[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional).

[User]
Complete the following proof.

```lean
theorem finrank_add_inf_finrank_orthogonal' {K₁ K₂ : Submodule 𝕜 E}
    [FiniteDimensional 𝕜 K₂] (h : K₁ ≤ K₂) {n : ℕ} (h_dim : finrank 𝕜 K₁ + n = finrank 𝕜 K₂) :
    finrank 𝕜 (K₁ᗮ ⊓ K₂ : Submodule 𝕜 E) = n
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem finrank_add_inf_finrank_orthogonal' {K₁ K₂ : Submodule 𝕜 E}
    [FiniteDimensional 𝕜 K₂] (h : K₁ ≤ K₂) {n : ℕ} (h_dim : finrank 𝕜 K₁ + n = finrank 𝕜 K₂) :
    finrank 𝕜 (K₁ᗮ ⊓ K₂ : Submodule 𝕜 E) = n
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]

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


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Group.Submonoid.Saturation).

[User]
Complete the following proof.

```lean
theorem saturation_sSup {f : Set (Submonoid M)} :
    (sSup f).saturation = ⨆ s ∈ f, s.saturation
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Group.Submonoid.Saturation).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem saturation_sSup {f : Set (Submonoid M)} :
    (sSup f).saturation = ⨆ s ∈ f, s.saturation
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Normed.Group.Basic).

[User]
Complete the following proof.

```lean
theorem mem_sphere_one_iff_norm : a ∈ sphere (1 : E) r ↔ ‖a‖ = r
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Normed.Group.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem mem_sphere_one_iff_norm : a ∈ sphere (1 : E) r ↔ ‖a‖ = r
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Stream.Init).

[User]
Complete the following proof.

```lean
theorem drop_map (n : ℕ) (s : Stream' α) : drop n (map f s) = map f (drop n s)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Stream.Init).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem drop_map (n : ℕ) (s : Stream' α) : drop n (map f s) = map f (drop n s)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.GroupWithZero.Units.Basic).

[User]
Complete the following proof.

```lean
lemma mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.GroupWithZero.Units.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma mul_div_mul_right (a b : G₀) (hc : c ≠ 0) : a * c / (b * c) = a / b
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.SimpleGraph.AdjMatrix).

[User]
Complete the following proof.

```lean
theorem adjMatrix_apply (v w : V) [Zero α] [One α] :
    G.adjMatrix α v w = if G.Adj v w then 1 else 0
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.SimpleGraph.AdjMatrix).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem adjMatrix_apply (v w : V) [Zero α] [One α] :
    G.adjMatrix α v w = if G.Adj v w then 1 else 0
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.PFilter).

[User]
Complete the following proof.

```lean
theorem sInf_gc :
    GaloisConnection (fun x => toDual (principal x)) fun F => sInf (ofDual F : PFilter P)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.PFilter).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem sInf_gc :
    GaloisConnection (fun x => toDual (principal x)) fun F => sInf (ofDual F : PFilter P)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence).

[User]
Complete the following proof.

```lean
lemma homologyFunctor_shiftMap
    {K L : CochainComplex C ℤ} {n : ℤ} (f : K ⟶ L⟦n⟧) (a a' : ℤ) (h : n + a = a') :
    (homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap
      (ShiftedHom.map f (quotient _ _)) a a' h =
        (homologyFunctorFactors _ _ a).hom.app K ≫
          (HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap f a a' h ≫
            (homologyFunctorFactors _ _ a').inv.app L
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma homologyFunctor_shiftMap
    {K L : CochainComplex C ℤ} {n : ℤ} (f : K ⟶ L⟦n⟧) (a a' : ℤ) (h : n + a = a') :
    (homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap
      (ShiftedHom.map f (quotient _ _)) a a' h =
        (homologyFunctorFactors _ _ a).hom.app K ≫
          (HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap f a a' h ≫
            (homologyFunctorFactors _ _ a').inv.app L
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Set.Subset).

[User]
Complete the following proof.

```lean
lemma image_val_union : (↑(D ∪ E) : Set α) = ↑D ∪ ↑E
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Set.Subset).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma image_val_union : (↑(D ∪ E) : Set α) = ↑D ∪ ↑E
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Group.Unbundled.Basic).

[User]
Complete the following proof.

```lean
theorem div_lt_div_iff' : a / b < c / d ↔ a * d < c * b
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Group.Unbundled.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem div_lt_div_iff' : a / b < c / d ↔ a * d < c * b
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.Matrix).

[User]
Complete the following proof.

```lean
theorem isUnit_toMatrix : IsUnit (b.toMatrix b₂)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.Matrix).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem isUnit_toMatrix : IsUnit (b.toMatrix b₂)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.UpperLower.Principal).

[User]
Complete the following proof.

```lean
theorem map_Ioi (f : α ≃o β) (a : α) : map f (Ioi a) = Ioi (f a)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.UpperLower.Principal).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem map_Ioi (f : α ≃o β) (a : α) : map f (Ioi a) = Ioi (f a)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs).

[User]
Complete the following proof.

```lean
theorem coe_GLPos_neg (g : SpecialLinearGroup n R) : ↑(-g) = -(↑g : GLPos n R)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem coe_GLPos_neg (g : SpecialLinearGroup n R) : ↑(-g) = -(↑g : GLPos n R)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.Combination).

[User]
Complete the following proof.

```lean
theorem weightedVSubOfPoint_erase [DecidableEq ι] (w : ι → k) (p : ι → P) (i : ι) :
    (s.erase i).weightedVSubOfPoint p (p i) w = s.weightedVSubOfPoint p (p i) w
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.Combination).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem weightedVSubOfPoint_erase [DecidableEq ι] (w : ι → k) (p : ι → P) (i : ι) :
    (s.erase i).weightedVSubOfPoint p (p i) w = s.weightedVSubOfPoint p (p i) w
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.AdicCompletion.LocalRing).

