_coe, ciSup_eq_ite, dite_eq_ite, Set.mem_range,  exists_exists_eq_and, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
State 3: ⊢ <goal after intro i>
Tactic: intro i hi
State 4: ⊢ <goal after intro i hi>
Tactic: refine ⟨i, ?_⟩
State 5: ⊢ <goal after refine ⟨i, ?_⟩>
Tactic: simp [hi]
```

Question: what is the next tactic after the last recorded state, and why?[/INST]

mistral-common:Tekken v15 (Mistral chat template, [INST]/[SYSTEM_PROMPT])


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Convex.Combination).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Analysis.Convex.Combination.affineCombination_mem_convexHull`:
State 0: ⊢ theorem affineCombination_mem_convexHull {s : Finset ι} {v : ι → E} {w : ι → R}
    (hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : s.sum w = 1) :
    s.affineCombination R v w ∈ convexHull R (range v)
Tactic: rw [affineCombination_eq_centerMass hw₁]
State 1: ⊢ <goal after rw [affineCombination_eq_centerMass hw₁]>
Tactic: apply s.centerMass_mem_convexHull hw₀  · simp [hw₁]  · simp  /-- The centroid can be regarded as a center of mass. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Convex.Combination).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Analysis.Convex.Combination.affineCombination_mem_convexHull`:
State 0: ⊢ theorem affineCombination_mem_convexHull {s : Finset ι} {v : ι → E} {w : ι → R}
    (hw₀ : ∀ i ∈ s, 0 ≤ w i) (hw₁ : s.sum w = 1) :
    s.affineCombination R v w ∈ convexHull R (range v)
Tactic: rw [affineCombination_eq_centerMass hw₁]
State 1: ⊢ <goal after rw [affineCombination_eq_centerMass hw₁]>
Tactic: apply s.centerMass_mem_convexHull hw₀  · simp [hw₁]  · simp  /-- The centroid can be regarded as a center of mass. -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.PushoutI).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.GroupTheory.PushoutI.ext_smul`:
State 0: ⊢ theorem ext_smul {w₁ w₂ : NormalWord d} (i : ι)
    (h : CoprodI.of (φ i w₁.head) • w₁.toWord =
         CoprodI.of (φ i w₂.head) • w₂.toWord) :
    w₁ = w₂
Tactic: rw [smul_eq_iff_eq_inv_smul, ← mul_smul] at h  subst h
State 1: ⊢ <goal after rw [smul_eq_iff_eq_inv_smul, ← mul_smul] at h>
Tactic: simp only [← map_inv, ← map_mul] at hw₁
State 2: ⊢ <goal after simp only [← map_inv, ← map_mul] at hw₁>
Tactic: have : h₁⁻¹ * h₂ = 1 := eq_one_of_smul_normalized w₂ (h₁⁻¹ * h₂) hw₂ hw₁
State 3: ⊢ <goal after have : h₁⁻¹ * h₂ = 1 := eq_one_of_smul_normalized w₂ (h₁⁻¹ * h₂) hw₂ hw₁>
Tactic: rw [inv_mul_eq_one] at this; subst this
State 4: ⊢ <goal after rw [inv_mul_eq_one] at this; subst this>
Tactic: simp  /-- Given a pair `(head, tail)`, we can form a word by prepending `head` to `tail`, but  putting head into normal form first, by making sure it is expressed as an element  of the base group multiplied by an element of the transversal. -/  noncomputable def rcons (i : ι) (p : Pair d i) : NormalWord d :=  letI n := (d.compl i).equiv p.head  let w := (Word.equivPair i).symm { p.toPair with head := n.2 }  { toWord := w  head := (MonoidHom.ofInjective (d.injective i)).symm n.1  normalized := fun i g hg => by  dsimp [w] at hg
State 5: ⊢ <goal after simp>
Tactic: rw [Word.equivPair_symm, Word.mem_rcons_iff] at hg  rcases hg with hg | ⟨_, rfl, rfl⟩  · exact p.normalized _ _ hg  · simp }
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.PushoutI).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.GroupTheory.PushoutI.ext_smul`:
State 0: ⊢ theorem ext_smul {w₁ w₂ : NormalWord d} (i : ι)
    (h : CoprodI.of (φ i w₁.head) • w₁.toWord =
         CoprodI.of (φ i w₂.head) • w₂.toWord) :
    w₁ = w₂
Tactic: rw [smul_eq_iff_eq_inv_smul, ← mul_smul] at h  subst h
State 1: ⊢ <goal after rw [smul_eq_iff_eq_inv_smul, ← mul_smul] at h>
Tactic: simp only [← map_inv, ← map_mul] at hw₁
State 2: ⊢ <goal after simp only [← map_inv, ← map_mul] at hw₁>
Tactic: have : h₁⁻¹ * h₂ = 1 := eq_one_of_smul_normalized w₂ (h₁⁻¹ * h₂) hw₂ hw₁
State 3: ⊢ <goal after have : h₁⁻¹ * h₂ = 1 := eq_one_of_smul_normalized w₂ (h₁⁻¹ * h₂) hw₂ hw₁>
Tactic: rw [inv_mul_eq_one] at this; subst this
State 4: ⊢ <goal after rw [inv_mul_eq_one] at this; subst this>
Tactic: simp  /-- Given a pair `(head, tail)`, we can form a word by prepending `head` to `tail`, but  putting head into normal form first, by making sure it is expressed as an element  of the base group multiplied by an element of the transversal. -/  noncomputable def rcons (i : ι) (p : Pair d i) : NormalWord d :=  letI n := (d.compl i).equiv p.head  let w := (Word.equivPair i).symm { p.toPair with head := n.2 }  { toWord := w  head := (MonoidHom.ofInjective (d.injective i)).symm n.1  normalized := fun i g hg => by  dsimp [w] at hg
State 5: ⊢ <goal after simp>
Tactic: rw [Word.equivPair_symm, Word.mem_rcons_iff] at hg  rcases hg with hg | ⟨_, rfl, rfl⟩  · exact p.normalized _ _ hg  · simp }
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.GradedAlgebra.Homogeneous.Submodule).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.GradedAlgebra.Homogeneous.Submodule.HomogeneousSubmodule`:
State 0: ⊢ theorem HomogeneousSubmodule.ext' {I J : HomogeneousSubmodule 𝒜 ℳ}
    (h : ∀ i, ∀ x ∈ ℳ i, x ∈ I ↔ x ∈ J) :
    I = J
Tactic: ext
State 1: ⊢ <goal after ext>
Tactic: rw [I.isHomogeneous.mem_iff, J.isHomogeneous.mem_iff]
State 2: ⊢ <goal after rw [I.isHomogeneous.mem_iff, J.isHomogeneous.mem_iff]>
Tactic: apply forall_congr'
State 3: ⊢ <goal after apply forall_congr'>
Tactic: exact fun i ↦ h i _ (decompose ℳ _ i).2
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.GradedAlgebra.Homogeneous.Submodule).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.GradedAlgebra.Homogeneous.Submodule.HomogeneousSubmodule`:
State 0: ⊢ theorem HomogeneousSubmodule.ext' {I J : HomogeneousSubmodule 𝒜 ℳ}
    (h : ∀ i, ∀ x ∈ ℳ i, x ∈ I ↔ x ∈ J) :
    I = J
Tactic: ext
State 1: ⊢ <goal after ext>
Tactic: rw [I.isHomogeneous.mem_iff, J.isHomogeneous.mem_iff]
State 2: ⊢ <goal after rw [I.isHomogeneous.mem_iff, J.isHomogeneous.mem_iff]>
Tactic: apply forall_congr'
State 3: ⊢ <goal after apply forall_congr'>
Tactic: exact fun i ↦ h i _ (decompose ℳ _ i).2
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.Matrix).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.LinearAlgebra.AffineSpace.Matrix.affineIndependent_of_toMatrix_right_inv`:
State 0: ⊢ theorem affineIndependent_of_toMatrix_right_inv [Fintype ι] [Finite ι'] [DecidableEq ι']
    (p : ι' → P) {A : Matrix ι ι' k} (hA : b.toMatrix p * A = 1) : AffineIndependent k p
Tactic: cases nonempty_fintype ι'
State 1: ⊢ <goal after cases nonempty_fintype ι'>
Tactic: rw [affineIndependent_iff_eq_of_fintype_affineCombination_eq]
State 2: ⊢ <goal after rw [affineIndependent_iff_eq_of_fintype_affineCombination_eq]>
Tactic: intro w₁ w₂ hw₁ hw₂ hweq
