[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.Nat.BinaryRec).

[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.Nat.BinaryRec.binaryRec_eq`:
State 0: ⊢ theorem binaryRec_eq {zero : motive 0} {bit : ∀ b n, motive n → motive (bit b n)}
    (b n) (h : bit false 0 zero = zero ∨ (n = 0 → b = true)) :
    binaryRec zero bit (n.bit b) = bit b n (binaryRec zero bit n)
Tactic: obtain ⟨rfl, rfl⟩ := bit_eq_zero_iff.mp h'
State 1: ⊢ <goal after obtain ⟨rfl, rfl⟩ := bit_eq_zero_iff.mp h'>
Tactic: simp only [Bool.false_eq_true, imp_false, not_true_eq_false, or_false] at h  unfold binaryRec
State 2: ⊢ <goal after simp only [Bool.false_eq_true, imp_false, not_true_eq_false, or_false] at h>
Tactic: exact h.symm  case neg =>
State 3: ⊢ <goal after exact h.symm>
Tactic: rw [binaryRec, dif_neg h']  change congrArg motive (n.bit b).bit_testBit_zero_shiftRight_one ▸ bit _ _ _ = _  generalize congrArg motive (n.bit b).bit_testBit_zero_shiftRight_one = e; revert e
State 4: ⊢ <goal after rw [binaryRec, dif_neg h']>
Tactic: rw [testBit_bit_zero, bit_shiftRight_one]  intros; rfl
```

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.Nat.BinaryRec).[/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.Nat.BinaryRec.binaryRec_eq`:
State 0: ⊢ theorem binaryRec_eq {zero : motive 0} {bit : ∀ b n, motive n → motive (bit b n)}
    (b n) (h : bit false 0 zero = zero ∨ (n = 0 → b = true)) :
    binaryRec zero bit (n.bit b) = bit b n (binaryRec zero bit n)
Tactic: obtain ⟨rfl, rfl⟩ := bit_eq_zero_iff.mp h'
State 1: ⊢ <goal after obtain ⟨rfl, rfl⟩ := bit_eq_zero_iff.mp h'>
Tactic: simp only [Bool.false_eq_true, imp_false, not_true_eq_false, or_false] at h  unfold binaryRec
State 2: ⊢ <goal after simp only [Bool.false_eq_true, imp_false, not_true_eq_false, or_false] at h>
Tactic: exact h.symm  case neg =>
State 3: ⊢ <goal after exact h.symm>
Tactic: rw [binaryRec, dif_neg h']  change congrArg motive (n.bit b).bit_testBit_zero_shiftRight_one ▸ bit _ _ _ = _  generalize congrArg motive (n.bit b).bit_testBit_zero_shiftRight_one = e; revert e
State 4: ⊢ <goal after rw [binaryRec, dif_neg h']>
Tactic: rw [testBit_bit_zero, bit_shiftRight_one]  intros; rfl
```

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.Circulant).

[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.Circulant.circulantGraph_eq_symm`:
State 0: ⊢ theorem circulantGraph_eq_symm : circulantGraph s = circulantGraph (s ∪ (-s))
Tactic: ext
State 1: ⊢ <goal after ext>
Tactic: simp only [circulantGraph_adj, Set.mem_union, Set.mem_neg, neg_sub]  grind
```

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.Circulant).[/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.Circulant.circulantGraph_eq_symm`:
State 0: ⊢ theorem circulantGraph_eq_symm : circulantGraph s = circulantGraph (s ∪ (-s))
Tactic: ext
State 1: ⊢ <goal after ext>
Tactic: simp only [circulantGraph_adj, Set.mem_union, Set.mem_neg, neg_sub]  grind
```

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.LinearAlgebra.RootSystem.Finite.Nondegenerate).

[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.RootSystem.Finite.Nondegenerate.ker_rootForm_eq_dualAnnihilator`:
State 0: ⊢ lemma ker_rootForm_eq_dualAnnihilator :
    P.RootForm.ker =
      (P.corootSpan R).dualAnnihilator.map (P.toPerfPair.symm : Dual R N →ₗ[R] M)
Tactic: have : IsReflexive R M := .of_isPerfPair P.toLinearMap
State 1: ⊢ <goal after have : IsReflexive R M := .of_isPerfPair P.toLinearMap>
Tactic: have : IsReflexive R N := .of_isPerfPair P.flip.toLinearMap  suffices finrank R (LinearMap.ker P.RootForm) = finrank R (P.corootSpan R).dualAnnihilator by
State 2: ⊢ <goal after have : IsReflexive R N := .of_isPerfPair P.flip.toLinearMap>
Tactic: refine (Submodule.eq_of_le_of_finrank_eq P.corootSpan_dualAnnihilator_le_ker_rootForm ?_).symm
State 3: ⊢ <goal after refine (Submodule.eq_of_le_of_finrank_eq P.corootSpan_dualAnnihilator_le_ker_rootForm ?_).symm>
Tactic: rw [this]
State 4: ⊢ <goal after rw [this]>
Tactic: apply LinearEquiv.finrank_map_eq
State 5: ⊢ <goal after apply LinearEquiv.finrank_map_eq>
Tactic: have aux0 := Subspace.finrank_add_finrank_dualAnnihilator_eq (P.corootSpan R)
State 6: ⊢ <goal after have aux0 := Subspace.finrank_add_finrank_dualAnnihilator_eq (P.corootSpan R)>
Tactic: have aux1 := Submodule.finrank_add_eq_of_isCompl P.isCompl_rootSpan_ker_rootForm
State 7: ⊢ <goal after have aux1 := Submodule.finrank_add_eq_of_isCompl P.isCompl_rootSpan_ker_rootForm>
Tactic: rw [← P.finrank_corootSpan_eq', P.toPerfPair.finrank_eq, Subspace.dual_finrank_eq] at aux1  lia
```

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.RootSystem.Finite.Nondegenerate).[/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.RootSystem.Finite.Nondegenerate.ker_rootForm_eq_dualAnnihilator`:
State 0: ⊢ lemma ker_rootForm_eq_dualAnnihilator :
    P.RootForm.ker =
      (P.corootSpan R).dualAnnihilator.map (P.toPerfPair.symm : Dual R N →ₗ[R] M)
Tactic: have : IsReflexive R M := .of_isPerfPair P.toLinearMap
State 1: ⊢ <goal after have : IsReflexive R M := .of_isPerfPair P.toLinearMap>
Tactic: have : IsReflexive R N := .of_isPerfPair P.flip.toLinearMap  suffices finrank R (LinearMap.ker P.RootForm) = finrank R (P.corootSpan R).dualAnnihilator by
State 2: ⊢ <goal after have : IsReflexive R N := .of_isPerfPair P.flip.toLinearMap>
Tactic: refine (Submodule.eq_of_le_of_finrank_eq P.corootSpan_dualAnnihilator_le_ker_rootForm ?_).symm
State 3: ⊢ <goal after refine (Submodule.eq_of_le_of_finrank_eq P.corootSpan_dualAnnihilator_le_ker_rootForm ?_).symm>
Tactic: rw [this]
State 4: ⊢ <goal after rw [this]>
Tactic: apply LinearEquiv.finrank_map_eq
State 5: ⊢ <goal after apply LinearEquiv.finrank_map_eq>
Tactic: have aux0 := Subspace.finrank_add_finrank_dualAnnihilator_eq (P.corootSpan R)
State 6: ⊢ <goal after have aux0 := Subspace.finrank_add_finrank_dualAnnihilator_eq (P.corootSpan R)>
Tactic: have aux1 := Submodule.finrank_add_eq_of_isCompl P.isCompl_rootSpan_ker_rootForm
State 7: ⊢ <goal after have aux1 := Submodule.finrank_add_eq_of_isCompl P.isCompl_rootSpan_ker_rootForm>
Tactic: rw [← P.finrank_corootSpan_eq', P.toPerfPair.finrank_eq, Subspace.dual_finrank_eq] at aux1  lia
```

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.CMField).