[User]
Complete the following proof.

```lean
theorem isLocalRing_of_isAdicComplete_maximal [IsAdicComplete m R] : IsLocalRing R where
  exists_pair_ne
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.AdicCompletion.LocalRing).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem isLocalRing_of_isAdicComplete_maximal [IsAdicComplete m R] : IsLocalRing R where
  exists_pair_ne
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Regular.Basic).

[User]
Complete the following proof.

```lean
lemma IsLeftRegular.pow (n : ℕ) (rla : IsLeftRegular a) : IsLeftRegular (a ^ n)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Regular.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma IsLeftRegular.pow (n : ℕ) (rla : IsLeftRegular a) : IsLeftRegular (a ^ n)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Ring.Unbundled.Basic).

[User]
Complete the following proof.

```lean
theorem mul_nonneg_iff_left_nonneg_of_pos [PosMulStrictMono R] [MulPosStrictMono R]
    (hb : 0 < b) : 0 ≤ a * b ↔ 0 ≤ a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Ring.Unbundled.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem mul_nonneg_iff_left_nonneg_of_pos [PosMulStrictMono R] [MulPosStrictMono R]
    (hb : 0 < b) : 0 ≤ a * b ↔ 0 ≤ a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.Module.FiniteDimension).

[User]
Complete the following proof.

```lean
theorem constrL_basis (v : Basis ι 𝕜 E) (f : ι → F) (i : ι) : v.constrL f (v i) = f i
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.Module.FiniteDimension).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem constrL_basis (v : Basis ι 𝕜 E) (f : ι → F) (i : ι) : v.constrL f (v i) = f i
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.NonUnitalAlgebra).

[User]
Complete the following proof.

```lean
theorem le_of_mem {x : A} {s : NonUnitalSubalgebra R A} (hs : IsClosed (s : Set A)) (hx : x ∈ s) :
    elemental R x ≤ s
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.NonUnitalAlgebra).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem le_of_mem {x : A} {s : NonUnitalSubalgebra R A} (hs : IsClosed (s : Set A)) (hx : x ∈ s) :
    elemental R x ≤ s
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.CompleteLattice.Basic).

[User]
Complete the following proof.

```lean
theorem fst_iSup [SupSet α] [SupSet β] (f : ι → α × β) : (iSup f).fst = ⨆ i, (f i).fst
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.CompleteLattice.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem fst_iSup [SupSet α] [SupSet β] (f : ι → α × β) : (iSup f).fst = ⨆ i, (f i).fst
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Ring.Divisibility.Lemmas).

[User]
Complete the following proof.

```lean
theorem associated_abs_left_iff :
    Associated |x| y ↔ Associated x y
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Ring.Divisibility.Lemmas).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem associated_abs_left_iff :
    Associated |x| y ↔ Associated x y
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.HahnSeries.HEval).

[User]
Complete the following proof.

```lean
theorem powerSeriesFamily_of_orderTop_pos {x : V⟦Γ⟧} (hx : 0 < x.orderTop)
    (f : PowerSeries R) (n : ℕ) :
    powerSeriesFamily x f n = f.coeff n • x ^ n
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.HahnSeries.HEval).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem powerSeriesFamily_of_orderTop_pos {x : V⟦Γ⟧} (hx : 0 < x.orderTop)
    (f : PowerSeries R) (n : ℕ) :
    powerSeriesFamily x f n = f.coeff n • x ^ n
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.GroupWithZero).

[User]
Complete the following proof.

```lean
theorem Filter.Tendsto.zpow₀ {f : α → G₀} {l : Filter α} {a : G₀} (hf : Tendsto f l (𝓝 a)) (m : ℤ)
    (h : a ≠ 0 ∨ 0 ≤ m) : Tendsto (fun x => f x ^ m) l (𝓝 (a ^ m))
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.GroupWithZero).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem Filter.Tendsto.zpow₀ {f : α → G₀} {l : Filter α} {a : G₀} (hf : Tendsto f l (𝓝 a)) (m : ℤ)
    (h : a ≠ 0 ∨ 0 ≤ m) : Tendsto (fun x => f x ^ m) l (𝓝 (a ^ m))
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex).

[User]
Complete the following proof.

```lean
lemma homologyMap_neg : homologyMap (-φ) i = -homologyMap φ i
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma homologyMap_neg : homologyMap (-φ) i = -homologyMap φ i
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.Ring.Compact).

[User]
Complete the following proof.

```lean
lemma IsDedekindDomain.isOpen_of_ne_bot
    [IsDedekindDomain R] {I : Ideal R} (hI : I ≠ ⊥) :
    IsOpen (X
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.Ring.Compact).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma IsDedekindDomain.isOpen_of_ne_bot
    [IsDedekindDomain R] {I : Ideal R} (hI : I ≠ ⊥) :
    IsOpen (X
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Lie.Semisimple.Basic).

[User]
Complete the following proof.

```lean
theorem subsingleton_of_hasTrivialRadical_lie_abelian [HasTrivialRadical R L] [h : IsLieAbelian L] :
    Subsingleton L
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Lie.Semisimple.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem subsingleton_of_hasTrivialRadical_lie_abelian [HasTrivialRadical R L] [h : IsLieAbelian L] :
    Subsingleton L
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Prod.Lex).