State 3: ⊢ <goal after intro w₁ w₂ hw₁ hw₂ hweq>
Tactic: have hweq' : w₁ ᵥ* b.toMatrix p = w₂ ᵥ* b.toMatrix p := by
State 4: ⊢ <goal after have hweq' : w₁ ᵥ* b.toMatrix p = w₂ ᵥ* b.toMatrix p := by>
Tactic: ext j  change (∑ i, w₁ i • b.coord j (p i)) = ∑ i, w₂ i • b.coord j (p i)
State 5: ⊢ <goal after ext j>
Tactic: rw [← Finset.univ.affineCombination_eq_linear_combination _ _ hw₁,  ← Finset.univ.affineCombination_eq_linear_combination _ _ hw₂,  ← Function.comp_def (b.coord j) p, ← Finset.univ.map_affineCombination p w₁ hw₁,  ← Finset.univ.map_affineCombination p w₂ hw₂, hweq]  replace hweq' := congr_arg (fun w => w ᵥ* A) hweq'
State 6: ⊢ <goal after rw [← Finset.univ.affineCombination_eq_linear_combination _ _ hw₁,>
Tactic: simpa only [Matrix.vecMul_vecMul, hA, Matrix.vecMul_one] using hweq'  /-- Given a family of points `p : ι' → P` and an affine basis `b`, if the matrix whose rows are the  coordinates of `p` with respect `b` has a left inverse, then `p` spans the entire space. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.Matrix).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.LinearAlgebra.AffineSpace.Matrix.affineIndependent_of_toMatrix_right_inv`:
State 0: ⊢ theorem affineIndependent_of_toMatrix_right_inv [Fintype ι] [Finite ι'] [DecidableEq ι']
    (p : ι' → P) {A : Matrix ι ι' k} (hA : b.toMatrix p * A = 1) : AffineIndependent k p
Tactic: cases nonempty_fintype ι'
State 1: ⊢ <goal after cases nonempty_fintype ι'>
Tactic: rw [affineIndependent_iff_eq_of_fintype_affineCombination_eq]
State 2: ⊢ <goal after rw [affineIndependent_iff_eq_of_fintype_affineCombination_eq]>
Tactic: intro w₁ w₂ hw₁ hw₂ hweq
State 3: ⊢ <goal after intro w₁ w₂ hw₁ hw₂ hweq>
Tactic: have hweq' : w₁ ᵥ* b.toMatrix p = w₂ ᵥ* b.toMatrix p := by
State 4: ⊢ <goal after have hweq' : w₁ ᵥ* b.toMatrix p = w₂ ᵥ* b.toMatrix p := by>
Tactic: ext j  change (∑ i, w₁ i • b.coord j (p i)) = ∑ i, w₂ i • b.coord j (p i)
State 5: ⊢ <goal after ext j>
Tactic: rw [← Finset.univ.affineCombination_eq_linear_combination _ _ hw₁,  ← Finset.univ.affineCombination_eq_linear_combination _ _ hw₂,  ← Function.comp_def (b.coord j) p, ← Finset.univ.map_affineCombination p w₁ hw₁,  ← Finset.univ.map_affineCombination p w₂ hw₂, hweq]  replace hweq' := congr_arg (fun w => w ᵥ* A) hweq'
State 6: ⊢ <goal after rw [← Finset.univ.affineCombination_eq_linear_combination _ _ hw₁,>
Tactic: simpa only [Matrix.vecMul_vecMul, hA, Matrix.vecMul_one] using hweq'  /-- Given a family of points `p : ι' → P` and an affine basis `b`, if the matrix whose rows are the  coordinates of `p` with respect `b` has a left inverse, then `p` spans the entire space. -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.Padics.PadicVal.Basic).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.NumberTheory.Padics.PadicVal.Basic.padicValNat_factorial_mul_add`:
State 0: ⊢ theorem padicValNat_factorial_mul_add {n : ℕ} (m : ℕ) [hp : Fact p.Prime] (h : n < p) :
    padicValNat p (p * m + n)! = padicValNat p (p * m)!
Tactic: induction n with  | zero => rw [add_zero]  | succ n hn =>
State 1: ⊢ <goal after induction n with>
Tactic: rw [add_succ, factorial_succ,  padicValNat.mul (succ_ne_zero (p * m + n)) <| factorial_ne_zero (p * m + _),  hn <| lt_of_succ_lt h, ← add_succ,  padicValNat_eq_zero_of_mem_Ioo ⟨(Nat.lt_add_of_pos_right <| succ_pos n),  (Nat.mul_add _ _ _▸ Nat.mul_one _ ▸ ((add_lt_add_iff_left (p * m)).mpr h))⟩,  zero_add]  /-- The `p`-adic valuation of `n!` is equal to the `p`-adic valuation of the factorial of the  largest multiple of `p` below `n`, i.e. `(p * ⌊n / p⌋)!`. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.Padics.PadicVal.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.NumberTheory.Padics.PadicVal.Basic.padicValNat_factorial_mul_add`:
State 0: ⊢ theorem padicValNat_factorial_mul_add {n : ℕ} (m : ℕ) [hp : Fact p.Prime] (h : n < p) :
    padicValNat p (p * m + n)! = padicValNat p (p * m)!
Tactic: induction n with  | zero => rw [add_zero]  | succ n hn =>
State 1: ⊢ <goal after induction n with>
Tactic: rw [add_succ, factorial_succ,  padicValNat.mul (succ_ne_zero (p * m + n)) <| factorial_ne_zero (p * m + _),  hn <| lt_of_succ_lt h, ← add_succ,  padicValNat_eq_zero_of_mem_Ioo ⟨(Nat.lt_add_of_pos_right <| succ_pos n),  (Nat.mul_add _ _ _▸ Nat.mul_one _ ▸ ((add_lt_add_iff_left (p * m)).mpr h))⟩,  zero_add]  /-- The `p`-adic valuation of `n!` is equal to the `p`-adic valuation of the factorial of the  largest multiple of `p` below `n`, i.e. `(p * ⌊n / p⌋)!`. -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Algebra.Subalgebra.Unitization).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Algebra.Algebra.Subalgebra.Unitization.unitization_range`:
State 0: ⊢ theorem unitization_range :
    (unitization s).range = subalgebraOfSubsemiring (.closure s)
Tactic: have := AddSubmonoidClass.nsmulMemClass (S := S)
State 1: ⊢ <goal after have := AddSubmonoidClass.nsmulMemClass (S := S)>
Tactic: rw [unitization, NonUnitalSubalgebra.unitization_range (hSRA := this), Algebra.adjoin_nat]  end NonUnitalSubsemiring  /-! ## Subrings -/  namespace NonUnitalSubring  variable {R S : Type*} [Ring R] [SetLike S R] [hSR : NonUnitalSubringClass S R] (s : S)  /-- The natural `ℤ`-algebra homomorphism from the unitization of a non-unital subring to  its `Subring.closure`. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Algebra.Subalgebra.Unitization).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Algebra.Algebra.Subalgebra.Unitization.unitization_range`:
State 0: ⊢ theorem unitization_range :
    (unitization s).range = subalgebraOfSubsemiring (.closure s)
Tactic: have := AddSubmonoidClass.nsmulMemClass (S := S)
State 1: ⊢ <goal after have := AddSubmonoidClass.nsmulMemClass (S := S)>
Tactic: rw [unitization, NonUnitalSubalgebra.unitization_range (hSRA := this), Algebra.adjoin_nat]  end NonUnitalSubsemiring  /-! ## Subrings -/  namespace NonUnitalSubring  variable {R S : Type*} [Ring R] [SetLike S R] [hSR : NonUnitalSubringClass S R] (s : S)  /-- The natural `ℤ`-algebra homomorphism from the unitization of a non-unital subring to  its `Subring.closure`. -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.QPF.Multivariate.Constructions.Fix).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Data.QPF.Multivariate.Constructions.Fix.recF_eq_of_wEquiv`:
State 0: ⊢ theorem recF_eq_of_wEquiv (α : TypeVec n) {β : Type u} (u : F (α.append1 β) → β) (x y : q.P.W α) :
    WEquiv x y → recF u x = recF u y
Tactic: induction x using q.P.wCases  case ih a₀ f'₀ f₀ =>
State 1: ⊢ <goal after induction x using q.P.wCases>
Tactic: induction y using q.P.wCases  case ih a₁ f'₁ f₁ =>
State 2: ⊢ <goal after induction y using q.P.wCases>
Tactic: intro h  -- Porting note: induction on h doesn't work.