[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.CMField.ringOfIntegersComplexConj_eq_self_iff`:
State 0: ⊢ theorem ringOfIntegersComplexConj_eq_self_iff (x : 𝓞 K) :
    ringOfIntegersComplexConj K x = x ↔ x ∈ Set.range (algebraMap (𝓞 K⁺) (𝓞 K))
Tactic: refine ⟨fun h ↦ ?_, ?_⟩  · rw [RingOfIntegers.ext_iff, coe_ringOfIntegersComplexConj,  RingOfIntegers.complexConj_eq_self_iff] at h
State 1: ⊢ <goal after refine ⟨fun h ↦ ?_, ?_⟩>
Tactic: obtain ⟨y, hy⟩ := h
State 2: ⊢ <goal after obtain ⟨y, hy⟩ := h>
Tactic: exact ⟨y, RingOfIntegers.ext_iff.mpr hy⟩  · rintro ⟨y, rfl⟩
State 3: ⊢ <goal after exact ⟨y, RingOfIntegers.ext_iff.mpr hy⟩>
Tactic: simp  end complexConj  section units  open Units  /--  The complex conjugation as an isomorphism of the units of `K`. -/  noncomputable abbrev unitsComplexConj [Algebra.IsIntegral ℚ K] : (𝓞 K)ˣ ≃* (𝓞 K)ˣ :=  Units.mapEquiv <| RingOfIntegers.mapRingEquiv (complexConj K).toRingEquiv  /--  The subgroup of `(𝓞 K)ˣ` generated by the units of `K⁺`. These units are exactly the units fixed  by the complex conjugation, see `IsCMField.unitsComplexConj_eq_self_iff`.  -/
```

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.CMField).[/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.CMField.ringOfIntegersComplexConj_eq_self_iff`:
State 0: ⊢ theorem ringOfIntegersComplexConj_eq_self_iff (x : 𝓞 K) :
    ringOfIntegersComplexConj K x = x ↔ x ∈ Set.range (algebraMap (𝓞 K⁺) (𝓞 K))
Tactic: refine ⟨fun h ↦ ?_, ?_⟩  · rw [RingOfIntegers.ext_iff, coe_ringOfIntegersComplexConj,  RingOfIntegers.complexConj_eq_self_iff] at h
State 1: ⊢ <goal after refine ⟨fun h ↦ ?_, ?_⟩>
Tactic: obtain ⟨y, hy⟩ := h
State 2: ⊢ <goal after obtain ⟨y, hy⟩ := h>
Tactic: exact ⟨y, RingOfIntegers.ext_iff.mpr hy⟩  · rintro ⟨y, rfl⟩
State 3: ⊢ <goal after exact ⟨y, RingOfIntegers.ext_iff.mpr hy⟩>
Tactic: simp  end complexConj  section units  open Units  /--  The complex conjugation as an isomorphism of the units of `K`. -/  noncomputable abbrev unitsComplexConj [Algebra.IsIntegral ℚ K] : (𝓞 K)ˣ ≃* (𝓞 K)ˣ :=  Units.mapEquiv <| RingOfIntegers.mapRingEquiv (complexConj K).toRingEquiv  /--  The subgroup of `(𝓞 K)ˣ` generated by the units of `K⁺`. These units are exactly the units fixed  by the complex conjugation, see `IsCMField.unitsComplexConj_eq_self_iff`.  -/
```

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.Set.Lattice.Image).

[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.Set.Lattice.Image.image2_sInter_subset_left`:
State 0: ⊢ theorem image2_sInter_subset_left (S : Set (Set α)) (t : Set β) :
    image2 f (⋂₀ S) t ⊆ ⋂ s ∈ S, image2 f s t
Tactic: rw [sInter_eq_biInter]
State 1: ⊢ <goal after rw [sInter_eq_biInter]>
Tactic: exact image2_iInter₂_subset_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.Data.Set.Lattice.Image).[/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.Set.Lattice.Image.image2_sInter_subset_left`:
State 0: ⊢ theorem image2_sInter_subset_left (S : Set (Set α)) (t : Set β) :
    image2 f (⋂₀ S) t ⊆ ⋂ s ∈ S, image2 f s t
Tactic: rw [sInter_eq_biInter]
State 1: ⊢ <goal after rw [sInter_eq_biInter]>
Tactic: exact image2_iInter₂_subset_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.Topology.Separation.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.Topology.Separation.Basic.singleton_mem_nhdsWithin_of_mem_discrete`:
State 0: ⊢ theorem singleton_mem_nhdsWithin_of_mem_discrete {s : Set X} (hs : IsDiscrete s) {x : X}
    (hx : x ∈ s) : {x} ∈ 𝓝[s] x
Tactic: rw [isDiscrete_iff_discreteTopology] at hs
State 1: ⊢ <goal after rw [isDiscrete_iff_discreteTopology] at hs>
Tactic: have : ({⟨x, hx⟩} : Set s) ∈ 𝓝 (⟨x, hx⟩ : s) := by simp [nhds_discrete]
State 2: ⊢ <goal after have : ({⟨x, hx⟩} : Set s) ∈ 𝓝 (⟨x, hx⟩ : s) := by simp [nhds_discrete]>
Tactic: simpa only [nhdsWithin_eq_map_subtype_coe hx, image_singleton] using  @image_mem_map _ _ _ ((↑) : s → X) _ this  /-- The neighbourhoods filter of `x` within `s`, under the discrete topology, is equal to  the pure `x` filter (which is the principal filter at the 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.Separation.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.Topology.Separation.Basic.singleton_mem_nhdsWithin_of_mem_discrete`:
State 0: ⊢ theorem singleton_mem_nhdsWithin_of_mem_discrete {s : Set X} (hs : IsDiscrete s) {x : X}
    (hx : x ∈ s) : {x} ∈ 𝓝[s] x
Tactic: rw [isDiscrete_iff_discreteTopology] at hs
State 1: ⊢ <goal after rw [isDiscrete_iff_discreteTopology] at hs>
Tactic: have : ({⟨x, hx⟩} : Set s) ∈ 𝓝 (⟨x, hx⟩ : s) := by simp [nhds_discrete]
State 2: ⊢ <goal after have : ({⟨x, hx⟩} : Set s) ∈ 𝓝 (⟨x, hx⟩ : s) := by simp [nhds_discrete]>
Tactic: simpa only [nhdsWithin_eq_map_subtype_coe hx, image_singleton] using  @image_mem_map _ _ _ ((↑) : s → X) _ this  /-- The neighbourhoods filter of `x` within `s`, under the discrete topology, is equal to  the pure `x` filter (which is the principal filter at the 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.RingTheory.Extension.Generators).