[User]
Complete the following proof.

```lean
theorem covBy_iff {a b : α ×ₗ β} :
    a ⋖ b ↔ (ofLex a).1 = (ofLex b).1 ∧ (ofLex a).2 ⋖ (ofLex b).2 ∨
      (ofLex a).1 ⋖ (ofLex b).1 ∧ IsMax (ofLex a).2 ∧ IsMin (ofLex b).2
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Prod.Lex).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem covBy_iff {a b : α ×ₗ β} :
    a ⋖ b ↔ (ofLex a).1 = (ofLex b).1 ∧ (ofLex a).2 ⋖ (ofLex b).2 ∨
      (ofLex a).1 ⋖ (ofLex b).1 ∧ IsMax (ofLex a).2 ∧ IsMin (ofLex b).2
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder).

[User]
Complete the following proof.

```lean
def LocallyFiniteOrder.addMonoidHom :
    G →+ ℤ where
  toFun a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def LocallyFiniteOrder.addMonoidHom :
    G →+ ℤ where
  toFun a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Set.Accumulate).

[User]
Complete the following proof.

```lean
theorem accumulate_eq_biInter_lt {s : ℕ → Set β} {n : ℕ} : accumulate s n = ⋃ k < n + 1, s k
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Set.Accumulate).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem accumulate_eq_biInter_lt {s : ℕ → Set β} {n : ℕ} : accumulate s n = ⋃ k < n + 1, s k
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.MvPolynomial.Basic).

[User]
Complete the following proof.

```lean
lemma restrictScalars_restrictSupportIdeal (s : Set (σ →₀ ℕ)) (hs) :
    (restrictSupportIdeal (R
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.MvPolynomial.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma restrictScalars_restrictSupportIdeal (s : Set (σ →₀ ℕ)) (hs) :
    (restrictSupportIdeal (R
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.UpperLower.Principal).

[User]
Complete the following proof.

```lean
theorem Ioi_eq_top [OrderTop α] {a : α} : Ioi a = ⊤ ↔ a = ⊤
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.UpperLower.Principal).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem Ioi_eq_top [OrderTop α] {a : α} : Ioi a = ⊤ ↔ a = ⊤
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.Transcendental.Liouville.Measure).

[User]
Complete the following proof.

```lean
theorem Real.disjoint_residual_ae : Disjoint (residual ℝ) (ae volume)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.Transcendental.Liouville.Measure).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem Real.disjoint_residual_ae : Disjoint (residual ℝ) (ae volume)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Multilinear.Curry).

[User]
Complete the following proof.

```lean
lemma currySum_uncurrySum
    (g : MultilinearMap R (fun i : ι ↦ N (.inl i))
      (MultilinearMap R (fun i : ι' ↦ N (.inr i)) M₂)) :
    currySum (uncurrySum g) = g
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Multilinear.Curry).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma currySum_uncurrySum
    (g : MultilinearMap R (fun i : ι ↦ N (.inl i))
      (MultilinearMap R (fun i : ι' ↦ N (.inr i)) M₂)) :
    currySum (uncurrySum g) = g
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.NhdsWithin).

[User]
Complete the following proof.

```lean
theorem closure_pi_set {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] (I : Set ι)
    (s : ∀ i, Set (α i)) : closure (pi I s) = pi I fun i => closure (s i)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.NhdsWithin).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem closure_pi_set {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] (I : Set ι)
    (s : ∀ i, Set (α i)) : closure (pi I s) = pi I fun i => closure (s i)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.FixedSubmodule).

[User]
Complete the following proof.

```lean
theorem mem_fixedSubmodule_iff {f : V →ₗ[R] V} {v : V} :
    v ∈ f.fixedSubmodule ↔ f v = v
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.FixedSubmodule).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem mem_fixedSubmodule_iff {f : V →ₗ[R] V} {v : V} :
    v ∈ f.fixedSubmodule ↔ f v = v
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Calculus.BumpFunction.Normed).

[User]
Complete the following proof.

```lean
theorem normed_le_div_measure_closedBall_rOut [IsAddHaarMeasure μ] (K : ℝ) (h : f.rOut ≤ K * f.rIn)
    (x : E) :
    f.normed μ x ≤ K ^ finrank ℝ E / μ.real (closedBall c f.rOut)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Calculus.BumpFunction.Normed).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem normed_le_div_measure_closedBall_rOut [IsAddHaarMeasure μ] (K : ℝ) (h : f.rOut ≤ K * f.rIn)
    (x : E) :
    f.normed μ x ≤ K ^ finrank ℝ E / μ.real (closedBall c f.rOut)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.SpecialFunctions.Exponential).

[User]
Complete the following proof.

```lean
theorem hasStrictDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕂)
    (htx : t • x ∈ Metric.eball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictDerivAt (fun u : 𝕂 => exp (u • x)) (x * exp (t • x)) t
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.SpecialFunctions.Exponential).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem hasStrictDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕂)
    (htx : t • x ∈ Metric.eball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictDerivAt (fun u : 𝕂 => exp (u • x)) (x * exp (t • x)) t
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.BigOperators.Field).

[User]
Complete the following proof.

```lean
lemma dens_biUnion_le [DecidableEq β] : (s.biUnion t).dens ≤ ∑ a ∈ s, (t a).dens
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.BigOperators.Field).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma dens_biUnion_le [DecidableEq β] : (s.biUnion t).dens ≤ ∑ a ∈ s, (t a).dens
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Compactness.Lindelof).

[User]
Complete the following proof.

```lean
theorem IsLindelof.inter_iInter_nonempty {ι : Type v} (hs : IsLindelof s) (t : ι → Set X)
    (htc : ∀ i, IsClosed (t i)) (hst : ∀ u : Set ι, u.Countable ∧ (s ∩ ⋂ i ∈ u, t i).Nonempty) :
    (s ∩ ⋂ i, t i).Nonempty
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Compactness.Lindelof).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem IsLindelof.inter_iInter_nonempty {ι : Type v} (hs : IsLindelof s) (t : ι → Set X)
    (htc : ∀ i, IsClosed (t i)) (hst : ∀ u : Set ι, u.Countable ∧ (s ∩ ⋂ i ∈ u, t i).Nonempty) :
    (s ∩ ⋂ i, t i).Nonempty
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.GroupWithZero.Units.Basic).

[User]
Complete the following proof.