State 3: ⊢ <goal after intro h>
Tactic: refine @WEquiv.recOn _ _ _ _ (fun a a' _ ↦ recF u a = recF u a') _ _ h ?_ ?_ ?_  · intro a f' f₀ f₁ _h ih; simp only [recF_eq]  congr 4; funext; apply ih  · intro a₀ f'₀ f₀ a₁ f'₁ f₁ h; simp only [recF_eq', abs_map, MvPFunctor.wDest'_wMk, h]  · intro x y z _e₁ _e₂ ih₁ ih₂; exact Eq.trans ih₁ ih₂
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.QPF.Multivariate.Constructions.Fix).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Data.QPF.Multivariate.Constructions.Fix.recF_eq_of_wEquiv`:
State 0: ⊢ theorem recF_eq_of_wEquiv (α : TypeVec n) {β : Type u} (u : F (α.append1 β) → β) (x y : q.P.W α) :
    WEquiv x y → recF u x = recF u y
Tactic: induction x using q.P.wCases  case ih a₀ f'₀ f₀ =>
State 1: ⊢ <goal after induction x using q.P.wCases>
Tactic: induction y using q.P.wCases  case ih a₁ f'₁ f₁ =>
State 2: ⊢ <goal after induction y using q.P.wCases>
Tactic: intro h  -- Porting note: induction on h doesn't work.
State 3: ⊢ <goal after intro h>
Tactic: refine @WEquiv.recOn _ _ _ _ (fun a a' _ ↦ recF u a = recF u a') _ _ h ?_ ?_ ?_  · intro a f' f₀ f₁ _h ih; simp only [recF_eq]  congr 4; funext; apply ih  · intro a₀ f'₀ f₀ a₁ f'₁ f₁ h; simp only [recF_eq', abs_map, MvPFunctor.wDest'_wMk, h]  · intro x y z _e₁ _e₂ ih₁ ih₂; exact Eq.trans ih₁ ih₂
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Monotone.Odd).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Order.Monotone.Odd.strictMono_of_odd_strictMonoOn_nonneg`:
State 0: ⊢ theorem strictMono_of_odd_strictMonoOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
    (h₂ : StrictMonoOn f (Ici 0)) : StrictMono f
Tactic: refine StrictMonoOn.Iic_union_Ici (fun x hx y hy hxy => neg_lt_neg_iff.1 ?_) h₂
State 1: ⊢ <goal after refine StrictMonoOn.Iic_union_Ici (fun x hx y hy hxy => neg_lt_neg_iff.1 ?_) h₂>
Tactic: rw [← h₁, ← h₁]
State 2: ⊢ <goal after rw [← h₁, ← h₁]>
Tactic: exact h₂ (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_lt_neg hxy)  /-- An odd function on a linear ordered additive commutative group is strictly antitone on the whole  group provided that it is strictly antitone on `Set.Ici 0`. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Monotone.Odd).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Order.Monotone.Odd.strictMono_of_odd_strictMonoOn_nonneg`:
State 0: ⊢ theorem strictMono_of_odd_strictMonoOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
    (h₂ : StrictMonoOn f (Ici 0)) : StrictMono f
Tactic: refine StrictMonoOn.Iic_union_Ici (fun x hx y hy hxy => neg_lt_neg_iff.1 ?_) h₂
State 1: ⊢ <goal after refine StrictMonoOn.Iic_union_Ici (fun x hx y hy hxy => neg_lt_neg_iff.1 ?_) h₂>
Tactic: rw [← h₁, ← h₁]
State 2: ⊢ <goal after rw [← h₁, ← h₁]>
Tactic: exact h₂ (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_lt_neg hxy)  /-- An odd function on a linear ordered additive commutative group is strictly antitone on the whole  group provided that it is strictly antitone on `Set.Ici 0`. -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Sets.VietorisTopology).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Topology.Sets.VietorisTopology.compactSpace_iff`:
State 0: ⊢ theorem compactSpace_iff : CompactSpace (Compacts α) ↔ CompactSpace α
Tactic: refine ⟨fun h => ⟨?_⟩, fun _ => inferInstance⟩
State 1: ⊢ <goal after refine ⟨fun h => ⟨?_⟩, fun _ => inferInstance⟩>
Tactic: convert isCompact_biUnion_coe_of_isCompact (α := α) isCompact_univ  symm  simp_rw [biUnion_univ, eq_univ_iff_forall, mem_iUnion]
State 2: ⊢ <goal after convert isCompact_biUnion_coe_of_isCompact (α := α) isCompact_univ>
Tactic: exact fun x => ⟨{x}, Set.mem_singleton x⟩
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Sets.VietorisTopology).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Topology.Sets.VietorisTopology.compactSpace_iff`:
State 0: ⊢ theorem compactSpace_iff : CompactSpace (Compacts α) ↔ CompactSpace α
Tactic: refine ⟨fun h => ⟨?_⟩, fun _ => inferInstance⟩
State 1: ⊢ <goal after refine ⟨fun h => ⟨?_⟩, fun _ => inferInstance⟩>
Tactic: convert isCompact_biUnion_coe_of_isCompact (α := α) isCompact_univ  symm  simp_rw [biUnion_univ, eq_univ_iff_forall, mem_iUnion]
State 2: ⊢ <goal after convert isCompact_biUnion_coe_of_isCompact (α := α) isCompact_univ>
Tactic: exact fun x => ⟨{x}, Set.mem_singleton x⟩
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.InnerProductSpace.Harmonic.Basic).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Analysis.InnerProductSpace.Harmonic.Basic.HarmonicAt`:
State 0: ⊢ theorem HarmonicAt.comp_CLM (h : HarmonicAt f x) (l : F →L[ℝ] G) :
    HarmonicAt (l ∘ f) x
Tactic: constructor  · exact h.1.continuousLinearMap_comp l  · filter_upwards [h.1.laplacian_CLM_comp_left_nhds (l := l), h.2]
State 1: ⊢ <goal after constructor>
Tactic: simp_all  /--  Compositions of continuous linear maps with harmonic functions are harmonic.  -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.InnerProductSpace.Harmonic.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Analysis.InnerProductSpace.Harmonic.Basic.HarmonicAt`:
State 0: ⊢ theorem HarmonicAt.comp_CLM (h : HarmonicAt f x) (l : F →L[ℝ] G) :
    HarmonicAt (l ∘ f) x
Tactic: constructor  · exact h.1.continuousLinearMap_comp l  · filter_upwards [h.1.laplacian_CLM_comp_left_nhds (l := l), h.2]
State 1: ⊢ <goal after constructor>
Tactic: simp_all  /--  Compositions of continuous linear maps with harmonic functions are harmonic.  -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.IntegralClosure.IntegralRestrict).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.IntegralClosure.IntegralRestrict.galLift`:
State 0: ⊢ def galLift (σ : B →ₐ[A] B₂) : L →ₐ[K] L₂
Tactic: simpa only [RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, AlgHom.commutes,  isUnit_iff_ne_zero, ne_eq, map_eq_zero_iff _ (FaithfulSMul.algebraMap_injective _ _),  ← IsScalarTower.algebraMap_apply] using nonZeroDivisors.ne_zero hx  haveI H_eq : (IsLocalization.lift (S := L) H).comp (algebraMap K L) = (algebraMap K L₂) := by
State 1: ⊢ <goal after simpa only [RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, AlgHom.commutes,>
Tactic: apply IsLocalization.ringHom_ext A⁰
State 2: ⊢ <goal after apply IsLocalization.ringHom_ext A⁰>
Tactic: ext
State 3: ⊢ <goal after ext>
