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

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

```text
Proof-state transcript for `Mathlib.FieldTheory.PurelyInseparable.Basic.isPurelyInseparable_of_finSepDegree_eq_one`:
State 0: ⊢ theorem isPurelyInseparable_of_finSepDegree_eq_one
    (hdeg : finSepDegree F E = 1) : IsPurelyInseparable F E
Tactic: refine fun x ↦ ⟨Algebra.IsIntegral.isIntegral x, fun hsep ↦ ?_⟩
State 1: ⊢ <goal after refine fun x ↦ ⟨Algebra.IsIntegral.isIntegral x, fun hsep ↦ ?_⟩>
Tactic: have := finSepDegree_mul_finSepDegree_of_isAlgebraic F F⟮x⟯ E
State 2: ⊢ <goal after have := finSepDegree_mul_finSepDegree_of_isAlgebraic F F⟮x⟯ E>
Tactic: rw [hdeg, mul_eq_one, (finSepDegree_adjoin_simple_eq_finrank_iff F E x  (Algebra.IsAlgebraic.isAlgebraic x)).2 hsep,  IntermediateField.finrank_eq_one_iff] at this
State 3: ⊢ <goal after rw [hdeg, mul_eq_one, (finSepDegree_adjoin_simple_eq_finrank_iff F E x>
Tactic: simpa only [this.1] using mem_adjoin_simple_self F x  · rw [← Algebra.transcendental_iff_not_isAlgebraic] at H
State 4: ⊢ <goal after simpa only [this.1] using mem_adjoin_simple_self F x>
Tactic: simp [finSepDegree_eq_zero_of_transcendental F E] at hdeg  namespace IsPurelyInseparable  variable [IsPurelyInseparable F E] (R L : Type*) [CommSemiring R] [Algebra R F] [Algebra R E]  /-- If `E / F` is purely inseparable, then for any reduced ring `L`, the map `(E →+* L) → (F →+* L)`  induced by `algebraMap F E` is injective. In particular, a purely inseparable field extension  is an epimorphism in the category of fields. -/
```

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

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

```text
Proof-state transcript for `Mathlib.FieldTheory.PurelyInseparable.Basic.isPurelyInseparable_of_finSepDegree_eq_one`:
State 0: ⊢ theorem isPurelyInseparable_of_finSepDegree_eq_one
    (hdeg : finSepDegree F E = 1) : IsPurelyInseparable F E
Tactic: refine fun x ↦ ⟨Algebra.IsIntegral.isIntegral x, fun hsep ↦ ?_⟩
State 1: ⊢ <goal after refine fun x ↦ ⟨Algebra.IsIntegral.isIntegral x, fun hsep ↦ ?_⟩>
Tactic: have := finSepDegree_mul_finSepDegree_of_isAlgebraic F F⟮x⟯ E
State 2: ⊢ <goal after have := finSepDegree_mul_finSepDegree_of_isAlgebraic F F⟮x⟯ E>
Tactic: rw [hdeg, mul_eq_one, (finSepDegree_adjoin_simple_eq_finrank_iff F E x  (Algebra.IsAlgebraic.isAlgebraic x)).2 hsep,  IntermediateField.finrank_eq_one_iff] at this
State 3: ⊢ <goal after rw [hdeg, mul_eq_one, (finSepDegree_adjoin_simple_eq_finrank_iff F E x>
Tactic: simpa only [this.1] using mem_adjoin_simple_self F x  · rw [← Algebra.transcendental_iff_not_isAlgebraic] at H
State 4: ⊢ <goal after simpa only [this.1] using mem_adjoin_simple_self F x>
Tactic: simp [finSepDegree_eq_zero_of_transcendental F E] at hdeg  namespace IsPurelyInseparable  variable [IsPurelyInseparable F E] (R L : Type*) [CommSemiring R] [Algebra R F] [Algebra R E]  /-- If `E / F` is purely inseparable, then for any reduced ring `L`, the map `(E →+* L) → (F →+* L)`  induced by `algebraMap F E` is injective. In particular, a purely inseparable field extension  is an epimorphism in the category of fields. -/
```