```lean
theorem eq_mul_inverse_iff_mul_eq (x y z : M₀) (h : IsUnit z) : x = y * z⁻¹ʳ ↔ x * z = y
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.GroupWithZero.Units.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem eq_mul_inverse_iff_mul_eq (x y z : M₀) (h : IsUnit z) : x = y * z⁻¹ʳ ↔ x * z = y
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.SimpleGraph.DegreeSum).

[User]
Complete the following proof.

```lean
theorem dart_fst_fiber_card_eq_degree [DecidableEq V] (v : V) :
    #{d : G.Dart | d.fst = v} = G.degree v
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.SimpleGraph.DegreeSum).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem dart_fst_fiber_card_eq_degree [DecidableEq V] (v : V) :
    #{d : G.Dart | d.fst = v} = G.degree v
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.AlternatingConst).

[User]
Complete the following proof.

```lean
def alternatingConstHomologyDataZero (X : C) (n : ℕ) (hn : n = 0) :
    ((alternatingConst.obj X).sc n).HomologyData
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.AlternatingConst).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def alternatingConstHomologyDataZero (X : C) (n : ℕ) (hn : n = 0) :
    ((alternatingConst.obj X).sc n).HomologyData
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Ring.IsNonarchimedean).

[User]
Complete the following proof.

```lean
lemma apply_natCast_le_one_of_isNonarchimedean {F α : Type*} [AddMonoidWithOne α] [FunLike F α R]
    [ZeroHomClass F α R] [NonnegHomClass F α R] [OneHomClass F α R] {f : F}
    (hna : IsNonarchimedean f) {n : ℕ} : f n ≤ 1
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Ring.IsNonarchimedean).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma apply_natCast_le_one_of_isNonarchimedean {F α : Type*} [AddMonoidWithOne α] [FunLike F α R]
    [ZeroHomClass F α R] [NonnegHomClass F α R] [OneHomClass F α R] {f : F}
    (hna : IsNonarchimedean f) {n : ℕ} : f n ≤ 1
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.CoprodI).

[User]
Complete the following proof.

```lean
theorem mem_smul_iff {i j : ι} {m₁ : M i} {m₂ : M j} {w : Word M} :
    ⟨_, m₁⟩ ∈ (of m₂ • w).toList ↔
      (¬i = j ∧ ⟨i, m₁⟩ ∈ w.toList)
      ∨ (m₁ ≠ 1 ∧ ∃ (hij : i = j), (⟨i, m₁⟩ ∈ w.toList.tail) ∨
        (∃ m', ⟨j, m'⟩ ∈ w.toList.head? ∧ m₁ = hij ▸ (m₂ * m')) ∨
        (w.fstIdx ≠ some j ∧ m₁ = hij ▸ m₂))
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.CoprodI).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem mem_smul_iff {i j : ι} {m₁ : M i} {m₂ : M j} {w : Word M} :
    ⟨_, m₁⟩ ∈ (of m₂ • w).toList ↔
      (¬i = j ∧ ⟨i, m₁⟩ ∈ w.toList)
      ∨ (m₁ ≠ 1 ∧ ∃ (hij : i = j), (⟨i, m₁⟩ ∈ w.toList.tail) ∨
        (∃ m', ⟨j, m'⟩ ∈ w.toList.head? ∧ m₁ = hij ▸ (m₂ * m')) ∨
        (w.fstIdx ≠ some j ∧ m₁ = hij ▸ m₂))
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Normed.Field.UnitBall).

[User]
Complete the following proof.

```lean
instance Metric.unitClosedBall.instCommMonoid [SeminormedCommRing 𝕜] [NormOneClass 𝕜] :
    CommMonoid (closedBall (0 : 𝕜) 1)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Normed.Field.UnitBall).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
instance Metric.unitClosedBall.instCommMonoid [SeminormedCommRing 𝕜] [NormOneClass 𝕜] :
    CommMonoid (closedBall (0 : 𝕜) 1)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Real.Hyperreal).

[User]
Complete the following proof.

```lean
theorem infinitePos_abs_iff_infinite {x : ℝ*} : InfinitePos |x| ↔ Infinite x
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Real.Hyperreal).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem infinitePos_abs_iff_infinite {x : ℝ*} : InfinitePos |x| ↔ Infinite x
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Polynomial.Cyclotomic.Eval).

[User]
Complete the following proof.

```lean
theorem eval_one_cyclotomic_not_prime_pow {R : Type*} [Ring R] {n : ℕ}
    (h : ∀ {p : ℕ}, p.Prime → ∀ k : ℕ, p ^ k ≠ n) : eval 1 (cyclotomic n R) = 1
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Polynomial.Cyclotomic.Eval).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem eval_one_cyclotomic_not_prime_pow {R : Type*} [Ring R] {n : ℕ}
    (h : ∀ {p : ℕ}, p.Prime → ∀ k : ℕ, p ^ k ≠ n) : eval 1 (cyclotomic n R) = 1
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.List.Basic).

[User]
Complete the following proof.

```lean
theorem mem_getLast?_append_of_mem_getLast? {l₁ l₂ : List α} {x : α} (h : x ∈ l₂.getLast?) :
    x ∈ (l₁ ++ l₂).getLast?
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.List.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem mem_getLast?_append_of_mem_getLast? {l₁ l₂ : List α} {x : α} (h : x ∈ l₂.getLast?) :
    x ∈ (l₁ ++ l₂).getLast?
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.InnerProductSpace.Spectrum).