Tactic: simp only [RingHom.coe_comp, Function.comp_apply, ← IsScalarTower.algebraMap_apply A K L,  ← IsScalarTower.algebraMap_apply A K L₂,  IsScalarTower.algebraMap_apply A B L, IsScalarTower.algebraMap_apply A B₂ L₂,  IsLocalization.lift_eq, RingHom.coe_coe, AlgHom.commutes]  { IsLocalization.lift (S := L) H with commutes' := DFunLike.congr_fun H_eq }  omit [IsIntegralClosure B₂ A L₂] in
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.IntegralClosure.IntegralRestrict).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.IntegralClosure.IntegralRestrict.galLift`:
State 0: ⊢ def galLift (σ : B →ₐ[A] B₂) : L →ₐ[K] L₂
Tactic: simpa only [RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, AlgHom.commutes,  isUnit_iff_ne_zero, ne_eq, map_eq_zero_iff _ (FaithfulSMul.algebraMap_injective _ _),  ← IsScalarTower.algebraMap_apply] using nonZeroDivisors.ne_zero hx  haveI H_eq : (IsLocalization.lift (S := L) H).comp (algebraMap K L) = (algebraMap K L₂) := by
State 1: ⊢ <goal after simpa only [RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, AlgHom.commutes,>
Tactic: apply IsLocalization.ringHom_ext A⁰
State 2: ⊢ <goal after apply IsLocalization.ringHom_ext A⁰>
Tactic: ext
State 3: ⊢ <goal after ext>
Tactic: simp only [RingHom.coe_comp, Function.comp_apply, ← IsScalarTower.algebraMap_apply A K L,  ← IsScalarTower.algebraMap_apply A K L₂,  IsScalarTower.algebraMap_apply A B L, IsScalarTower.algebraMap_apply A B₂ L₂,  IsLocalization.lift_eq, RingHom.coe_coe, AlgHom.commutes]  { IsLocalization.lift (S := L) H with commutes' := DFunLike.congr_fun H_eq }  omit [IsIntegralClosure B₂ A L₂] in
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.SpecialFunctions.Complex.Arg).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Analysis.SpecialFunctions.Complex.Arg.arg_eq_zero_iff`:
State 0: ⊢ theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0
Tactic: refine ⟨fun h => ?_, ?_⟩  · rw [← norm_mul_cos_add_sin_mul_I z, h]
State 1: ⊢ <goal after refine ⟨fun h => ?_, ?_⟩>
Tactic: simp [norm_nonneg]  · obtain ⟨x, y⟩ := z  rintro ⟨h, rfl : y = 0⟩
State 2: ⊢ <goal after simp [norm_nonneg]>
Tactic: exact arg_ofReal_of_nonneg h  open ComplexOrder in
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.SpecialFunctions.Complex.Arg).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Analysis.SpecialFunctions.Complex.Arg.arg_eq_zero_iff`:
State 0: ⊢ theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0
Tactic: refine ⟨fun h => ?_, ?_⟩  · rw [← norm_mul_cos_add_sin_mul_I z, h]
State 1: ⊢ <goal after refine ⟨fun h => ?_, ?_⟩>
Tactic: simp [norm_nonneg]  · obtain ⟨x, y⟩ := z  rintro ⟨h, rfl : y = 0⟩
State 2: ⊢ <goal after simp [norm_nonneg]>
Tactic: exact arg_ofReal_of_nonneg h  open ComplexOrder in
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.MvPowerSeries.Order).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.MvPowerSeries.Order.coeff_mul_left_one_sub_of_lt_order`:
State 0: ⊢ theorem coeff_mul_left_one_sub_of_lt_order (d : σ →₀ ℕ) (h : degree d < g.order) :
    coeff d (f * (1 - g)) = coeff d f
Tactic: rw [degree_eq_weight_one] at h
State 1: ⊢ <goal after rw [degree_eq_weight_one] at h>
Tactic: exact coeff_mul_left_one_sub_of_lt_weightedOrder _ h
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.MvPowerSeries.Order).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.MvPowerSeries.Order.coeff_mul_left_one_sub_of_lt_order`:
State 0: ⊢ theorem coeff_mul_left_one_sub_of_lt_order (d : σ →₀ ℕ) (h : degree d < g.order) :
    coeff d (f * (1 - g)) = coeff d f
Tactic: rw [degree_eq_weight_one] at h
State 1: ⊢ <goal after rw [degree_eq_weight_one] at h>
Tactic: exact coeff_mul_left_one_sub_of_lt_weightedOrder _ h
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable.auxbound2`:
State 0: ⊢ lemma auxbound2 (c : ℝ) {d : ℝ} (hd : 1 ≤ d ^ 2) : r z ≤ ‖c * (z : ℂ) + d‖
Tactic: have H1 : √(r1 z) ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2) :=  (Real.sqrt_le_sqrt_iff (by positivity)).mpr (r1_aux_bound _ _ hd)
State 1: ⊢ <goal after have H1 : √(r1 z) ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2) :=>
Tactic: simpa only [r, norm_def, normSq_apply, add_re, re_ofReal_mul, coe_re, ofReal_re, ← pow_two,  add_im, im_ofReal_mul, coe_im, ofReal_im, add_zero, min_le_iff] using Or.inr H1
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable.auxbound2`:
State 0: ⊢ lemma auxbound2 (c : ℝ) {d : ℝ} (hd : 1 ≤ d ^ 2) : r z ≤ ‖c * (z : ℂ) + d‖
Tactic: have H1 : √(r1 z) ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2) :=  (Real.sqrt_le_sqrt_iff (by positivity)).mpr (r1_aux_bound _ _ hd)
State 1: ⊢ <goal after have H1 : √(r1 z) ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2) :=>
Tactic: simpa only [r, norm_def, normSq_apply, add_re, re_ofReal_mul, coe_re, ofReal_re, ← pow_two,  add_im, im_ofReal_mul, coe_im, ofReal_im, add_zero, min_le_iff] using Or.inr H1
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.OpenPartialHomeomorph.Continuity).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Topology.OpenPartialHomeomorph.Continuity.eventually_nhds'`:
State 0: ⊢ theorem eventually_nhds' {x : X} (p : X → Prop) (hx : x ∈ e.source) :
    (∀ᶠ y in 𝓝 (e x), p (e.symm y)) ↔ ∀ᶠ x in 𝓝 x, p x
Tactic: rw [e.eventually_nhds _ hx]
State 1: ⊢ <goal after rw [e.eventually_nhds _ hx]>
Tactic: refine eventually_congr ((e.eventually_left_inverse hx).mono fun y hy => ?_)
State 2: ⊢ <goal after refine eventually_congr ((e.eventually_left_inverse hx).mono fun y hy => ?_)>
Tactic: rw [hy]
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.OpenPartialHomeomorph.Continuity).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Topology.OpenPartialHomeomorph.Continuity.eventually_nhds'`:
State 0: ⊢ theorem eventually_nhds' {x : X} (p : X → Prop) (hx : x ∈ e.source) :
    (∀ᶠ y in 𝓝 (e x), p (e.symm y)) ↔ ∀ᶠ x in 𝓝 x, p x
Tactic: rw [e.eventually_nhds _ hx]
State 1: ⊢ <goal after rw [e.eventually_nhds _ hx]>
Tactic: refine eventually_congr ((e.eventually_left_inverse hx).mono fun y hy => ?_)
State 2: ⊢ <goal after refine eventually_congr ((e.eventually_left_inverse hx).mono fun y hy => ?_)>
Tactic: rw [hy]
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.Enumerative.DyckWord).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Combinatorics.Enumerative.DyckWord.count_D_lt_count_U_of_lt_firstReturn`:
State 0: ⊢ lemma count_D_lt_count_U_of_lt_firstReturn {i : ℕ} (hi : i < p.firstReturn) :