[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.Extension.Generators.map_toComp_ker`:
State 0: ⊢ lemma map_toComp_ker (Q : Generators S T ι') (P : Generators R S ι) :
    P.ker.map (Q.toComp P).toAlgHom = RingHom.ker (Q.ofComp P).toAlgHom
Tactic: apply le_antisymm  · rw [Ideal.map_le_iff_le_comap]  rintro x (hx : algebraMap P.Ring S x = 0)
State 1: ⊢ <goal after apply le_antisymm>
Tactic: have : (Q.ofComp P).toAlgHom.comp (Q.toComp P).toAlgHom = IsScalarTower.toAlgHom R _ _ := by  ext1; simp
State 2: ⊢ <goal after have : (Q.ofComp P).toAlgHom.comp (Q.toComp P).toAlgHom = IsScalarTower.toAlgHom R _ _ := by>
Tactic: simp only [Ideal.mem_comap,  RingHom.mem_ker, ← AlgHom.comp_apply, this, IsScalarTower.toAlgHom_apply]
State 3: ⊢ <goal after simp only [Ideal.mem_comap,>
Tactic: rw [IsScalarTower.algebraMap_apply P.Ring S, hx, map_zero]  · rintro x (h₂ : (Q.ofComp P).toAlgHom x = 0)  let e : (ι' ⊕ ι →₀ ℕ) ≃+ (ι' →₀ ℕ) × (ι →₀ ℕ) :=  Finsupp.sumFinsuppAddEquivProdFinsupp  suffices ∑ v ∈ (support x).map e, (monomial (e.symm v)) (coeff (e.symm v) x) ∈  Ideal.map (Q.toComp P).toAlgHom.toRingHom P.ker by
State 4: ⊢ <goal after rw [IsScalarTower.algebraMap_apply P.Ring S, hx, map_zero]>
Tactic: simpa only [AlgHom.toRingHom_eq_coe, Finset.sum_map, Equiv.coe_toEmbedding,  EquivLike.coe_coe, AddEquiv.symm_apply_apply, support_sum_monomial_coeff] using this
State 5: ⊢ <goal after simpa only [AlgHom.toRingHom_eq_coe, Finset.sum_map, Equiv.coe_toEmbedding,>
Tactic: rw [← Finset.sum_fiberwise_of_maps_to (fun i ↦ Finset.mem_image_of_mem Prod.fst)]
State 6: ⊢ <goal after rw [← Finset.sum_fiberwise_of_maps_to (fun i ↦ Finset.mem_image_of_mem Prod.fst)]>
Tactic: refine sum_mem fun i hi ↦ ?_  convert_to monomial (e.symm (i, 0)) 1 * (Q.toComp P).toAlgHom.toRingHom  (∑ j ∈ (support x).map e.toEmbedding with j.1 = i, monomial j.2 (coeff (e.symm j) x)) ∈ _  · rw [map_sum, Finset.mul_sum]
State 7: ⊢ <goal after refine sum_mem fun i hi ↦ ?_>
Tactic: refine Finset.sum_congr rfl fun j hj ↦ ?_
State 8: ⊢ <goal after refine Finset.sum_congr rfl fun j hj ↦ ?_>
Tactic: obtain rfl := (Finset.mem_filter.mp hj).2
State 9: ⊢ <goal after obtain rfl := (Finset.mem_filter.mp hj).2>
Tactic: obtain ⟨i, j⟩ := j  clear hj hi
State 10: ⊢ <goal after obtain ⟨i, j⟩ := j>
Tactic: have : (Q.toComp P).toAlgHom (monomial j (coeff (e.symm (i, j)) x)) =  monomial (e.symm (0, j)) (coeff (e.symm (i, j)) x) :=  toComp_toAlgHom_monomial ..
State 11: ⊢ <goal after have : (Q.toComp P).toAlgHom (monomial j (coeff (e.symm (i, j)) x)) =>
Tactic: simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe,  this]
State 12: ⊢ <goal after simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe,>
Tactic: rw [monomial_mul, ← map_add, Prod.mk_add_mk, add_zero, zero_add, one_mul]  · apply Ideal.mul_mem_left
State 13: ⊢ <goal after rw [monomial_mul, ← map_add, Prod.mk_add_mk, add_zero, zero_add, one_mul]>
Tactic: refine Ideal.mem_map_of_mem _ ?_
State 14: ⊢ <goal after refine Ideal.mem_map_of_mem _ ?_>
Tactic: simp only [ker_eq_ker_aeval_val, AddEquiv.toEquiv_eq_coe, RingHom.mem_ker, map_sum]
State 15: ⊢ <goal after simp only [ker_eq_ker_aeval_val, AddEquiv.toEquiv_eq_coe, RingHom.mem_ker, map_sum]>
Tactic: rw [← coeff_zero i, ← h₂]  clear h₂ hi
State 16: ⊢ <goal after rw [← coeff_zero i, ← h₂]>
Tactic: have (x : (Q.comp P).Ring) : (Function.support fun a ↦ if a.1 = i then aeval P.val  (monomial a.2 (coeff (e.symm a) x)) else 0) ⊆ SetLike.coe ((support x).map e) := by
State 17: ⊢ <goal after have (x : (Q.comp P).Ring) : (Function.support fun a ↦ if a.1 = i then aeval P.val>
Tactic: rw [← Set.compl_subset_compl]
State 18: ⊢ <goal after rw [← Set.compl_subset_compl]>
Tactic: intro j
State 19: ⊢ <goal after intro j>
Tactic: obtain ⟨j, rfl⟩ := e.surjective j
State 20: ⊢ <goal after obtain ⟨j, rfl⟩ := e.surjective j>
Tactic: simp_all
State 21: ⊢ <goal after simp_all>
Tactic: rw [Finset.sum_filter, ← finsum_eq_sum_of_support_subset _ (this x)]
State 22: ⊢ <goal after rw [Finset.sum_filter, ← finsum_eq_sum_of_support_subset _ (this x)]>
Tactic: induction x using MvPolynomial.induction_on' with  | monomial v a =>
State 23: ⊢ <goal after induction x using MvPolynomial.induction_on' with>
Tactic: rw [finsum_eq_sum_of_support_subset _ (this _), ← Finset.sum_filter]
State 24: ⊢ <goal after rw [finsum_eq_sum_of_support_subset _ (this _), ← Finset.sum_filter]>
Tactic: obtain ⟨v, rfl⟩ := e.symm.surjective v  -- Rewrite `e` in the right-hand side only.  conv_rhs => simp only [e, Finsupp.sumFinsuppAddEquivProdFinsupp,  Finsupp.sumFinsuppEquivProdFinsupp, AddEquiv.symm_mk, AddEquiv.coe_mk,  Equiv.coe_fn_symm_mk, ofComp_toAlgHom_monomial_sumElim]  classical
State 25: ⊢ <goal after obtain ⟨v, rfl⟩ := e.symm.surjective v>
Tactic: simp only [coeff_monomial, ← e.injective.eq_iff,  map_zero, AddEquiv.apply_symm_apply, apply_ite]
State 26: ⊢ <goal after simp only [coeff_monomial, ← e.injective.eq_iff,>
Tactic: rw [← apply_ite, Finset.sum_ite_eq]
State 27: ⊢ <goal after rw [← apply_ite, Finset.sum_ite_eq]>
Tactic: simp only [Finset.mem_filter, Finset.mem_map_equiv, AddEquiv.coe_toEquiv_symm,  mem_support_iff, coeff_monomial, ↓reduceIte, ne_eq, ite_and, ite_not]
State 28: ⊢ <goal after simp only [Finset.mem_filter, Finset.mem_map_equiv, AddEquiv.coe_toEquiv_symm,>
Tactic: split  · simp only [*, map_zero, ite_self]  · congr  | add p q hp hq =>
State 29: ⊢ <goal after split>
Tactic: simp only [coeff_add, map_add, ite_add_zero]
State 30: ⊢ <goal after simp only [coeff_add, map_add, ite_add_zero]>
Tactic: rw [finsum_add_distrib, hp, hq]  · refine (((support p).map e).finite_toSet.subset ?_)
State 31: ⊢ <goal after rw [finsum_add_distrib, hp, hq]>
Tactic: convert this p  · refine (((support q).map e).finite_toSet.subset ?_)
State 32: ⊢ <goal after convert this p>
Tactic: convert this q  /--  Given `R[X] → S` and `S[Y] → T`, this is the lift of an element in `ker(S[Y] → T)`  to `ker(R[X][Y] → S[Y] → T)` constructed from `P.σ`.  -/  noncomputable
```

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.Extension.Generators).[/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.Extension.Generators.map_toComp_ker`:
State 0: ⊢ lemma map_toComp_ker (Q : Generators S T ι') (P : Generators R S ι) :
    P.ker.map (Q.toComp P).toAlgHom = RingHom.ker (Q.ofComp P).toAlgHom
Tactic: apply le_antisymm  · rw [Ideal.map_le_iff_le_comap]  rintro x (hx : algebraMap P.Ring S x = 0)
State 1: ⊢ <goal after apply le_antisymm>
Tactic: have : (Q.ofComp P).toAlgHom.comp (Q.toComp P).toAlgHom = IsScalarTower.toAlgHom R _ _ := by  ext1; simp
State 2: ⊢ <goal after have : (Q.ofComp P).toAlgHom.comp (Q.toComp P).toAlgHom = IsScalarTower.toAlgHom R _ _ := by>
Tactic: simp only [Ideal.mem_comap,  RingHom.mem_ker, ← AlgHom.comp_apply, this, IsScalarTower.toAlgHom_apply]
State 3: ⊢ <goal after simp only [Ideal.mem_comap,>
Tactic: rw [IsScalarTower.algebraMap_apply P.Ring S, hx, map_zero]  · rintro x (h₂ : (Q.ofComp P).toAlgHom x = 0)  let e : (ι' ⊕ ι →₀ ℕ) ≃+ (ι' →₀ ℕ) × (ι →₀ ℕ) :=  Finsupp.sumFinsuppAddEquivProdFinsupp  suffices ∑ v ∈ (support x).map e, (monomial (e.symm v)) (coeff (e.symm v) x) ∈  Ideal.map (Q.toComp P).toAlgHom.toRingHom P.ker by
State 4: ⊢ <goal after rw [IsScalarTower.algebraMap_apply P.Ring S, hx, map_zero]>
Tactic: simpa only [AlgHom.toRingHom_eq_coe, Finset.sum_map, Equiv.coe_toEmbedding,  EquivLike.coe_coe, AddEquiv.symm_apply_apply, support_sum_monomial_coeff] using this
State 5: ⊢ <goal after simpa only [AlgHom.toRingHom_eq_coe, Finset.sum_map, Equiv.coe_toEmbedding,>
Tactic: rw [← Finset.sum_fiberwise_of_maps_to (fun i ↦ Finset.mem_image_of_mem Prod.fst)]
State 6: ⊢ <goal after rw [← Finset.sum_fiberwise_of_maps_to (fun i ↦ Finset.mem_image_of_mem Prod.fst)]>
Tactic: refine sum_mem fun i hi ↦ ?_  convert_to monomial (e.symm (i, 0)) 1 * (Q.toComp P).toAlgHom.toRingHom  (∑ j ∈ (support x).map e.toEmbedding with j.1 = i, monomial j.2 (coeff (e.symm j) x)) ∈ _  · rw [map_sum, Finset.mul_sum]
State 7: ⊢ <goal after refine sum_mem fun i hi ↦ ?_>
Tactic: refine Finset.sum_congr rfl fun j hj ↦ ?_
State 8: ⊢ <goal after refine Finset.sum_congr rfl fun j hj ↦ ?_>
Tactic: obtain rfl := (Finset.mem_filter.mp hj).2
State 9: ⊢ <goal after obtain rfl := (Finset.mem_filter.mp hj).2>
Tactic: obtain ⟨i, j⟩ := j  clear hj hi
State 10: ⊢ <goal after obtain ⟨i, j⟩ := j>
Tactic: have : (Q.toComp P).toAlgHom (monomial j (coeff (e.symm (i, j)) x)) =  monomial (e.symm (0, j)) (coeff (e.symm (i, j)) x) :=  toComp_toAlgHom_monomial ..
State 11: ⊢ <goal after have : (Q.toComp P).toAlgHom (monomial j (coeff (e.symm (i, j)) x)) =>
Tactic: simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe,  this]
State 12: ⊢ <goal after simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe,>
Tactic: rw [monomial_mul, ← map_add, Prod.mk_add_mk, add_zero, zero_add, one_mul]  · apply Ideal.mul_mem_left
State 13: ⊢ <goal after rw [monomial_mul, ← map_add, Prod.mk_add_mk, add_zero, zero_add, one_mul]>
Tactic: refine Ideal.mem_map_of_mem _ ?_
State 14: ⊢ <goal after refine Ideal.mem_map_of_mem _ ?_>
Tactic: simp only [ker_eq_ker_aeval_val, AddEquiv.toEquiv_eq_coe, RingHom.mem_ker, map_sum]
State 15: ⊢ <goal after simp only [ker_eq_ker_aeval_val, AddEquiv.toEquiv_eq_coe, RingHom.mem_ker, map_sum]>
Tactic: rw [← coeff_zero i, ← h₂]  clear h₂ hi
State 16: ⊢ <goal after rw [← coeff_zero i, ← h₂]>
Tactic: have (x : (Q.comp P).Ring) : (Function.support fun a ↦ if a.1 = i then aeval P.val  (monomial a.2 (coeff (e.symm a) x)) else 0) ⊆ SetLike.coe ((support x).map e) := by
State 17: ⊢ <goal after have (x : (Q.comp P).Ring) : (Function.support fun a ↦ if a.1 = i then aeval P.val>
Tactic: rw [← Set.compl_subset_compl]
State 18: ⊢ <goal after rw [← Set.compl_subset_compl]>
Tactic: intro j
State 19: ⊢ <goal after intro j>
Tactic: obtain ⟨j, rfl⟩ := e.surjective j
State 20: ⊢ <goal after obtain ⟨j, rfl⟩ := e.surjective j>
Tactic: simp_all
State 21: ⊢ <goal after simp_all>
Tactic: rw [Finset.sum_filter, ← finsum_eq_sum_of_support_subset _ (this x)]
State 22: ⊢ <goal after rw [Finset.sum_filter, ← finsum_eq_sum_of_support_subset _ (this x)]>
Tactic: induction x using MvPolynomial.induction_on' with  | monomial v a =>
State 23: ⊢ <goal after induction x using MvPolynomial.induction_on' with>
Tactic: rw [finsum_eq_sum_of_support_subset _ (this _), ← Finset.sum_filter]
State 24: ⊢ <goal after rw [finsum_eq_sum_of_support_subset _ (this _), ← Finset.sum_filter]>
Tactic: obtain ⟨v, rfl⟩ := e.symm.surjective v  -- Rewrite `e` in the right-hand side only.  conv_rhs => simp only [e, Finsupp.sumFinsuppAddEquivProdFinsupp,  Finsupp.sumFinsuppEquivProdFinsupp, AddEquiv.symm_mk, AddEquiv.coe_mk,  Equiv.coe_fn_symm_mk, ofComp_toAlgHom_monomial_sumElim]  classical
State 25: ⊢ <goal after obtain ⟨v, rfl⟩ := e.symm.surjective v>
Tactic: simp only [coeff_monomial, ← e.injective.eq_iff,  map_zero, AddEquiv.apply_symm_apply, apply_ite]
State 26: ⊢ <goal after simp only [coeff_monomial, ← e.injective.eq_iff,>
Tactic: rw [← apply_ite, Finset.sum_ite_eq]
State 27: ⊢ <goal after rw [← apply_ite, Finset.sum_ite_eq]>
Tactic: simp only [Finset.mem_filter, Finset.mem_map_equiv, AddEquiv.coe_toEquiv_symm,  mem_support_iff, coeff_monomial, ↓reduceIte, ne_eq, ite_and, ite_not]
State 28: ⊢ <goal after simp only [Finset.mem_filter, Finset.mem_map_equiv, AddEquiv.coe_toEquiv_symm,>
Tactic: split  · simp only [*, map_zero, ite_self]  · congr  | add p q hp hq =>
State 29: ⊢ <goal after split>
Tactic: simp only [coeff_add, map_add, ite_add_zero]
State 30: ⊢ <goal after simp only [coeff_add, map_add, ite_add_zero]>
Tactic: rw [finsum_add_distrib, hp, hq]  · refine (((support p).map e).finite_toSet.subset ?_)
State 31: ⊢ <goal after rw [finsum_add_distrib, hp, hq]>
Tactic: convert this p  · refine (((support q).map e).finite_toSet.subset ?_)
State 32: ⊢ <goal after convert this p>
Tactic: convert this q  /--  Given `R[X] → S` and `S[Y] → T`, this is the lift of an element in `ker(S[Y] → T)`  to `ker(R[X][Y] → S[Y] → T)` constructed from `P.σ`.  -/  noncomputable
```