[User]
Complete the following proof.

```lean
theorem orthogonalFamily_eigenspaces' (hT : T.IsSymmetric) :
    OrthogonalFamily 𝕜 (fun μ : Eigenvalues T => eigenspace T μ) fun μ =>
      (eigenspace T μ).subtypeₗᵢ
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.InnerProductSpace.Spectrum).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem orthogonalFamily_eigenspaces' (hT : T.IsSymmetric) :
    OrthogonalFamily 𝕜 (fun μ : Eigenvalues T => eigenspace T μ) fun μ =>
      (eigenspace T μ).subtypeₗᵢ
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.Independent).

[User]
Complete the following proof.

```lean
theorem affineIndependent_of_ne_of_notMem_of_mem_of_mem {s : AffineSubspace k P} {p₁ p₂ p₃ : P}
    (hp₂p₃ : p₂ ≠ p₃) (hp₁ : p₁ ∉ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) :
    AffineIndependent k ![p₁, p₂, p₃]
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.Independent).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem affineIndependent_of_ne_of_notMem_of_mem_of_mem {s : AffineSubspace k P} {p₁ p₂ p₃ : P}
    (hp₂p₃ : p₂ ≠ p₃) (hp₁ : p₁ ∉ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) :
    AffineIndependent k ![p₁, p₂, p₃]
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.SimpleGraph.Walks.Operations).

[User]
Complete the following proof.

```lean
lemma drop_add_heq (p : G.Walk u v) (n m : ℕ) : p.drop (n + m) ≍ (p.drop n).drop m
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.SimpleGraph.Walks.Operations).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma drop_add_heq (p : G.Walk u v) (n m : ℕ) : p.drop (n + m) ≍ (p.drop n).drop m
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Quandle).

[User]
Complete the following proof.

```lean
theorem invAct_act_eq (x y : R) : x ◃⁻¹ x ◃ y = y
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Quandle).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem invAct_act_eq (x y : R) : x ◃⁻¹ x ◃ y = y
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.EMetricSpace.Lipschitz).

[User]
Complete the following proof.

```lean
theorem edist_iterate_succ_le_geometric {f : α → α} (hf : LipschitzWith K f) (x n) :
    edist (f^[n] x) (f^[n + 1] x) ≤ edist x (f x) * (K : ℝ≥0∞) ^ n
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.EMetricSpace.Lipschitz).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem edist_iterate_succ_le_geometric {f : α → α} (hf : LipschitzWith K f) (x n) :
    edist (f^[n] x) (f^[n + 1] x) ≤ edist x (f x) * (K : ℝ≥0∞) ^ n
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Seq.Computation).

[User]
Complete the following proof.

```lean
theorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Seq.Computation).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Order.Basic).

[User]
Complete the following proof.

```lean
theorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]
    [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)
    (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)
    (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :
    induced f ‹TopologicalSpace β› = Preorder.topology α
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Order.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β]
    [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y)
    (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y)
    (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) :
    induced f ‹TopologicalSpace β› = Preorder.topology α
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.MonoidAlgebra.Defs).

[User]
Complete the following proof.

```lean
lemma mul_def (x y : R[M]) :
    x * y = x.sum fun m₁ r₁ ↦ y.sum fun m₂ r₂ ↦ single (m₁ * m₂) (r₁ * r₂)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.MonoidAlgebra.Defs).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma mul_def (x y : R[M]) :
    x * y = x.sum fun m₁ r₁ ↦ y.sum fun m₂ r₂ ↦ single (m₁ * m₂) (r₁ * r₂)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Ideal.Operations).

[User]
Complete the following proof.

```lean
theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) :
    (∏ i ∈ s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i ∈ s, I i}
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Ideal.Operations).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) :
    (∏ i ∈ s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i ∈ s, I i}
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Multiset).

[User]
Complete the following proof.

```lean
theorem Icc_self (a : α) : Icc a a = {a}
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Multiset).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem Icc_self (a : α) : Icc a a = {a}
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Matrix.ZPow).

[User]
Complete the following proof.

```lean
theorem Commute.zpow_right {A B : M} (h : Commute A B) (m : ℤ) : Commute A (B ^ m)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Matrix.ZPow).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem Commute.zpow_right {A B : M} (h : Commute A B) (m : ℤ) : Commute A (B ^ m)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Ring.Opposite).

[User]
Complete the following proof.

```lean
instance instRing [Ring R] : Ring Rᵃᵒᵖ where
  __
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Ring.Opposite).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
instance instRing [Ring R] : Ring Rᵃᵒᵖ where
  __
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.GroupAction.Jordan).

[User]
Complete the following proof.

```lean
theorem MulAction.IsPreprimitive.isMultiplyPreprimitive
    (hG : IsPreprimitive G α) {s : Set α} {n : ℕ}
    (hsn : s.ncard = n + 1) (hsn' : n + 2 < Nat.card α)
    (hprim : IsPreprimitive (fixingSubgroup G s) (ofFixingSubgroup G s)) :
    IsMultiplyPreprimitive G α (n + 2)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.GroupAction.Jordan).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem MulAction.IsPreprimitive.isMultiplyPreprimitive
    (hG : IsPreprimitive G α) {s : Set α} {n : ℕ}
    (hsn : s.ncard = n + 1) (hsn' : n + 2 < Nat.card α)
    (hprim : IsPreprimitive (fixingSubgroup G s) (ofFixingSubgroup G s)) :
    IsMultiplyPreprimitive G α (n + 2)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.ExactSequenceFour).

[User]
Complete the following proof.

```lean
lemma epi_cokerToKer' (hS' : (S.sc hS (k + 1)).Exact) :
    Epi (hS.cokerToKer' k hk cc kf hcc hkf)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.ExactSequenceFour).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma epi_cokerToKer' (hS' : (S.sc hS (k + 1)).Exact) :
    Epi (hS.cokerToKer' k hk cc kf hcc hkf)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Set.Function).