    (p.toList.take (i + 1)).count D < (p.toList.take (i + 1)).count U
Tactic: have ne := not_of_lt_findIdx hi
State 1: ⊢ <goal after have ne := not_of_lt_findIdx hi>
Tactic: rw [decide_eq_false_iff_not, ← ne_eq, getElem_range] at ne
State 2: ⊢ <goal after rw [decide_eq_false_iff_not, ← ne_eq, getElem_range] at ne>
Tactic: exact lt_of_le_of_ne (p.count_D_le_count_U (i + 1)) ne.symm
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.Enumerative.DyckWord).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Combinatorics.Enumerative.DyckWord.count_D_lt_count_U_of_lt_firstReturn`:
State 0: ⊢ lemma count_D_lt_count_U_of_lt_firstReturn {i : ℕ} (hi : i < p.firstReturn) :
    (p.toList.take (i + 1)).count D < (p.toList.take (i + 1)).count U
Tactic: have ne := not_of_lt_findIdx hi
State 1: ⊢ <goal after have ne := not_of_lt_findIdx hi>
Tactic: rw [decide_eq_false_iff_not, ← ne_eq, getElem_range] at ne
State 2: ⊢ <goal after rw [decide_eq_false_iff_not, ← ne_eq, getElem_range] at ne>
Tactic: exact lt_of_le_of_ne (p.count_D_le_count_U (i + 1)) ne.symm
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Matrix.PosDef).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.LinearAlgebra.Matrix.PosDef.conjTranspose_mul_mul_same`:
State 0: ⊢ lemma conjTranspose_mul_mul_same {A : Matrix n n R} {B : Matrix n m R} (hA : A.PosDef)
    (hB : Function.Injective B.mulVec) :
    (Bᴴ * A * B).PosDef
Tactic: refine of_dotProduct_mulVec_pos (isHermitian_conjTranspose_mul_mul _ hA.1) fun x hx => ?_
State 1: ⊢ <goal after refine of_dotProduct_mulVec_pos (isHermitian_conjTranspose_mul_mul _ hA.1) fun x hx => ?_>
Tactic: have : B *ᵥ x ≠ 0 := fun h => hx <| hB.eq_iff' (mulVec_zero _) |>.1 h
State 2: ⊢ <goal after have : B *ᵥ x ≠ 0 := fun h => hx <| hB.eq_iff' (mulVec_zero _) |>.1 h>
Tactic: simpa only [star_mulVec, dotProduct_mulVec, vecMul_vecMul] using hA.dotProduct_mulVec_pos this
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Matrix.PosDef).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.LinearAlgebra.Matrix.PosDef.conjTranspose_mul_mul_same`:
State 0: ⊢ lemma conjTranspose_mul_mul_same {A : Matrix n n R} {B : Matrix n m R} (hA : A.PosDef)
    (hB : Function.Injective B.mulVec) :
    (Bᴴ * A * B).PosDef
Tactic: refine of_dotProduct_mulVec_pos (isHermitian_conjTranspose_mul_mul _ hA.1) fun x hx => ?_
State 1: ⊢ <goal after refine of_dotProduct_mulVec_pos (isHermitian_conjTranspose_mul_mul _ hA.1) fun x hx => ?_>
Tactic: have : B *ᵥ x ≠ 0 := fun h => hx <| hB.eq_iff' (mulVec_zero _) |>.1 h
State 2: ⊢ <goal after have : B *ᵥ x ≠ 0 := fun h => hx <| hB.eq_iff' (mulVec_zero _) |>.1 h>
Tactic: simpa only [star_mulVec, dotProduct_mulVec, vecMul_vecMul] using hA.dotProduct_mulVec_pos this
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.SimpleGraph.Diam).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Combinatorics.SimpleGraph.Diam.ediam_le_two_mul_radius`:
State 0: ⊢ lemma ediam_le_two_mul_radius : G.ediam ≤ 2 * G.radius
Tactic: cases isEmpty_or_nonempty α  · rw [radius_eq_top_of_isEmpty]
State 1: ⊢ <goal after cases isEmpty_or_nonempty α>
Tactic: exact le_top  · by_cases hdiam : G.ediam = ⊤  · simp [hdiam, ediam_eq_top_iff_radius_eq_top.mp hdiam]  · obtain ⟨w, hw⟩ := G.exists_eccent_eq_radius
State 2: ⊢ <goal after exact le_top>
Tactic: obtain ⟨_, _, h⟩ := G.exists_edist_eq_ediam_of_ne_top hdiam
State 3: ⊢ <goal after obtain ⟨_, _, h⟩ := G.exists_edist_eq_ediam_of_ne_top hdiam>
Tactic: apply le_trans (h ▸ G.edist_triangle (v := w))
State 4: ⊢ <goal after apply le_trans (h ▸ G.edist_triangle (v := w))>
Tactic: rw [two_mul]
State 5: ⊢ <goal after rw [two_mul]>
Tactic: exact hw ▸ add_le_add (G.edist_comm ▸ G.edist_le_eccent) G.edist_le_eccent
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.SimpleGraph.Diam).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Combinatorics.SimpleGraph.Diam.ediam_le_two_mul_radius`:
State 0: ⊢ lemma ediam_le_two_mul_radius : G.ediam ≤ 2 * G.radius
Tactic: cases isEmpty_or_nonempty α  · rw [radius_eq_top_of_isEmpty]
State 1: ⊢ <goal after cases isEmpty_or_nonempty α>
Tactic: exact le_top  · by_cases hdiam : G.ediam = ⊤  · simp [hdiam, ediam_eq_top_iff_radius_eq_top.mp hdiam]  · obtain ⟨w, hw⟩ := G.exists_eccent_eq_radius
State 2: ⊢ <goal after exact le_top>
Tactic: obtain ⟨_, _, h⟩ := G.exists_edist_eq_ediam_of_ne_top hdiam
State 3: ⊢ <goal after obtain ⟨_, _, h⟩ := G.exists_edist_eq_ediam_of_ne_top hdiam>
Tactic: apply le_trans (h ▸ G.edist_triangle (v := w))
State 4: ⊢ <goal after apply le_trans (h ▸ G.edist_triangle (v := w))>
Tactic: rw [two_mul]
State 5: ⊢ <goal after rw [two_mul]>
Tactic: exact hw ▸ add_le_add (G.edist_comm ▸ G.edist_le_eccent) G.edist_le_eccent
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Polynomial.Vieta).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.Polynomial.Vieta.prod_X_add_C_eq_sum_esymm`:
State 0: ⊢ theorem prod_X_add_C_eq_sum_esymm (s : Multiset R) :
    (s.map fun r => X + C r).prod =
      ∑ j ∈ Finset.range (Multiset.card s + 1), (C (s.esymm j) * X ^ (Multiset.card s - j))
Tactic: rw [prod_map_add, antidiagonal_eq_map_powerset, map_map, ← bind_powerset_len,  map_bind, sum_bind, Finset.sum_eq_multiset_sum, Finset.range_val, map_congr (Eq.refl _)]
State 1: ⊢ <goal after rw [prod_map_add, antidiagonal_eq_map_powerset, map_map, ← bind_powerset_len,>
Tactic: intro _ _
State 2: ⊢ <goal after intro _ _>
Tactic: rw [esymm, ← sum_hom', ← sum_map_mul_right, map_congr (Eq.refl _)]
State 3: ⊢ <goal after rw [esymm, ← sum_hom', ← sum_map_mul_right, map_congr (Eq.refl _)]>
Tactic: intro s ht
State 4: ⊢ <goal after intro s ht>
Tactic: rw [mem_powersetCard] at ht  dsimp
State 5: ⊢ <goal after rw [mem_powersetCard] at ht>
Tactic: rw [prod_hom' s (Polynomial.C : R →+* R[X])]
State 6: ⊢ <goal after rw [prod_hom' s (Polynomial.C : R →+* R[X])]>
Tactic: simp [ht, prod_replicate, map_id', card_sub]  /-- Vieta's formula for the coefficients of the product of linear terms `X + λ` where `λ` runs  through a multiset `s` : the `k`th coefficient is the symmetric function `esymm (card s - k) s`. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Polynomial.Vieta).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.Polynomial.Vieta.prod_X_add_C_eq_sum_esymm`:
State 0: ⊢ theorem prod_X_add_C_eq_sum_esymm (s : Multiset R) :