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.Cyclotomic.PrimitiveRoots).

[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.Cyclotomic.PrimitiveRoots.norm_pow_sub_one_two`:
State 0: ⊢ theorem norm_pow_sub_one_two {k : ℕ} (hζ : IsPrimitiveRoot ζ (2 ^ (k + 1)))
    [IsCyclotomicExtension {2 ^ (k + 1)} K L]
    (hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)) :
    norm K (ζ ^ 2 ^ k - 1) = (-2 : K) ^ 2 ^ k
Tactic: have := hζ.pow_of_dvd  (fun h => two_ne_zero (eq_zero_of_pow_eq_zero h)) (pow_dvd_pow 2 (le_succ k))
State 1: ⊢ <goal after have := hζ.pow_of_dvd>
Tactic: rw [Nat.pow_div (le_succ k) zero_lt_two, Nat.succ_sub (le_refl k), Nat.sub_self, pow_one] at this
State 2: ⊢ <goal after rw [Nat.pow_div (le_succ k) zero_lt_two, Nat.succ_sub (le_refl k), Nat.sub_self, pow_one] at this>
Tactic: have H : (-1 : L) - (1 : L) = algebraMap K L (-2) := by
State 3: ⊢ <goal after have H : (-1 : L) - (1 : L) = algebraMap K L (-2) := by>
Tactic: simp only [map_neg, map_ofNat]
State 4: ⊢ <goal after simp only [map_neg, map_ofNat]>
Tactic: ring  replace hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K) := by simp [hirr]
State 5: ⊢ <goal after ring>
Tactic: rw [this.eq_neg_one_of_two_right, H, Algebra.norm_algebraMap,  IsCyclotomicExtension.finrank L hirr, totient_prime_pow Nat.prime_two (zero_lt_succ k),  succ_sub_succ_eq_sub, tsub_zero]
State 6: ⊢ <goal after rw [this.eq_neg_one_of_two_right, H, Algebra.norm_algebraMap,>
Tactic: simp  /-- If `Irreducible (cyclotomic (2 ^ k) K)` (in particular for `K = ℚ`) and `k` is at least `2`,  then the norm of `ζ - 1` is `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.NumberTheory.Cyclotomic.PrimitiveRoots).[/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.Cyclotomic.PrimitiveRoots.norm_pow_sub_one_two`:
State 0: ⊢ theorem norm_pow_sub_one_two {k : ℕ} (hζ : IsPrimitiveRoot ζ (2 ^ (k + 1)))
    [IsCyclotomicExtension {2 ^ (k + 1)} K L]
    (hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)) :
    norm K (ζ ^ 2 ^ k - 1) = (-2 : K) ^ 2 ^ k
Tactic: have := hζ.pow_of_dvd  (fun h => two_ne_zero (eq_zero_of_pow_eq_zero h)) (pow_dvd_pow 2 (le_succ k))
State 1: ⊢ <goal after have := hζ.pow_of_dvd>
Tactic: rw [Nat.pow_div (le_succ k) zero_lt_two, Nat.succ_sub (le_refl k), Nat.sub_self, pow_one] at this
State 2: ⊢ <goal after rw [Nat.pow_div (le_succ k) zero_lt_two, Nat.succ_sub (le_refl k), Nat.sub_self, pow_one] at this>
Tactic: have H : (-1 : L) - (1 : L) = algebraMap K L (-2) := by
State 3: ⊢ <goal after have H : (-1 : L) - (1 : L) = algebraMap K L (-2) := by>
Tactic: simp only [map_neg, map_ofNat]
State 4: ⊢ <goal after simp only [map_neg, map_ofNat]>
Tactic: ring  replace hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K) := by simp [hirr]
State 5: ⊢ <goal after ring>
Tactic: rw [this.eq_neg_one_of_two_right, H, Algebra.norm_algebraMap,  IsCyclotomicExtension.finrank L hirr, totient_prime_pow Nat.prime_two (zero_lt_succ k),  succ_sub_succ_eq_sub, tsub_zero]
State 6: ⊢ <goal after rw [this.eq_neg_one_of_two_right, H, Algebra.norm_algebraMap,>
Tactic: simp  /-- If `Irreducible (cyclotomic (2 ^ k) K)` (in particular for `K = ℚ`) and `k` is at least `2`,  then the norm of `ζ - 1` is `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.AdicCompletion.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.RingTheory.AdicCompletion.Basic.bot`:
State 0: ⊢ instance bot : IsPrecomplete (⊥ : Ideal R) M
Tactic: refine ⟨fun f hf => ⟨f 1, fun n => ?_⟩⟩  rcases n with - | n  · rw [pow_zero, Ideal.one_eq_top, top_smul]
State 1: ⊢ <goal after refine ⟨fun f hf => ⟨f 1, fun n => ?_⟩⟩>
Tactic: exact SModEq.top  specialize hf (Nat.le_add_left 1 n)
State 2: ⊢ <goal after exact SModEq.top>
Tactic: rw [pow_one, bot_smul, SModEq.bot] at hf; rw [hf]
```

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.AdicCompletion.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.RingTheory.AdicCompletion.Basic.bot`:
State 0: ⊢ instance bot : IsPrecomplete (⊥ : Ideal R) M
Tactic: refine ⟨fun f hf => ⟨f 1, fun n => ?_⟩⟩  rcases n with - | n  · rw [pow_zero, Ideal.one_eq_top, top_smul]
State 1: ⊢ <goal after refine ⟨fun f hf => ⟨f 1, fun n => ?_⟩⟩>
Tactic: exact SModEq.top  specialize hf (Nat.le_add_left 1 n)
State 2: ⊢ <goal after exact SModEq.top>
Tactic: rw [pow_one, bot_smul, SModEq.bot] at hf; rw [hf]
```