[User]
Complete the following proof.

```lean
theorem injOn_subtype_val {p : α → Prop} {s : Set {x // p x}} : Set.InjOn Subtype.val s
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Set.Function).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem injOn_subtype_val {p : α → Prop} {s : Set {x // p x}} : Set.InjOn Subtype.val s
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.AffineMap).

[User]
Complete the following proof.

```lean
theorem linear_injective_iff (f : P1 →ᵃ[k] P2) :
    Function.Injective f.linear ↔ Function.Injective f
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.AffineSpace.AffineMap).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem linear_injective_iff (f : P1 →ᵃ[k] P2) :
    Function.Injective f.linear ↔ Function.Injective f
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Valuation.ValuationSubring).

[User]
Complete the following proof.

```lean
theorem valuationSubring_eq_top_iff : v.valuationSubring = ⊤ ↔ ¬ v.IsNontrivial
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Valuation.ValuationSubring).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem valuationSubring_eq_top_iff : v.valuationSubring = ⊤ ↔ ¬ v.IsNontrivial
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Ideal.Maps).

[User]
Complete the following proof.

```lean
theorem annihilator_span_singleton (g : M) :
    (Submodule.span R {g}).annihilator = ker (toSpanSingleton R M g)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Ideal.Maps).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem annihilator_span_singleton (g : M) :
    (Submodule.span R {g}).annihilator = ker (toSpanSingleton R M g)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Matrix.Diagonal).

[User]
Complete the following proof.

```lean
theorem diagonal_apply [Zero α] (d : n → α) (i j) : diagonal d i j = if i = j then d i else 0
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Matrix.Diagonal).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem diagonal_apply [Zero α] (d : n → α) (i j) : diagonal d i j = if i = j then d i else 0
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Finset.Basic).

[User]
Complete the following proof.

```lean
lemma nonempty_Ioi : (Ioi a).Nonempty ↔ ¬ IsMax a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Finset.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma nonempty_Ioi : (Ioi a).Nonempty ↔ ¬ IsMax a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Group.Pointwise.Set.Scalar).

[User]
Complete the following proof.

```lean
lemma image_vsub_prod : (fun x : β × β ↦ x.fst -ᵥ x.snd) '' s ×ˢ t = s -ᵥ t
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Group.Pointwise.Set.Scalar).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma image_vsub_prod : (fun x : β × β ↦ x.fst -ᵥ x.snd) '' s ×ˢ t = s -ᵥ t
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart).

[User]
Complete the following proof.

```lean
lemma quasispectrum_realPart' (a : A) (ha : IsStarNormal a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma quasispectrum_realPart' (a : A) (ha : IsStarNormal a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.UpperLower.Closure).

[User]
Complete the following proof.

```lean
theorem upperClosure_sUnion (S : Set (Set α)) : upperClosure (⋃₀ S) = ⨅ s ∈ S, upperClosure s
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.UpperLower.Closure).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem upperClosure_sUnion (S : Set (Set α)) : upperClosure (⋃₀ S) = ⨅ s ∈ S, upperClosure s
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.Module.Determinant).

[User]
Complete the following proof.

```lean
theorem det_smulRight {𝕜 : Type*} [CommRing 𝕜] [TopologicalSpace 𝕜] [ContinuousMul 𝕜]
    (f : 𝕜 →L[𝕜] 𝕜) (v : 𝕜) : (smulRight f v).det = f 1 * v
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.Module.Determinant).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem det_smulRight {𝕜 : Type*} [CommRing 𝕜] [TopologicalSpace 𝕜] [ContinuousMul 𝕜]
    (f : 𝕜 →L[𝕜] 𝕜) (v : 𝕜) : (smulRight f v).det = f 1 * v
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Order).

[User]
Complete the following proof.

```lean
theorem closure.mono (h : t₁ ≤ t₂) : closure[t₁] s ⊆ closure[t₂] s
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Order).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem closure.mono (h : t₁ ≤ t₂) : closure[t₁] s ⊆ closure[t₂] s
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv).

[User]
Complete the following proof.

```lean
theorem localInverse_apply_image (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    hf.localInverse f f' a (f a) = a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem localInverse_apply_image (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
    hf.localInverse f f' a (f a) = a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Hom.CompleteLattice).

[User]
Complete the following proof.

```lean
lemma coe_mk (f : α → β) (hf) : ⇑(mk f hf) = f
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Hom.CompleteLattice).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma coe_mk (f : α → β) (hf) : ⇑(mk f hf) = f
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Fintype.EquivFin).

[User]
Complete the following proof.

```lean
theorem card_eq_one_iff : card α = 1 ↔ ∃ x : α, ∀ y, y = x
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Fintype.EquivFin).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem card_eq_one_iff : card α = 1 ↔ ∃ x : α, ∀ y, y = x
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Polynomial.Basic).

[User]
Complete the following proof.

```lean
theorem ext {p q : R[X]} : (∀ n, coeff p n = coeff q n) → p = q
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Polynomial.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem ext {p q : R[X]} : (∀ n, coeff p n = coeff q n) → p = q
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Ideal.Height).

[User]
Complete the following proof.

```lean
lemma Ideal.exists_ltSeries_length_eq_height (p : Ideal R) [p.IsPrime] [p.FiniteHeight] :
    ∃ (l : LTSeries (PrimeSpectrum R)),
      RelSeries.last l = ⟨p, inferInstance⟩ ∧ l.length = p.height
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Ideal.Height).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma Ideal.exists_ltSeries_length_eq_height (p : Ideal R) [p.IsPrime] [p.FiniteHeight] :
    ∃ (l : LTSeries (PrimeSpectrum R)),
      RelSeries.last l = ⟨p, inferInstance⟩ ∧ l.length = p.height
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.List.TakeDrop).