    (s.map fun r => X + C r).prod =
      ∑ j ∈ Finset.range (Multiset.card s + 1), (C (s.esymm j) * X ^ (Multiset.card s - j))
Tactic: rw [prod_map_add, antidiagonal_eq_map_powerset, map_map, ← bind_powerset_len,  map_bind, sum_bind, Finset.sum_eq_multiset_sum, Finset.range_val, map_congr (Eq.refl _)]
State 1: ⊢ <goal after rw [prod_map_add, antidiagonal_eq_map_powerset, map_map, ← bind_powerset_len,>
Tactic: intro _ _
State 2: ⊢ <goal after intro _ _>
Tactic: rw [esymm, ← sum_hom', ← sum_map_mul_right, map_congr (Eq.refl _)]
State 3: ⊢ <goal after rw [esymm, ← sum_hom', ← sum_map_mul_right, map_congr (Eq.refl _)]>
Tactic: intro s ht
State 4: ⊢ <goal after intro s ht>
Tactic: rw [mem_powersetCard] at ht  dsimp
State 5: ⊢ <goal after rw [mem_powersetCard] at ht>
Tactic: rw [prod_hom' s (Polynomial.C : R →+* R[X])]
State 6: ⊢ <goal after rw [prod_hom' s (Polynomial.C : R →+* R[X])]>
Tactic: simp [ht, prod_replicate, map_id', card_sub]  /-- Vieta's formula for the coefficients of the product of linear terms `X + λ` where `λ` runs  through a multiset `s` : the `k`th coefficient is the symmetric function `esymm (card s - k) s`. -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.PellMatiyasevic).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.NumberTheory.PellMatiyasevic.eq_pell_lem`:
State 0: ⊢ theorem eq_pell_lem : ∀ (n) (b : ℤ√(d a1)), 1 ≤ b → IsPell b →
    b ≤ pellZd a1 n → ∃ n, b = pellZd a1 n
  | 0, _ => fun h1 _ hl => ⟨0, le_antisymm hl h1⟩
  | n + 1, b => fun h1 hp h =>
    have a1p : (0 : ℤ√(d a1)) ≤ ⟨a, 1⟩
Tactic: have am1p : (0 : ℤ√(d a1)) ≤ ⟨a, -1⟩ := show (_ : Nat) ≤ _ by simp [d]
State 1: ⊢ <goal after have am1p : (0 : ℤ√(d a1)) ≤ ⟨a, -1⟩ := show (_ : Nat) ≤ _ by simp [d]>
Tactic: have a1m : (⟨a, 1⟩ * ⟨a, -1⟩ : ℤ√(d a1)) = 1 := isPell_norm.1 (isPell_one a1)  if ha : (⟨↑a, 1⟩ : ℤ√(d a1)) ≤ b then  let ⟨m, e⟩ :=  eq_pell_lem n (b * ⟨a, -1⟩) (by rw [← a1m]; exact mul_le_mul_of_nonneg_right ha am1p)  (isPell_mul hp (isPell_star.1 (isPell_one a1)))  (by
State 2: ⊢ <goal after have a1m : (⟨a, 1⟩ * ⟨a, -1⟩ : ℤ√(d a1)) = 1 := isPell_norm.1 (isPell_one a1)>
Tactic: have t := mul_le_mul_of_nonneg_right h am1p  rwa [pellZd_succ, mul_assoc, a1m, mul_one] at t)  ⟨m + 1, by
State 3: ⊢ <goal after have t := mul_le_mul_of_nonneg_right h am1p>
Tactic: rw [show b = b * ⟨a, -1⟩ * ⟨a, 1⟩ by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp,  pellZd_succ, e]⟩  else  suffices ¬1 < b from ⟨0, show b = 1 from (Or.resolve_left (lt_or_eq_of_le h1) this).symm⟩  fun h1l => by
State 4: ⊢ <goal after rw [show b = b * ⟨a, -1⟩ * ⟨a, 1⟩ by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp,>
Tactic: obtain ⟨x, y⟩ := b
State 5: ⊢ <goal after obtain ⟨x, y⟩ := b>
Tactic: exact by
State 6: ⊢ <goal after exact by>
Tactic: have bm : (_ * ⟨_, _⟩ : ℤ√d a1) = 1 := Pell.isPell_norm.1 hp
State 7: ⊢ <goal after have bm : (_ * ⟨_, _⟩ : ℤ√d a1) = 1 := Pell.isPell_norm.1 hp>
Tactic: have y0l : (0 : ℤ√d a1) < ⟨x - x, y - -y⟩ :=  sub_lt_sub h1l fun hn : (1 : ℤ√d a1) ≤ ⟨x, -y⟩ => by
State 8: ⊢ <goal after have y0l : (0 : ℤ√d a1) < ⟨x - x, y - -y⟩ :=>
Tactic: have t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)
State 9: ⊢ <goal after have t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)>
Tactic: rw [bm, mul_one] at t
State 10: ⊢ <goal after rw [bm, mul_one] at t>
Tactic: exact h1l t
State 11: ⊢ <goal after exact h1l t>
Tactic: have yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩ :=  show (⟨x, y⟩ - ⟨x, -y⟩ : ℤ√d a1) < ⟨a, 1⟩ - ⟨a, -1⟩ from  sub_lt_sub ha fun hn : (⟨x, -y⟩ : ℤ√d a1) ≤ ⟨a, -1⟩ => by
State 12: ⊢ <goal after have yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩ :=>
Tactic: have t := mul_le_mul_of_nonneg_right  (mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)) a1p
State 13: ⊢ <goal after have t := mul_le_mul_of_nonneg_right>
Tactic: rw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t
State 14: ⊢ <goal after rw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t>
Tactic: exact ha t
State 15: ⊢ <goal after exact ha t>
Tactic: simp only [sub_self, sub_neg_eq_add] at y0l; simp only [Zsqrtd.re_neg, add_neg_cancel,  Zsqrtd.im_neg, neg_neg] at yl2
State 16: ⊢ <goal after simp only [sub_self, sub_neg_eq_add] at y0l; simp only [Zsqrtd.re_neg, add_neg_cancel,>
Tactic: exact  match y, y0l, (yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩) with  | 0, y0l, _ => y0l (le_refl 0)  | (y + 1 : ℕ), _, yl2 =>  yl2  (Zsqrtd.le_of_le_le (by simp)  (let t := Int.ofNat_le_ofNat_of_le (Nat.succ_pos y)  add_le_add t t))  | Int.negSucc _, y0l, _ => y0l trivial  set_option backward.privateInPublic true in  set_option backward.privateInPublic.warn false in
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.PellMatiyasevic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.NumberTheory.PellMatiyasevic.eq_pell_lem`:
State 0: ⊢ theorem eq_pell_lem : ∀ (n) (b : ℤ√(d a1)), 1 ≤ b → IsPell b →
    b ≤ pellZd a1 n → ∃ n, b = pellZd a1 n
  | 0, _ => fun h1 _ hl => ⟨0, le_antisymm hl h1⟩
  | n + 1, b => fun h1 hp h =>
    have a1p : (0 : ℤ√(d a1)) ≤ ⟨a, 1⟩
Tactic: have am1p : (0 : ℤ√(d a1)) ≤ ⟨a, -1⟩ := show (_ : Nat) ≤ _ by simp [d]
State 1: ⊢ <goal after have am1p : (0 : ℤ√(d a1)) ≤ ⟨a, -1⟩ := show (_ : Nat) ≤ _ by simp [d]>
Tactic: have a1m : (⟨a, 1⟩ * ⟨a, -1⟩ : ℤ√(d a1)) = 1 := isPell_norm.1 (isPell_one a1)  if ha : (⟨↑a, 1⟩ : ℤ√(d a1)) ≤ b then  let ⟨m, e⟩ :=  eq_pell_lem n (b * ⟨a, -1⟩) (by rw [← a1m]; exact mul_le_mul_of_nonneg_right ha am1p)  (isPell_mul hp (isPell_star.1 (isPell_one a1)))  (by
State 2: ⊢ <goal after have a1m : (⟨a, 1⟩ * ⟨a, -1⟩ : ℤ√(d a1)) = 1 := isPell_norm.1 (isPell_one a1)>
Tactic: have t := mul_le_mul_of_nonneg_right h am1p  rwa [pellZd_succ, mul_assoc, a1m, mul_one] at t)  ⟨m + 1, by
State 3: ⊢ <goal after have t := mul_le_mul_of_nonneg_right h am1p>
Tactic: rw [show b = b * ⟨a, -1⟩ * ⟨a, 1⟩ by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp,  pellZd_succ, e]⟩  else  suffices ¬1 < b from ⟨0, show b = 1 from (Or.resolve_left (lt_or_eq_of_le h1) this).symm⟩  fun h1l => by
State 4: ⊢ <goal after rw [show b = b * ⟨a, -1⟩ * ⟨a, 1⟩ by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp,>
Tactic: obtain ⟨x, y⟩ := b
State 5: ⊢ <goal after obtain ⟨x, y⟩ := b>
Tactic: exact by
State 6: ⊢ <goal after exact by>
Tactic: have bm : (_ * ⟨_, _⟩ : ℤ√d a1) = 1 := Pell.isPell_norm.1 hp