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.Homology.DerivedCategory.Ext.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.Algebra.Homology.DerivedCategory.Ext.Basic.hasExt_iff_small_ext`:
State 0: ⊢ lemma hasExt_iff_small_ext :
    HasExt.{w'} C ↔ ∀ (X Y : C) (n : ℕ), Small.{w'} (Ext.{w} X Y n)
Tactic: simp only [hasExt_iff, small_congr Ext.homEquiv]
State 1: ⊢ <goal after simp only [hasExt_iff, small_congr Ext.homEquiv]>
Tactic: constructor  · intro h X Y n
State 2: ⊢ <goal after constructor>
Tactic: exact h X Y n (by simp)  · intro h X Y n hn  lift n to ℕ using hn
State 3: ⊢ <goal after exact h X Y n (by simp)>
Tactic: exact h X Y n  end CategoryTheory
```

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.Homology.DerivedCategory.Ext.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.Algebra.Homology.DerivedCategory.Ext.Basic.hasExt_iff_small_ext`:
State 0: ⊢ lemma hasExt_iff_small_ext :
    HasExt.{w'} C ↔ ∀ (X Y : C) (n : ℕ), Small.{w'} (Ext.{w} X Y n)
Tactic: simp only [hasExt_iff, small_congr Ext.homEquiv]
State 1: ⊢ <goal after simp only [hasExt_iff, small_congr Ext.homEquiv]>
Tactic: constructor  · intro h X Y n
State 2: ⊢ <goal after constructor>
Tactic: exact h X Y n (by simp)  · intro h X Y n hn  lift n to ℕ using hn
State 3: ⊢ <goal after exact h X Y n (by simp)>
Tactic: exact h X Y n  end CategoryTheory
```

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.Cyclotomic.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.NumberField.Cyclotomic.Basic.finite_quotient_span_sub_one`:
State 0: ⊢ theorem finite_quotient_span_sub_one [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ (p ^ (k + 1))) :
    Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1})
Tactic: have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
State 1: ⊢ <goal after have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K>
Tactic: refine Ideal.finiteQuotientOfFreeOfNeBot _ (fun h ↦ ?_)
State 2: ⊢ <goal after refine Ideal.finiteQuotientOfFreeOfNeBot _ (fun h ↦ ?_)>
Tactic: simp only [Ideal.span_singleton_eq_bot, sub_eq_zero] at h
State 3: ⊢ <goal after simp only [Ideal.span_singleton_eq_bot, sub_eq_zero] at h>
Tactic: exact hζ.ne_one (one_lt_pow₀ hp.1.one_lt (Nat.zero_ne_add_one k).symm)  (RingOfIntegers.ext_iff.1 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.NumberTheory.NumberField.Cyclotomic.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.NumberField.Cyclotomic.Basic.finite_quotient_span_sub_one`:
State 0: ⊢ theorem finite_quotient_span_sub_one [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ (p ^ (k + 1))) :
    Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1})
Tactic: have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
State 1: ⊢ <goal after have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K>
Tactic: refine Ideal.finiteQuotientOfFreeOfNeBot _ (fun h ↦ ?_)
State 2: ⊢ <goal after refine Ideal.finiteQuotientOfFreeOfNeBot _ (fun h ↦ ?_)>
Tactic: simp only [Ideal.span_singleton_eq_bot, sub_eq_zero] at h
State 3: ⊢ <goal after simp only [Ideal.span_singleton_eq_bot, sub_eq_zero] at h>
Tactic: exact hζ.ne_one (one_lt_pow₀ hp.1.one_lt (Nat.zero_ne_add_one k).symm)  (RingOfIntegers.ext_iff.1 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.Data.ENNReal.Holder).

[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.ENNReal.Holder.eq_top_iff_eq_one`:
State 0: ⊢ lemma eq_top_iff_eq_one : p = ∞ ↔ q = 1
Tactic: constructor  · rintro rfl
State 1: ⊢ <goal after constructor>
Tactic: rw [← inv_inv q, ← one_sub_inv ∞ q]
State 2: ⊢ <goal after rw [← inv_inv q, ← one_sub_inv ∞ q]>
Tactic: simp  · rintro rfl
State 3: ⊢ <goal after simp>
Tactic: rw [← inv_inv p, ← one_sub_inv 1 p]
State 4: ⊢ <goal after rw [← inv_inv p, ← one_sub_inv 1 p]>
Tactic: 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.Data.ENNReal.Holder).[/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.ENNReal.Holder.eq_top_iff_eq_one`:
State 0: ⊢ lemma eq_top_iff_eq_one : p = ∞ ↔ q = 1
Tactic: constructor  · rintro rfl
State 1: ⊢ <goal after constructor>
Tactic: rw [← inv_inv q, ← one_sub_inv ∞ q]
State 2: ⊢ <goal after rw [← inv_inv q, ← one_sub_inv ∞ q]>
Tactic: simp  · rintro rfl
State 3: ⊢ <goal after simp>
Tactic: rw [← inv_inv p, ← one_sub_inv 1 p]
State 4: ⊢ <goal after rw [← inv_inv p, ← one_sub_inv 1 p]>
Tactic: 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.Analysis.Distribution.SchwartzSpace.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.Distribution.SchwartzSpace.Basic.smulLeftCLM_compCLMOfContinuousLinearEquiv`:
State 0: ⊢ theorem smulLeftCLM_compCLMOfContinuousLinearEquiv {u : D → 𝕜'} (hu : u.HasTemperateGrowth)
    (g : D ≃L[ℝ] E) (f : 𝓢(E, F)) :
    smulLeftCLM F u (compCLMOfContinuousLinearEquiv 𝕜 g f) =
    compCLMOfContinuousLinearEquiv 𝕜 g (smulLeftCLM F (u ∘ g.symm) f)
Tactic: ext x
State 1: ⊢ <goal after ext x>
Tactic: have hu' : (u ∘ g.symm).HasTemperateGrowth := by fun_prop
State 2: ⊢ <goal after have hu' : (u ∘ g.symm).HasTemperateGrowth := by fun_prop>
Tactic: simp [smulLeftCLM_apply_apply hu, smulLeftCLM_apply_apply hu']  end Comp  section Postcomp  variable [RCLike 𝕜]  [NormedSpace 𝕜 F]  [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedSpace 𝕜 G]  [NormedAddCommGroup H] [NormedSpace ℝ H] [NormedSpace 𝕜 H]  /-- Postcomposition with a continuous linear map is a continuous linear map on Schwartz  functions. -/
```

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.Distribution.SchwartzSpace.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.Distribution.SchwartzSpace.Basic.smulLeftCLM_compCLMOfContinuousLinearEquiv`:
State 0: ⊢ theorem smulLeftCLM_compCLMOfContinuousLinearEquiv {u : D → 𝕜'} (hu : u.HasTemperateGrowth)
    (g : D ≃L[ℝ] E) (f : 𝓢(E, F)) :
    smulLeftCLM F u (compCLMOfContinuousLinearEquiv 𝕜 g f) =
    compCLMOfContinuousLinearEquiv 𝕜 g (smulLeftCLM F (u ∘ g.symm) f)
Tactic: ext x
State 1: ⊢ <goal after ext x>
Tactic: have hu' : (u ∘ g.symm).HasTemperateGrowth := by fun_prop
State 2: ⊢ <goal after have hu' : (u ∘ g.symm).HasTemperateGrowth := by fun_prop>
Tactic: simp [smulLeftCLM_apply_apply hu, smulLeftCLM_apply_apply hu']  end Comp  section Postcomp  variable [RCLike 𝕜]  [NormedSpace 𝕜 F]  [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedSpace 𝕜 G]  [NormedAddCommGroup H] [NormedSpace ℝ H] [NormedSpace 𝕜 H]  /-- Postcomposition with a continuous linear map is a continuous linear map on Schwartz  functions. -/
```

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.Localization.Ideal).

[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.Localization.Ideal.map_comap`:
State 0: ⊢ theorem map_comap (J : Ideal S) :
    Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) J) = J
Tactic: obtain ⟨r, s, hx⟩ := exists_mk'_eq M x
State 1: ⊢ <goal after obtain ⟨r, s, hx⟩ := exists_mk'_eq M x>
Tactic: rw [← hx] at hJ ⊢
State 2: ⊢ <goal after rw [← hx] at hJ ⊢>
Tactic: exact  Ideal.mul_mem_right _ _  (Ideal.mem_map_of_mem _  (show (algebraMap R S) r ∈ J from  mk'_spec S r s ▸ J.mul_mem_right ((algebraMap R S) s) 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.RingTheory.Localization.Ideal).[/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.Localization.Ideal.map_comap`:
State 0: ⊢ theorem map_comap (J : Ideal S) :
    Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) J) = J
Tactic: obtain ⟨r, s, hx⟩ := exists_mk'_eq M x
State 1: ⊢ <goal after obtain ⟨r, s, hx⟩ := exists_mk'_eq M x>
Tactic: rw [← hx] at hJ ⊢
State 2: ⊢ <goal after rw [← hx] at hJ ⊢>
Tactic: exact  Ideal.mul_mem_right _ _  (Ideal.mem_map_of_mem _  (show (algebraMap R S) r ∈ J from  mk'_spec S r s ▸ J.mul_mem_right ((algebraMap R S) s) 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.SpecialFunctions.Log.Deriv).