[User]
Complete the following proof.

```lean
theorem takeI_left' {l₁ l₂ : List α} {n} (h : length l₁ = n) : takeI n (l₁ ++ l₂) = l₁
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.List.TakeDrop).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem takeI_left' {l₁ l₂ : List α} {n} (h : length l₁ = n) : takeI n (l₁ ++ l₂) = l₁
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.InnerProductSpace.LinearPMap).

[User]
Complete the following proof.

```lean
theorem adjointDomainMkCLMExtend_apply (hT : Dense (T.domain : Set E)) (y : T.adjointDomain)
    (x : T.domain) : adjointDomainMkCLMExtend T y (x : E) = ⟪(y : F), T x⟫
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.InnerProductSpace.LinearPMap).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem adjointDomainMkCLMExtend_apply (hT : Dense (T.domain : Set E)) (y : T.adjointDomain)
    (x : T.domain) : adjointDomainMkCLMExtend T y (x : E) = ⟪(y : F), T x⟫
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.NNReal.Defs).

[User]
Complete the following proof.

```lean
theorem mul_lt_of_lt_div {a b r : ℝ≥0} (h : a < b / r) : a * r < b
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.NNReal.Defs).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem mul_lt_of_lt_div {a b r : ℝ≥0} (h : a < b / r) : a * r < b
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual).

[User]
Complete the following proof.

```lean
instance mulLeftReflectLE [LE α] [Mul α] [c : MulLeftReflectLE α] : MulLeftReflectLE αᵒᵈ
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
instance mulLeftReflectLE [LE α] [Mul α] [c : MulLeftReflectLE α] : MulLeftReflectLE αᵒᵈ
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.UniformSpace.Equiv).

[User]
Complete the following proof.

```lean
def changeInv (f : α ≃ᵤ β) (g : β → α) (hg : Function.RightInverse g f) : α ≃ᵤ β
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.UniformSpace.Equiv).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def changeInv (f : α ≃ᵤ β) (g : β → α) (hg : Function.RightInverse g f) : α ≃ᵤ β
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.NumberField.Discriminant.Different).

[User]
Complete the following proof.

```lean
theorem discr_dvd_discr [Algebra K L] :
    discr K ∣ discr L
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.NumberField.Discriminant.Different).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem discr_dvd_discr [Algebra K L] :
    discr K ∣ discr L
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Finsupp.Order).

[User]
Complete the following proof.

```lean
theorem subset_support_tsub [DecidableEq ι] {f1 f2 : ι →₀ α} :
    f1.support \ f2.support ⊆ (f1 - f2).support
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Finsupp.Order).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem subset_support_tsub [DecidableEq ι] {f1 f2 : ι →₀ α} :
    f1.support \ f2.support ⊆ (f1 - f2).support
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Filter.AtTopBot.CountablyGenerated).

[User]
Complete the following proof.

```lean
theorem subseq_tendsto_of_neBot {f : Filter α} [IsCountablyGenerated f] {u : ℕ → α}
    (hx : NeBot (f ⊓ map u atTop)) : ∃ θ : ℕ → ℕ, StrictMono θ ∧ Tendsto (u ∘ θ) atTop f
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Filter.AtTopBot.CountablyGenerated).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem subseq_tendsto_of_neBot {f : Filter α} [IsCountablyGenerated f] {u : ℕ → α}
    (hx : NeBot (f ⊓ map u atTop)) : ∃ θ : ℕ → ℕ, StrictMono θ ∧ Tendsto (u ∘ θ) atTop f
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne).

[User]
Complete the following proof.

```lean
theorem expMap_smul (c : ℝ) (x : realSpace K) :
    expMap (c • x) = (expMap x) ^ c
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem expMap_smul (c : ℝ) (x : realSpace K) :
    expMap (c • x) = (expMap x) ^ c
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Asymptotics.Lemmas).

[User]
Complete the following proof.

```lean
theorem isLittleO_one_left_iff : (fun _x => 1 : α → F) =o[l] f ↔ Tendsto (fun x => ‖f x‖) l atTop
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Asymptotics.Lemmas).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem isLittleO_one_left_iff : (fun _x => 1 : α → F) =o[l] f ↔ Tendsto (fun x => ‖f x‖) l atTop
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Finset.SuccPred).

[User]
Complete the following proof.

```lean
lemma insert_Ioc_right_eq_Ioc_succ_of_not_isMax (h : a ≤ b) (hb : ¬ IsMax b) :
    insert (succ b) (Ioc a b) = Ioc a (succ b)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Finset.SuccPred).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma insert_Ioc_right_eq_Ioc_succ_of_not_isMax (h : a ≤ b) (hb : ¬ IsMax b) :
    insert (succ b) (Ioc a b) = Ioc a (succ b)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Flat.EquationalCriterion).

[User]
Complete the following proof.

```lean
theorem projective_of_finitePresentation [Flat R M] [FinitePresentation R M] : Projective R M
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Flat.EquationalCriterion).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem projective_of_finitePresentation [Flat R M] [FinitePresentation R M] : Projective R M
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.TotalComplex).

[User]
Complete the following proof.

```lean
lemma d₂_eq_zero' (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂)
    (h' : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂'⟩ ≠ i₁₂) :
    K.d₂ c₁₂ i₁ i₂ i₁₂ = 0
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Homology.TotalComplex).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma d₂_eq_zero' (i₁ : I₁) {i₂ i₂' : I₂} (h : c₂.Rel i₂ i₂') (i₁₂ : I₁₂)
    (h' : ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂'⟩ ≠ i₁₂) :
    K.d₂ c₁₂ i₁ i₂ i₁₂ = 0
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Holor).

[User]
Complete the following proof.