State 7: ⊢ <goal after have bm : (_ * ⟨_, _⟩ : ℤ√d a1) = 1 := Pell.isPell_norm.1 hp>
Tactic: have y0l : (0 : ℤ√d a1) < ⟨x - x, y - -y⟩ :=  sub_lt_sub h1l fun hn : (1 : ℤ√d a1) ≤ ⟨x, -y⟩ => by
State 8: ⊢ <goal after have y0l : (0 : ℤ√d a1) < ⟨x - x, y - -y⟩ :=>
Tactic: have t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)
State 9: ⊢ <goal after have t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)>
Tactic: rw [bm, mul_one] at t
State 10: ⊢ <goal after rw [bm, mul_one] at t>
Tactic: exact h1l t
State 11: ⊢ <goal after exact h1l t>
Tactic: have yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩ :=  show (⟨x, y⟩ - ⟨x, -y⟩ : ℤ√d a1) < ⟨a, 1⟩ - ⟨a, -1⟩ from  sub_lt_sub ha fun hn : (⟨x, -y⟩ : ℤ√d a1) ≤ ⟨a, -1⟩ => by
State 12: ⊢ <goal after have yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩ :=>
Tactic: have t := mul_le_mul_of_nonneg_right  (mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)) a1p
State 13: ⊢ <goal after have t := mul_le_mul_of_nonneg_right>
Tactic: rw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t
State 14: ⊢ <goal after rw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t>
Tactic: exact ha t
State 15: ⊢ <goal after exact ha t>
Tactic: simp only [sub_self, sub_neg_eq_add] at y0l; simp only [Zsqrtd.re_neg, add_neg_cancel,  Zsqrtd.im_neg, neg_neg] at yl2
State 16: ⊢ <goal after simp only [sub_self, sub_neg_eq_add] at y0l; simp only [Zsqrtd.re_neg, add_neg_cancel,>
Tactic: exact  match y, y0l, (yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩) with  | 0, y0l, _ => y0l (le_refl 0)  | (y + 1 : ℕ), _, yl2 =>  yl2  (Zsqrtd.le_of_le_le (by simp)  (let t := Int.ofNat_le_ofNat_of_le (Nat.succ_pos y)  add_le_add t t))  | Int.negSucc _, y0l, _ => y0l trivial  set_option backward.privateInPublic true in  set_option backward.privateInPublic.warn false in
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.InitialSeg).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Order.InitialSeg.top_rel_top`:
State 0: ⊢ theorem top_rel_top {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} [IsWellOrder γ t]
    (f : r ≺i s) (g : s ≺i t) (h : r ≺i t) : t h.top g.top
Tactic: rw [Subsingleton.elim h (f.trans g)]
State 1: ⊢ <goal after rw [Subsingleton.elim h (f.trans g)]>
Tactic: apply PrincipalSeg.lt_top  /-- Any element of a well order yields a principal segment. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.InitialSeg).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Order.InitialSeg.top_rel_top`:
State 0: ⊢ theorem top_rel_top {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} [IsWellOrder γ t]
    (f : r ≺i s) (g : s ≺i t) (h : r ≺i t) : t h.top g.top
Tactic: rw [Subsingleton.elim h (f.trans g)]
State 1: ⊢ <goal after rw [Subsingleton.elim h (f.trans g)]>
Tactic: apply PrincipalSeg.lt_top  /-- Any element of a well order yields a principal segment. -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Homotopy.HomotopyGroup).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Topology.Homotopy.HomotopyGroup.fromLoop`:
State 0: ⊢ def fromLoop (i : N) (p : Ω (Ω^ { j // j ≠ i } X x) const) : Ω^ N X x
Tactic: simp only [ContinuousMap.comp_apply, ContinuousMap.coe_coe,  funSplitAt_apply, ContinuousMap.uncurry_apply, ContinuousMap.coe_mk,  Function.uncurry_apply_pair]
State 1: ⊢ <goal after simp only [ContinuousMap.comp_apply, ContinuousMap.coe_coe,>
Tactic: obtain rfl | Hne := eq_or_ne j i  · rcases Hj with Hj | Hj <;> simp only [Hj, p.coe_toContinuousMap, p.source, p.target] <;> rfl  · exact GenLoop.boundary _ _ ⟨⟨j, Hne⟩, Hj⟩⟩
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Homotopy.HomotopyGroup).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Topology.Homotopy.HomotopyGroup.fromLoop`:
State 0: ⊢ def fromLoop (i : N) (p : Ω (Ω^ { j // j ≠ i } X x) const) : Ω^ N X x
Tactic: simp only [ContinuousMap.comp_apply, ContinuousMap.coe_coe,  funSplitAt_apply, ContinuousMap.uncurry_apply, ContinuousMap.coe_mk,  Function.uncurry_apply_pair]
State 1: ⊢ <goal after simp only [ContinuousMap.comp_apply, ContinuousMap.coe_coe,>
Tactic: obtain rfl | Hne := eq_or_ne j i  · rcases Hj with Hj | Hj <;> simp only [Hj, p.coe_toContinuousMap, p.source, p.target] <;> rfl  · exact GenLoop.boundary _ _ ⟨⟨j, Hne⟩, Hj⟩⟩
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Normed.Ring.Units).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Analysis.Normed.Ring.Units.closure_ne_top`:
State 0: ⊢ theorem closure_ne_top (I : Ideal R) (hI : I ≠ ⊤) : I.closure ≠ ⊤
Tactic: have h := closure_minimal (coe_subset_nonunits hI) nonunits.isClosed
State 1: ⊢ <goal after have h := closure_minimal (coe_subset_nonunits hI) nonunits.isClosed>
Tactic: simpa only [I.closure.eq_top_iff_one, Ne] using mt (@h 1) one_notMem_nonunits  /-- The `Ideal.closure` of a maximal ideal in a normed ring with summable  geometric series is the ideal itself. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Normed.Ring.Units).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Analysis.Normed.Ring.Units.closure_ne_top`:
State 0: ⊢ theorem closure_ne_top (I : Ideal R) (hI : I ≠ ⊤) : I.closure ≠ ⊤
Tactic: have h := closure_minimal (coe_subset_nonunits hI) nonunits.isClosed
State 1: ⊢ <goal after have h := closure_minimal (coe_subset_nonunits hI) nonunits.isClosed>
Tactic: simpa only [I.closure.eq_top_iff_one, Ne] using mt (@h 1) one_notMem_nonunits  /-- The `Ideal.closure` of a maximal ideal in a normed ring with summable  geometric series is the ideal itself. -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.Complement).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.GroupTheory.Complement.IsComplement'`:
State 0: ⊢ theorem IsComplement'.isCompl (h : IsComplement' H K) : IsCompl H K
Tactic: refine  ⟨disjoint_iff_inf_le.mpr fun g ⟨p, q⟩ =>  let x : H × K := ⟨⟨g, p⟩, 1⟩  let y : H × K := ⟨1, g, q⟩  Subtype.ext_iff.mp  (Prod.ext_iff.mp (show x = y from h.1 ((mul_one g).trans (one_mul g).symm))).1,  codisjoint_iff_le_sup.mpr fun g _ => ?_⟩
State 1: ⊢ <goal after refine>
Tactic: obtain ⟨⟨h, k⟩, rfl⟩ := h.2 g
State 2: ⊢ <goal after obtain ⟨⟨h, k⟩, rfl⟩ := h.2 g>
Tactic: exact Subgroup.mul_mem_sup h.2 k.2
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.Complement).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.GroupTheory.Complement.IsComplement'`:
State 0: ⊢ theorem IsComplement'.isCompl (h : IsComplement' H K) : IsCompl H K