[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.Log.Deriv.hasSum_pow_div_log_of_abs_lt_one`:
State 0: ⊢ theorem hasSum_pow_div_log_of_abs_lt_one {x : ℝ} (h : |x| < 1) :
    HasSum (fun n : ℕ => x ^ (n + 1) / (n + 1)) (-log (1 - x))
Tactic: rw [Summable.hasSum_iff_tendsto_nat]  · show Tendsto (fun n : ℕ => ∑ i ∈ range n, x ^ (i + 1) / (i + 1)) atTop (𝓝 (-log (1 - x)))
State 1: ⊢ <goal after rw [Summable.hasSum_iff_tendsto_nat]>
Tactic: rw [tendsto_iff_norm_sub_tendsto_zero]
State 2: ⊢ <goal after rw [tendsto_iff_norm_sub_tendsto_zero]>
Tactic: simp only [norm_eq_abs, sub_neg_eq_add]
State 3: ⊢ <goal after simp only [norm_eq_abs, sub_neg_eq_add]>
Tactic: refine squeeze_zero (fun n => abs_nonneg _) (abs_log_sub_add_sum_range_le h) ?_  suffices Tendsto (fun t : ℕ => |x| ^ (t + 1) / (1 - |x|)) atTop (𝓝 (|x| * 0 / (1 - |x|))) by
State 4: ⊢ <goal after refine squeeze_zero (fun n => abs_nonneg _) (abs_log_sub_add_sum_range_le h) ?_>
Tactic: simpa
State 5: ⊢ <goal after simpa>
Tactic: simp only [pow_succ']
State 6: ⊢ <goal after simp only [pow_succ']>
Tactic: refine (tendsto_const_nhds.mul ?_).div_const _
State 7: ⊢ <goal after refine (tendsto_const_nhds.mul ?_).div_const _>
Tactic: exact tendsto_pow_atTop_nhds_zero_of_lt_one (abs_nonneg _) h  show Summable fun n : ℕ => x ^ (n + 1) / (n + 1)
State 8: ⊢ <goal after exact tendsto_pow_atTop_nhds_zero_of_lt_one (abs_nonneg _) h>
Tactic: refine .of_norm_bounded (summable_geometric_of_lt_one (abs_nonneg _) h) fun i => ?_
State 9: ⊢ <goal after refine .of_norm_bounded (summable_geometric_of_lt_one (abs_nonneg _) h) fun i => ?_>
Tactic: calc  ‖x ^ (i + 1) / (i + 1)‖ = |x| ^ (i + 1) / (i + 1) := by
State 10: ⊢ <goal after calc>
Tactic: have : (0 : ℝ) ≤ i + 1 := le_of_lt (Nat.cast_add_one_pos i)
State 11: ⊢ <goal after have : (0 : ℝ) ≤ i + 1 := le_of_lt (Nat.cast_add_one_pos i)>
Tactic: rw [norm_eq_abs, abs_div, ← pow_abs, abs_of_nonneg this]  _ ≤ |x| ^ (i + 1) / (0 + 1) := by
State 12: ⊢ <goal after rw [norm_eq_abs, abs_div, ← pow_abs, abs_of_nonneg this]>
Tactic: gcongr
State 13: ⊢ <goal after gcongr>
Tactic: exact i.cast_nonneg  _ ≤ |x| ^ i := by
State 14: ⊢ <goal after exact i.cast_nonneg>
Tactic: simpa [pow_succ] using mul_le_of_le_one_right (pow_nonneg (abs_nonneg x) i) (le_of_lt h)  /-- Power series expansion of `log(1 + x) - log(1 - x)` for `|x| < 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.Analysis.SpecialFunctions.Log.Deriv).[/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.Log.Deriv.hasSum_pow_div_log_of_abs_lt_one`:
State 0: ⊢ theorem hasSum_pow_div_log_of_abs_lt_one {x : ℝ} (h : |x| < 1) :
    HasSum (fun n : ℕ => x ^ (n + 1) / (n + 1)) (-log (1 - x))
Tactic: rw [Summable.hasSum_iff_tendsto_nat]  · show Tendsto (fun n : ℕ => ∑ i ∈ range n, x ^ (i + 1) / (i + 1)) atTop (𝓝 (-log (1 - x)))
State 1: ⊢ <goal after rw [Summable.hasSum_iff_tendsto_nat]>
Tactic: rw [tendsto_iff_norm_sub_tendsto_zero]
State 2: ⊢ <goal after rw [tendsto_iff_norm_sub_tendsto_zero]>
Tactic: simp only [norm_eq_abs, sub_neg_eq_add]
State 3: ⊢ <goal after simp only [norm_eq_abs, sub_neg_eq_add]>
Tactic: refine squeeze_zero (fun n => abs_nonneg _) (abs_log_sub_add_sum_range_le h) ?_  suffices Tendsto (fun t : ℕ => |x| ^ (t + 1) / (1 - |x|)) atTop (𝓝 (|x| * 0 / (1 - |x|))) by
State 4: ⊢ <goal after refine squeeze_zero (fun n => abs_nonneg _) (abs_log_sub_add_sum_range_le h) ?_>
Tactic: simpa
State 5: ⊢ <goal after simpa>
Tactic: simp only [pow_succ']
State 6: ⊢ <goal after simp only [pow_succ']>
Tactic: refine (tendsto_const_nhds.mul ?_).div_const _
State 7: ⊢ <goal after refine (tendsto_const_nhds.mul ?_).div_const _>
Tactic: exact tendsto_pow_atTop_nhds_zero_of_lt_one (abs_nonneg _) h  show Summable fun n : ℕ => x ^ (n + 1) / (n + 1)
State 8: ⊢ <goal after exact tendsto_pow_atTop_nhds_zero_of_lt_one (abs_nonneg _) h>
Tactic: refine .of_norm_bounded (summable_geometric_of_lt_one (abs_nonneg _) h) fun i => ?_
State 9: ⊢ <goal after refine .of_norm_bounded (summable_geometric_of_lt_one (abs_nonneg _) h) fun i => ?_>
Tactic: calc  ‖x ^ (i + 1) / (i + 1)‖ = |x| ^ (i + 1) / (i + 1) := by
State 10: ⊢ <goal after calc>
Tactic: have : (0 : ℝ) ≤ i + 1 := le_of_lt (Nat.cast_add_one_pos i)
State 11: ⊢ <goal after have : (0 : ℝ) ≤ i + 1 := le_of_lt (Nat.cast_add_one_pos i)>
Tactic: rw [norm_eq_abs, abs_div, ← pow_abs, abs_of_nonneg this]  _ ≤ |x| ^ (i + 1) / (0 + 1) := by
State 12: ⊢ <goal after rw [norm_eq_abs, abs_div, ← pow_abs, abs_of_nonneg this]>
Tactic: gcongr
State 13: ⊢ <goal after gcongr>
Tactic: exact i.cast_nonneg  _ ≤ |x| ^ i := by
State 14: ⊢ <goal after exact i.cast_nonneg>
Tactic: simpa [pow_succ] using mul_le_of_le_one_right (pow_nonneg (abs_nonneg x) i) (le_of_lt h)  /-- Power series expansion of `log(1 + x) - log(1 - x)` for `|x| < 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.RingTheory.FiniteType).

[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.FiniteType.freeAlgebra_lift_of_surjective_of_closure`:
State 0: ⊢ theorem freeAlgebra_lift_of_surjective_of_closure [CommSemiring R] {S : Set M}
    (hS : closure S = ⊤) :
    Function.Surjective
      (FreeAlgebra.lift R fun s : S => of R M ↑s : FreeAlgebra R S → R[M])
Tactic: intro f
State 1: ⊢ <goal after intro f>
Tactic: induction f using induction_on with  | hM m =>
State 2: ⊢ <goal after induction f using induction_on with>
Tactic: have : m ∈ closure S := hS.symm ▸ mem_top _
State 3: ⊢ <goal after have : m ∈ closure S := hS.symm ▸ mem_top _>
Tactic: refine Submonoid.closure_induction (fun m hm => ?_) ?_ ?_ this  · exact ⟨FreeAlgebra.ι R ⟨m, hm⟩, FreeAlgebra.lift_ι_apply _ _⟩  · exact ⟨1, map_one _⟩  · rintro m₁ m₂ _ _ ⟨P₁, hP₁⟩ ⟨P₂, hP₂⟩
State 4: ⊢ <goal after refine Submonoid.closure_induction (fun m hm => ?_) ?_ ?_ this>
Tactic: exact  ⟨P₁ * P₂, by
State 5: ⊢ <goal after exact>
Tactic: rw [map_mul, hP₁, hP₂, of_apply, of_apply, of_apply, single_mul_single, one_mul]⟩  | hadd f g ihf ihg =>  rcases ihf with ⟨P, rfl⟩  rcases ihg with ⟨Q, rfl⟩
State 6: ⊢ <goal after rw [map_mul, hP₁, hP₂, of_apply, of_apply, of_apply, single_mul_single, one_mul]⟩>
Tactic: exact ⟨P + Q, map_add _ _ _⟩  | hsmul r f ih =>  rcases ih with ⟨P, rfl⟩
State 7: ⊢ <goal after exact ⟨P + Q, map_add _ _ _⟩>
Tactic: exact ⟨r • P, map_smul _ _ _⟩  /-- If a monoid `M` is finitely generated then `R[M]` is of finite type. -/
```

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.FiniteType).[/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.FiniteType.freeAlgebra_lift_of_surjective_of_closure`:
State 0: ⊢ theorem freeAlgebra_lift_of_surjective_of_closure [CommSemiring R] {S : Set M}
    (hS : closure S = ⊤) :
    Function.Surjective
      (FreeAlgebra.lift R fun s : S => of R M ↑s : FreeAlgebra R S → R[M])
Tactic: intro f
State 1: ⊢ <goal after intro f>
Tactic: induction f using induction_on with  | hM m =>
State 2: ⊢ <goal after induction f using induction_on with>
Tactic: have : m ∈ closure S := hS.symm ▸ mem_top _
State 3: ⊢ <goal after have : m ∈ closure S := hS.symm ▸ mem_top _>
Tactic: refine Submonoid.closure_induction (fun m hm => ?_) ?_ ?_ this  · exact ⟨FreeAlgebra.ι R ⟨m, hm⟩, FreeAlgebra.lift_ι_apply _ _⟩  · exact ⟨1, map_one _⟩  · rintro m₁ m₂ _ _ ⟨P₁, hP₁⟩ ⟨P₂, hP₂⟩
State 4: ⊢ <goal after refine Submonoid.closure_induction (fun m hm => ?_) ?_ ?_ this>
Tactic: exact  ⟨P₁ * P₂, by
State 5: ⊢ <goal after exact>
Tactic: rw [map_mul, hP₁, hP₂, of_apply, of_apply, of_apply, single_mul_single, one_mul]⟩  | hadd f g ihf ihg =>  rcases ihf with ⟨P, rfl⟩  rcases ihg with ⟨Q, rfl⟩
State 6: ⊢ <goal after rw [map_mul, hP₁, hP₂, of_apply, of_apply, of_apply, single_mul_single, one_mul]⟩>
Tactic: exact ⟨P + Q, map_add _ _ _⟩  | hsmul r f ih =>  rcases ih with ⟨P, rfl⟩
State 7: ⊢ <goal after exact ⟨P + Q, map_add _ _ _⟩>
Tactic: exact ⟨r • P, map_smul _ _ _⟩  /-- If a monoid `M` is finitely generated then `R[M]` is of finite type. -/
```

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.JacobiTheta.Bounds).