```lean
def slice (x : Holor α (d :: ds)) (i : ℕ) (h : i < d) : Holor α ds
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.Holor).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def slice (x : Holor α (d :: ds)) (i : ℕ) (h : i < d) : Holor α ds
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Asymptotics.Defs).

[User]
Complete the following proof.

```lean
theorem isBigOWith_snd_prod : IsBigOWith 1 l g' fun x => (f' x, g' x)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Asymptotics.Defs).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem isBigOWith_snd_prod : IsBigOWith 1 l g' fun x => (f' x, g' x)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Calculus.BumpFunction.Basic).

[User]
Complete the following proof.

```lean
theorem zero_of_le_dist (hx : f.rOut ≤ dist x c) : f x = 0
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Calculus.BumpFunction.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem zero_of_le_dist (hx : f.rOut ≤ dist x c) : f x = 0
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.Nilpotent).

[User]
Complete the following proof.

```lean
theorem nilpotent_iff_finite_ascending_central_series :
    IsNilpotent G ↔ ∃ n : ℕ, ∃ H : ℕ → Subgroup G, IsAscendingCentralSeries H ∧ H n = ⊤
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.Nilpotent).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem nilpotent_iff_finite_ascending_central_series :
    IsNilpotent G ↔ ∃ n : ℕ, ∃ H : ℕ → Subgroup G, IsAscendingCentralSeries H ∧ H n = ⊤
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.CStarAlgebra.Multiplier).

[User]
Complete the following proof.

```lean
theorem algebraMap_fst (k : 𝕜) : (algebraMap 𝕜 𝓜(𝕜, A) k).fst = algebraMap 𝕜 _ k
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.CStarAlgebra.Multiplier).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem algebraMap_fst (k : 𝕜) : (algebraMap 𝕜 𝓜(𝕜, A) k).fst = algebraMap 𝕜 _ k
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Set.Pi).

[User]
Complete the following proof.

```lean
theorem image_mulSingle_Ioc (i : ι) (a b : α i) :
    Pi.mulSingle i '' Ioc a b = Ioc (Pi.mulSingle i a) (Pi.mulSingle i b)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Interval.Set.Pi).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem image_mulSingle_Ioc (i : ι) (a b : α i) :
    Pi.mulSingle i '' Ioc a b = Ioc (Pi.mulSingle i a) (Pi.mulSingle i b)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Category.DeltaGenerated).

[User]
Complete the following proof.

```lean
def fullyFaithfulDeltaGeneratedToTop : deltaGeneratedToTop.{u}.FullyFaithful
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Category.DeltaGenerated).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def fullyFaithfulDeltaGeneratedToTop : deltaGeneratedToTop.{u}.FullyFaithful
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.Congruence.Hom).

[User]
Complete the following proof.

```lean
lemma hom_ext {f g : c.Quotient →* P} (h : f.comp c.mk' = g.comp c.mk') : f = g
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.GroupTheory.Congruence.Hom).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma hom_ext {f g : c.Quotient →* P} (h : f.comp c.mk' = g.comp c.mk') : f = g
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Field.MinimalAxioms).

[User]
Complete the following proof.

```lean
abbrev Field.ofMinimalAxioms (K : Type u)
    [Add K] [Mul K] [Neg K] [Inv K] [Zero K] [One K]
    (add_assoc : ∀ a b c : K, a + b + c = a + (b + c))
    (zero_add : ∀ a : K, 0 + a = a)
    (neg_add_cancel : ∀ a : K, -a + a = 0)
    (mul_assoc : ∀ a b c : K, a * b * c = a * (b * c))
    (mul_comm : ∀ a b : K, a * b = b * a)
    (one_mul : ∀ a : K, 1 * a = a)
    (mul_inv_cancel : ∀ a : K, a ≠ 0 → a * a⁻¹ = 1)
    (inv_zero : (0 : K)⁻¹ = 0)
    (left_distrib : ∀ a b c : K, a * (b + c) = a * b + a * c)
    (exists_pair_ne : ∃ x y : K, x ≠ y) : Field K
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Field.MinimalAxioms).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
abbrev Field.ofMinimalAxioms (K : Type u)
    [Add K] [Mul K] [Neg K] [Inv K] [Zero K] [One K]
    (add_assoc : ∀ a b c : K, a + b + c = a + (b + c))
    (zero_add : ∀ a : K, 0 + a = a)
    (neg_add_cancel : ∀ a : K, -a + a = 0)
    (mul_assoc : ∀ a b c : K, a * b * c = a * (b * c))
    (mul_comm : ∀ a b : K, a * b = b * a)
    (one_mul : ∀ a : K, 1 * a = a)
    (mul_inv_cancel : ∀ a : K, a ≠ 0 → a * a⁻¹ = 1)
    (inv_zero : (0 : K)⁻¹ = 0)
    (left_distrib : ∀ a b c : K, a * (b + c) = a * b + a * c)
    (exists_pair_ne : ∃ x y : K, x ≠ y) : Field K
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Hom.Basic).

[User]
Complete the following proof.

```lean
theorem map_inv_le_iff (f : F) {a : α} {b : β} : EquivLike.inv f b ≤ a ↔ b ≤ f a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Hom.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem map_inv_le_iff (f : F) {a : α} {b : β} : EquivLike.inv f b ≤ a ↔ b ≤ f a
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.MetricSpace.IsometricSMul).

[User]
Complete the following proof.

```lean
theorem constSMul_symm (c : G) : (constSMul c : X ≃ᵢ X).symm = constSMul c⁻¹
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.MetricSpace.IsometricSMul).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem constSMul_symm (c : G) : (constSMul c : X ≃ᵢ X).symm = constSMul c⁻¹
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/INST]