Tactic: refine  ⟨disjoint_iff_inf_le.mpr fun g ⟨p, q⟩ =>  let x : H × K := ⟨⟨g, p⟩, 1⟩  let y : H × K := ⟨1, g, q⟩  Subtype.ext_iff.mp  (Prod.ext_iff.mp (show x = y from h.1 ((mul_one g).trans (one_mul g).symm))).1,  codisjoint_iff_le_sup.mpr fun g _ => ?_⟩
State 1: ⊢ <goal after refine>
Tactic: obtain ⟨⟨h, k⟩, rfl⟩ := h.2 g
State 2: ⊢ <goal after obtain ⟨⟨h, k⟩, rfl⟩ := h.2 g>
Tactic: exact Subgroup.mul_mem_sup h.2 k.2
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.MvPolynomial.MonomialOrder).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.MvPolynomial.MonomialOrder.degree_sub_leadingTerm_lt_degree`:
State 0: ⊢ theorem degree_sub_leadingTerm_lt_degree {f : MvPolynomial σ R} (h : m.degree f ≠ 0) :
    m.degree (f - m.leadingTerm f) ≺[m] m.degree f
Tactic: simp only [EmbeddingLike.apply_eq_iff_eq] at h'
State 1: ⊢ <goal after simp only [EmbeddingLike.apply_eq_iff_eq] at h'>
Tactic: apply m.degree_mem_support at hl
State 2: ⊢ <goal after apply m.degree_mem_support at hl>
Tactic: rw [h', mem_support_iff] at hl
State 3: ⊢ <goal after rw [h', mem_support_iff] at hl>
Tactic: simp [leadingTerm, leadingCoeff] at hl
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.MvPolynomial.MonomialOrder).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.MvPolynomial.MonomialOrder.degree_sub_leadingTerm_lt_degree`:
State 0: ⊢ theorem degree_sub_leadingTerm_lt_degree {f : MvPolynomial σ R} (h : m.degree f ≠ 0) :
    m.degree (f - m.leadingTerm f) ≺[m] m.degree f
Tactic: simp only [EmbeddingLike.apply_eq_iff_eq] at h'
State 1: ⊢ <goal after simp only [EmbeddingLike.apply_eq_iff_eq] at h'>
Tactic: apply m.degree_mem_support at hl
State 2: ⊢ <goal after apply m.degree_mem_support at hl>
Tactic: rw [h', mem_support_iff] at hl
State 3: ⊢ <goal after rw [h', mem_support_iff] at hl>
Tactic: simp [leadingTerm, leadingCoeff] at hl
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody.convexBodySum_volume_eq_zero_of_le_zero`:
State 0: ⊢ theorem convexBodySum_volume_eq_zero_of_le_zero {B} (hB : B ≤ 0) :
    volume (convexBodySum K B) = 0
Tactic: obtain hB | hB := lt_or_eq_of_le hB  · suffices convexBodySum K B = ∅ by rw [this, measure_empty]
State 1: ⊢ <goal after obtain hB | hB := lt_or_eq_of_le hB>
Tactic: ext x
State 2: ⊢ <goal after ext x>
Tactic: refine ⟨fun hx => ?_, fun h => h.elim⟩
State 3: ⊢ <goal after refine ⟨fun hx => ?_, fun h => h.elim⟩>
Tactic: rw [Set.mem_setOf] at hx
State 4: ⊢ <goal after rw [Set.mem_setOf] at hx>
Tactic: linarith [convexBodySumFun_nonneg x]  · suffices convexBodySum K B = { 0 } by rw [this, measure_singleton]
State 5: ⊢ <goal after linarith [convexBodySumFun_nonneg x]>
Tactic: ext
State 6: ⊢ <goal after ext>
Tactic: rw [convexBodySum, Set.mem_setOf_eq, Set.mem_singleton_iff, hB, ← convexBodySumFun_eq_zero_iff]
State 7: ⊢ <goal after rw [convexBodySum, Set.mem_setOf_eq, Set.mem_singleton_iff, hB, ← convexBodySumFun_eq_zero_iff]>
Tactic: exact (convexBodySumFun_nonneg _).ge_iff_eq'
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody.convexBodySum_volume_eq_zero_of_le_zero`:
State 0: ⊢ theorem convexBodySum_volume_eq_zero_of_le_zero {B} (hB : B ≤ 0) :
    volume (convexBodySum K B) = 0
Tactic: obtain hB | hB := lt_or_eq_of_le hB  · suffices convexBodySum K B = ∅ by rw [this, measure_empty]
State 1: ⊢ <goal after obtain hB | hB := lt_or_eq_of_le hB>
Tactic: ext x
State 2: ⊢ <goal after ext x>
Tactic: refine ⟨fun hx => ?_, fun h => h.elim⟩
State 3: ⊢ <goal after refine ⟨fun hx => ?_, fun h => h.elim⟩>
Tactic: rw [Set.mem_setOf] at hx
State 4: ⊢ <goal after rw [Set.mem_setOf] at hx>
Tactic: linarith [convexBodySumFun_nonneg x]  · suffices convexBodySum K B = { 0 } by rw [this, measure_singleton]
State 5: ⊢ <goal after linarith [convexBodySumFun_nonneg x]>
Tactic: ext
State 6: ⊢ <goal after ext>
Tactic: rw [convexBodySum, Set.mem_setOf_eq, Set.mem_singleton_iff, hB, ← convexBodySumFun_eq_zero_iff]
State 7: ⊢ <goal after rw [convexBodySum, Set.mem_setOf_eq, Set.mem_singleton_iff, hB, ← convexBodySumFun_eq_zero_iff]>
Tactic: exact (convexBodySumFun_nonneg _).ge_iff_eq'
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Convex.Side).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Analysis.Convex.Side.wOppSide_of_left_mem`:
State 0: ⊢ theorem wOppSide_of_left_mem {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s) :
    s.WOppSide x y
Tactic: refine ⟨x, hx, x, hx, ?_⟩
State 1: ⊢ <goal after refine ⟨x, hx, x, hx, ?_⟩>
Tactic: rw [vsub_self]
State 2: ⊢ <goal after rw [vsub_self]>
Tactic: apply SameRay.zero_left
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Convex.Side).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.Analysis.Convex.Side.wOppSide_of_left_mem`:
State 0: ⊢ theorem wOppSide_of_left_mem {s : AffineSubspace R P} {x : P} (y : P) (hx : x ∈ s) :
    s.WOppSide x y
Tactic: refine ⟨x, hx, x, hx, ?_⟩
State 1: ⊢ <goal after refine ⟨x, hx, x, hx, ?_⟩>
Tactic: rw [vsub_self]
State 2: ⊢ <goal after rw [vsub_self]>
Tactic: apply SameRay.zero_left
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.LaurentSeries).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.LaurentSeries.eq_coeff_of_valuation_sub_lt`:
State 0: ⊢ theorem eq_coeff_of_valuation_sub_lt {d n : ℤ} {f g : K⸨X⸩}
    (H : Valued.v (g - f) ≤ exp (-d)) : n < d → g.coeff n = f.coeff n
Tactic: apply eq_of_sub_eq_zero
State 1: ⊢ <goal after apply eq_of_sub_eq_zero>
Tactic: rw [← HahnSeries.coeff_sub]
State 2: ⊢ <goal after rw [← HahnSeries.coeff_sub]>
Tactic: apply coeff_zero_of_lt_valuation K H hn  /- Every Laurent series of valuation less than `(1 : ℤᵐ⁰)` comes from a power series. -/
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.LaurentSeries).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.LaurentSeries.eq_coeff_of_valuation_sub_lt`:
State 0: ⊢ theorem eq_coeff_of_valuation_sub_lt {d n : ℤ} {f g : K⸨X⸩}
    (H : Valued.v (g - f) ≤ exp (-d)) : n < d → g.coeff n = f.coeff n
Tactic: apply eq_of_sub_eq_zero
State 1: ⊢ <goal after apply eq_of_sub_eq_zero>
Tactic: rw [← HahnSeries.coeff_sub]
State 2: ⊢ <goal after rw [← HahnSeries.coeff_sub]>
Tactic: apply coeff_zero_of_lt_valuation K H hn  /- Every Laurent series of valuation less than `(1 : ℤᵐ⁰)` comes from a power series. -/
```

Question: what is the next tactic after the last recorded state, and why?[/INST]