[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.JacobiTheta.Bounds.F_int`:
State 0: ⊢ def F_int (k : ℕ) (a : UnitAddCircle) (t : ℝ) : ℝ
Tactic: intro b  simp_rw [← (Equiv.addRight (1 : ℤ)).tsum_eq (f := fun n ↦ f_int k b t n)]
State 1: ⊢ <goal after intro b>
Tactic: simp only [f_int, ← add_assoc, add_comm, Equiv.coe_addRight, Int.cast_add, Int.cast_one]).lift a
```

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.JacobiTheta.Bounds).[/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.JacobiTheta.Bounds.F_int`:
State 0: ⊢ def F_int (k : ℕ) (a : UnitAddCircle) (t : ℝ) : ℝ
Tactic: intro b  simp_rw [← (Equiv.addRight (1 : ℤ)).tsum_eq (f := fun n ↦ f_int k b t n)]
State 1: ⊢ <goal after intro b>
Tactic: simp only [f_int, ← add_assoc, add_comm, Equiv.coe_addRight, Int.cast_add, Int.cast_one]).lift a
```

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.Interval.Set.ProjIcc).

[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.Interval.Set.ProjIcc.IccExtend_eq_self`:
State 0: ⊢ theorem IccExtend_eq_self (f : α → β) (ha : ∀ x < a, f x = f a) (hb : ∀ x, b < x → f x = f b) :
    IccExtend h (f ∘ (↑)) = f
Tactic: ext x  rcases lt_or_ge x a with hxa | hax  · simp [IccExtend_of_le_left _ _ hxa.le, ha x hxa]  · rcases le_or_gt x b with hxb | hbx  · lift x to Icc a b using ⟨hax, hxb⟩
State 1: ⊢ <goal after ext x>
Tactic: rw [IccExtend_val, comp_apply]  · simp [IccExtend_of_right_le _ _ hbx.le, hb x hbx]  end Set  open Set  variable [Preorder β] {s t : Set α} {a b : α} (h : a ≤ b) {f : Icc a b → β}  protected theorem Monotone.IciExtend {f : Ici a → β} (hf : Monotone f) : Monotone (IciExtend f) :=  hf.comp monotone_projIci  protected theorem Monotone.IicExtend {f : Iic b → β} (hf : Monotone f) : Monotone (IicExtend f) :=  hf.comp monotone_projIic  protected theorem Monotone.IccExtend (hf : Monotone f) : Monotone (IccExtend h f) :=  hf.comp <| monotone_projIcc 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.Order.Interval.Set.ProjIcc).[/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.Interval.Set.ProjIcc.IccExtend_eq_self`:
State 0: ⊢ theorem IccExtend_eq_self (f : α → β) (ha : ∀ x < a, f x = f a) (hb : ∀ x, b < x → f x = f b) :
    IccExtend h (f ∘ (↑)) = f
Tactic: ext x  rcases lt_or_ge x a with hxa | hax  · simp [IccExtend_of_le_left _ _ hxa.le, ha x hxa]  · rcases le_or_gt x b with hxb | hbx  · lift x to Icc a b using ⟨hax, hxb⟩
State 1: ⊢ <goal after ext x>
Tactic: rw [IccExtend_val, comp_apply]  · simp [IccExtend_of_right_le _ _ hbx.le, hb x hbx]  end Set  open Set  variable [Preorder β] {s t : Set α} {a b : α} (h : a ≤ b) {f : Icc a b → β}  protected theorem Monotone.IciExtend {f : Ici a → β} (hf : Monotone f) : Monotone (IciExtend f) :=  hf.comp monotone_projIci  protected theorem Monotone.IicExtend {f : Iic b → β} (hf : Monotone f) : Monotone (IicExtend f) :=  hf.comp monotone_projIic  protected theorem Monotone.IccExtend (hf : Monotone f) : Monotone (IccExtend h f) :=  hf.comp <| monotone_projIcc 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.Combinatorics.SimpleGraph.Connectivity.Connected).

[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.Connectivity.Connected.connectedComponentMk_supp_subset_supp`:
State 0: ⊢ lemma connectedComponentMk_supp_subset_supp {G'} {v : V} (h : G ≤ G') (c' : G'.ConnectedComponent)
    (hc' : v ∈ c'.supp) : (G.connectedComponentMk v).supp ⊆ c'.supp
Tactic: intro v' hv'
State 1: ⊢ <goal after intro v' hv'>
Tactic: simp only [mem_supp_iff, ConnectedComponent.eq] at hv' ⊢
State 2: ⊢ <goal after simp only [mem_supp_iff, ConnectedComponent.eq] at hv' ⊢>
Tactic: rw [ConnectedComponent.sound (hv'.mono h)]
State 3: ⊢ <goal after rw [ConnectedComponent.sound (hv'.mono h)]>
Tactic: exact hc'
```

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.Connectivity.Connected).[/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.Connectivity.Connected.connectedComponentMk_supp_subset_supp`:
State 0: ⊢ lemma connectedComponentMk_supp_subset_supp {G'} {v : V} (h : G ≤ G') (c' : G'.ConnectedComponent)
    (hc' : v ∈ c'.supp) : (G.connectedComponentMk v).supp ⊆ c'.supp
Tactic: intro v' hv'
State 1: ⊢ <goal after intro v' hv'>
Tactic: simp only [mem_supp_iff, ConnectedComponent.eq] at hv' ⊢
State 2: ⊢ <goal after simp only [mem_supp_iff, ConnectedComponent.eq] at hv' ⊢>
Tactic: rw [ConnectedComponent.sound (hv'.mono h)]
State 3: ⊢ <goal after rw [ConnectedComponent.sound (hv'.mono h)]>
Tactic: exact hc'
```

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.LegendreSymbol.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.LegendreSymbol.Basic.prime_dvd_of_eq_neg_one`:
State 0: ⊢ theorem prime_dvd_of_eq_neg_one {p : ℕ} [Fact p.Prime] {a : ℤ} (h : legendreSym p a = -1) {x y : ℤ}
    (hxy : (p : ℤ) ∣ x ^ 2 - a * y ^ 2) : ↑p ∣ x ∧ ↑p ∣ y
Tactic: push_cast at hxy
State 1: ⊢ <goal after push_cast at hxy>
Tactic: exact eq_zero_mod_of_eq_neg_one h hxy  end legendreSym  end QuadraticForm  section Values  /-!  ### The value of the Legendre symbol at `-1`  See `jacobiSym.at_neg_one` for the corresponding statement for the Jacobi symbol.  -/  variable {p : ℕ} [Fact p.Prime]  open ZMod  /-- `legendreSym p (-1)` is given by `χ₄ 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.LegendreSymbol.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.LegendreSymbol.Basic.prime_dvd_of_eq_neg_one`:
State 0: ⊢ theorem prime_dvd_of_eq_neg_one {p : ℕ} [Fact p.Prime] {a : ℤ} (h : legendreSym p a = -1) {x y : ℤ}
    (hxy : (p : ℤ) ∣ x ^ 2 - a * y ^ 2) : ↑p ∣ x ∧ ↑p ∣ y
Tactic: push_cast at hxy
State 1: ⊢ <goal after push_cast at hxy>
Tactic: exact eq_zero_mod_of_eq_neg_one h hxy  end legendreSym  end QuadraticForm  section Values  /-!  ### The value of the Legendre symbol at `-1`  See `jacobiSym.at_neg_one` for the corresponding statement for the Jacobi symbol.  -/  variable {p : ℕ} [Fact p.Prime]  open ZMod  /-- `legendreSym p (-1)` is given by `χ₄ 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.Data.Nat.Choose.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.Data.Nat.Choose.Basic.choose_two_right`:
State 0: ⊢ theorem choose_two_right (n : ℕ) : choose n 2 = n * (n - 1) / 2
Tactic: induction n with  | zero => simp  | succ n ih =>
State 1: ⊢ <goal after induction n with>
Tactic: rw [triangle_succ n, choose, ih]
State 2: ⊢ <goal after rw [triangle_succ n, choose, ih]>
Tactic: simp [Nat.add_comm]
```

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.Nat.Choose.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.Data.Nat.Choose.Basic.choose_two_right`:
State 0: ⊢ theorem choose_two_right (n : ℕ) : choose n 2 = n * (n - 1) / 2
Tactic: induction n with  | zero => simp  | succ n ih =>
State 1: ⊢ <goal after induction n with>
Tactic: rw [triangle_succ n, choose, ih]
State 2: ⊢ <goal after rw [triangle_succ n, choose, ih]>
Tactic: simp [Nat.add_comm]
```

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.Affine.Simplex).

[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.Affine.Simplex.scalene_iff_dist_ne_and_dist_ne_and_dist_ne`:
State 0: ⊢ lemma scalene_iff_dist_ne_and_dist_ne_and_dist_ne {t : Triangle R P} :
    t.Scalene ↔ dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
      dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
Tactic: refine ⟨fun h ↦  ⟨h.dist_ne (by decide : (0 : Fin 3) ≠ 1) (by decide : (0 : Fin 3) ≠ 2) (by decide) (by decide),  h.dist_ne (by decide : (0 : Fin 3) ≠ 1) (by decide : (1 : Fin 3) ≠ 2) (by decide) (by decide),  h.dist_ne (by decide : (0 : Fin 3) ≠ 2) (by decide : (1 : Fin 3) ≠ 2) (by decide) (by decide)⟩,  fun ⟨h₁, h₂, h₃⟩ ↦ ?_⟩
State 1: ⊢ <goal after refine ⟨fun h ↦>
Tactic: intro ⟨⟨x₁, x₂⟩, hx⟩ ⟨⟨y₁, y₂⟩, hy⟩ hxy
State 2: ⊢ <goal after intro ⟨⟨x₁, x₂⟩, hx⟩ ⟨⟨y₁, y₂⟩, hy⟩ hxy>
Tactic: simp only at hx hy hxy
State 3: ⊢ <goal after simp only at hx hy hxy>
Tactic: simp only [Subtype.mk.injEq, Prod.mk.injEq]  fin_cases x₁ <;> fin_cases x₂ <;> simp +decide only at hx <;>  fin_cases y₁ <;> fin_cases y₂ <;> simp +decide only at hy <;>
State 4: ⊢ <goal after simp only [Subtype.mk.injEq, Prod.mk.injEq]>
Tactic: simp [h₁, h₂, h₃, h₁.symm, h₂.symm, h₃.symm] at hxy ⊢
```

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.Affine.Simplex).[/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.Affine.Simplex.scalene_iff_dist_ne_and_dist_ne_and_dist_ne`:
State 0: ⊢ lemma scalene_iff_dist_ne_and_dist_ne_and_dist_ne {t : Triangle R P} :
    t.Scalene ↔ dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧
      dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧
      dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
Tactic: refine ⟨fun h ↦  ⟨h.dist_ne (by decide : (0 : Fin 3) ≠ 1) (by decide : (0 : Fin 3) ≠ 2) (by decide) (by decide),  h.dist_ne (by decide : (0 : Fin 3) ≠ 1) (by decide : (1 : Fin 3) ≠ 2) (by decide) (by decide),  h.dist_ne (by decide : (0 : Fin 3) ≠ 2) (by decide : (1 : Fin 3) ≠ 2) (by decide) (by decide)⟩,  fun ⟨h₁, h₂, h₃⟩ ↦ ?_⟩
State 1: ⊢ <goal after refine ⟨fun h ↦>
Tactic: intro ⟨⟨x₁, x₂⟩, hx⟩ ⟨⟨y₁, y₂⟩, hy⟩ hxy
State 2: ⊢ <goal after intro ⟨⟨x₁, x₂⟩, hx⟩ ⟨⟨y₁, y₂⟩, hy⟩ hxy>
Tactic: simp only at hx hy hxy
State 3: ⊢ <goal after simp only at hx hy hxy>
Tactic: simp only [Subtype.mk.injEq, Prod.mk.injEq]  fin_cases x₁ <;> fin_cases x₂ <;> simp +decide only at hx <;>  fin_cases y₁ <;> fin_cases y₂ <;> simp +decide only at hy <;>
State 4: ⊢ <goal after simp only [Subtype.mk.injEq, Prod.mk.injEq]>
Tactic: simp [h₁, h₂, h₃, h₁.symm, h₂.symm, h₃.symm] at hxy ⊢
```

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.SumTwoSquares).

[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.SumTwoSquares.Nat`:
State 0: ⊢ theorem Nat.mod_four_ne_three_of_mem_primeFactors_of_isSquare_neg_one {p n : ℕ}
    (hp : p ∈ n.primeFactors) (hs : IsSquare (-1 : ZMod n)) : p % 4 ≠ 3
Tactic: obtain ⟨y, h⟩ := ZMod.isSquare_neg_one_of_dvd (Nat.dvd_of_mem_primeFactors hp) hs
State 1: ⊢ <goal after obtain ⟨y, h⟩ := ZMod.isSquare_neg_one_of_dvd (Nat.dvd_of_mem_primeFactors hp) hs>
Tactic: rw [← sq, eq_comm, show (-1 : ZMod p) = -1 ^ 2 by ring] at h
State 2: ⊢ <goal after rw [← sq, eq_comm, show (-1 : ZMod p) = -1 ^ 2 by ring] at h>
Tactic: have : Fact p.Prime := ⟨Nat.prime_of_mem_primeFactors hp⟩
State 3: ⊢ <goal after have : Fact p.Prime := ⟨Nat.prime_of_mem_primeFactors hp⟩>
Tactic: exact ZMod.mod_four_ne_three_of_sq_eq_neg_sq' one_ne_zero 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.NumberTheory.SumTwoSquares).[/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.SumTwoSquares.Nat`:
State 0: ⊢ theorem Nat.mod_four_ne_three_of_mem_primeFactors_of_isSquare_neg_one {p n : ℕ}
    (hp : p ∈ n.primeFactors) (hs : IsSquare (-1 : ZMod n)) : p % 4 ≠ 3
Tactic: obtain ⟨y, h⟩ := ZMod.isSquare_neg_one_of_dvd (Nat.dvd_of_mem_primeFactors hp) hs
State 1: ⊢ <goal after obtain ⟨y, h⟩ := ZMod.isSquare_neg_one_of_dvd (Nat.dvd_of_mem_primeFactors hp) hs>
Tactic: rw [← sq, eq_comm, show (-1 : ZMod p) = -1 ^ 2 by ring] at h
State 2: ⊢ <goal after rw [← sq, eq_comm, show (-1 : ZMod p) = -1 ^ 2 by ring] at h>
Tactic: have : Fact p.Prime := ⟨Nat.prime_of_mem_primeFactors hp⟩
State 3: ⊢ <goal after have : Fact p.Prime := ⟨Nat.prime_of_mem_primeFactors hp⟩>
Tactic: exact ZMod.mod_four_ne_three_of_sq_eq_neg_sq' one_ne_zero 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.Algebra.Polynomial.Splits).

[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.Polynomial.Splits.Splits`:
State 0: ⊢ theorem Splits.of_dvd (hg : Splits g) (hg₀ : g ≠ 0) (hfg : f ∣ g) : Splits f
Tactic: obtain ⟨g, rfl⟩ := hfg
State 1: ⊢ <goal after obtain ⟨g, rfl⟩ := hfg>
Tactic: exact ((splits_mul_iff (by simp_all) (by simp_all)).mp hg).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.Algebra.Polynomial.Splits).[/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.Polynomial.Splits.Splits`:
State 0: ⊢ theorem Splits.of_dvd (hg : Splits g) (hg₀ : g ≠ 0) (hfg : f ∣ g) : Splits f
Tactic: obtain ⟨g, rfl⟩ := hfg
State 1: ⊢ <goal after obtain ⟨g, rfl⟩ := hfg>
Tactic: exact ((splits_mul_iff (by simp_all) (by simp_all)).mp hg).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.Atoms).

[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.Atoms.covBy_iff`:
State 0: ⊢ theorem covBy_iff {K L : A} :
    K ⋖ L ↔ K < L ∧ ∀ H g, K ≤ H → H ≤ L → g ∉ K → g ∈ H → H = L
Tactic: refine and_congr_right fun _ ↦ forall_congr' fun H ↦ ?_
State 1: ⊢ <goal after refine and_congr_right fun _ ↦ forall_congr' fun H ↦ ?_>
Tactic: contrapose!
State 2: ⊢ <goal after contrapose!>
Tactic: rw [lt_iff_le_not_ge, lt_iff_le_and_ne, and_and_and_comm]  simp_rw [exists_and_left, and_assoc, and_congr_right_iff, ← and_assoc, and_comm, exists_and_left,  SetLike.not_le_iff_exists, and_comm, implies_true]  /-- Dual variant of `SetLike.covBy_iff` -/
```

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.Atoms).[/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.Atoms.covBy_iff`:
State 0: ⊢ theorem covBy_iff {K L : A} :
    K ⋖ L ↔ K < L ∧ ∀ H g, K ≤ H → H ≤ L → g ∉ K → g ∈ H → H = L
Tactic: refine and_congr_right fun _ ↦ forall_congr' fun H ↦ ?_
State 1: ⊢ <goal after refine and_congr_right fun _ ↦ forall_congr' fun H ↦ ?_>
Tactic: contrapose!
State 2: ⊢ <goal after contrapose!>
Tactic: rw [lt_iff_le_not_ge, lt_iff_le_and_ne, and_and_and_comm]  simp_rw [exists_and_left, and_assoc, and_congr_right_iff, ← and_assoc, and_comm, exists_and_left,  SetLike.not_le_iff_exists, and_comm, implies_true]  /-- Dual variant of `SetLike.covBy_iff` -/
```

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.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]


[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]