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


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

[User]
The following Lean code fails to compile. Fix it.

```lean
theorem IsIdempotentElem.spectrum_subset (𝕜 : Type*) {A : Type*} [Field 𝕜] [Ring A] [Algebra 𝕜 A]
    {p : A} (hp : IsIdempotentElem p) : spectrum 𝕜 p ⊆ {0, 1}
:= by thisIsWrongLemmaNameXYZ
nontriviality A
  apply Set.image_subset_iff.mp (spectrum.subset_polynomial_aeval p (X ^ 2 - X)) |>.trans
  refine fun a ha => eq_zero_or_one_of_sq_eq_self ?_
  simpa [pow_two p, hp.eq, sub_eq_zero] using ha
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=181, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=181, i_end=411, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.FieldTheory.IsAlgClosed.Spectrum).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem IsIdempotentElem.spectrum_subset (𝕜 : Type*) {A : Type*} [Field 𝕜] [Ring A] [Algebra 𝕜 A]
    {p : A} (hp : IsIdempotentElem p) : spectrum 𝕜 p ⊆ {0, 1}
:= by thisIsWrongLemmaNameXYZ
nontriviality A
  apply Set.image_subset_iff.mp (spectrum.subset_polynomial_aeval p (X ^ 2 - X)) |>.trans
  refine fun a ha => eq_zero_or_one_of_sq_eq_self ?_
  simpa [pow_two p, hp.eq, sub_eq_zero] using ha
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=181, messages=[Message(data='unknown tactic', pos=Position(line=3, column=7), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=181, i_end=411, messages=[], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.[/INST]


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

[User]
Complete the following proof.

```lean
theorem domain_nontrivial [Nontrivial β] : Nontrivial 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.Algebra.Ring.Hom.Defs).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem domain_nontrivial [Nontrivial β] : Nontrivial 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. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.CategoryTheory.Quotient).

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

```text
Proof-state transcript for `Mathlib.CategoryTheory.Quotient.compClosure_iff_self`:
State 0: ⊢ lemma compClosure_iff_self {X Y : C} (f g : X ⟶ Y) :
    CompClosure r f g ↔ r f g
Tactic: refine ⟨?_, CompClosure.of⟩  rintro ⟨_, _, _, _, _, _, h⟩
State 1: ⊢ <goal after refine ⟨?_, CompClosure.of⟩>
Tactic: exact HomRel.comp_left _ (HomRel.comp_right _ h)
```

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

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

```text
Proof-state transcript for `Mathlib.CategoryTheory.Quotient.compClosure_iff_self`:
State 0: ⊢ lemma compClosure_iff_self {X Y : C} (f g : X ⟶ Y) :
    CompClosure r f g ↔ r f g
Tactic: refine ⟨?_, CompClosure.of⟩  rintro ⟨_, _, _, _, _, _, h⟩
State 1: ⊢ <goal after refine ⟨?_, CompClosure.of⟩>
Tactic: exact HomRel.comp_left _ (HomRel.comp_right _ h)
```

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


Instruction: Edit `map_hom_inv` so it still compiles and the proof is simplified where possible. Return the full updated `map_hom_inv` block.

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

[User]
You are editing a file in `Mathlib/CategoryTheory/Iso.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Iso.map_hom_inv`, L507-L512)
```lean
theorem map_hom_inv (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [IsIso f] :
    F.map f ≫ F.map (inv f) = 𝟙 (F.obj X) := by simp
end Functor

end CategoryTheory
```

## Edit instruction
Edit `map_hom_inv` so it still compiles and the proof is simplified where possible. Return the full updated `map_hom_inv` block.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.CategoryTheory.Iso).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/CategoryTheory/Iso.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.CategoryTheory.Iso.map_hom_inv`, L507-L512)
```lean
theorem map_hom_inv (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [IsIso f] :
    F.map f ≫ F.map (inv f) = 𝟙 (F.obj X) := by simp
end Functor

end CategoryTheory
```

## Edit instruction
Edit `map_hom_inv` so it still compiles and the proof is simplified where possible. Return the full updated `map_hom_inv` block.
[/INST]


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

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem bounded_le_Iic [Preorder α] (a : α) : Bounded (· ≤ ·) (Iic a)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Order.Bounded.Bounded (L29-L31)
theorem Bounded.mono (hst : s ⊆ t) (hs : Bounded r t) : Bounded r s
```

-- snippet 2
```lean
-- Mathlib.Order.Bounded.Unbounded (L32-L38)
theorem Unbounded.mono (hst : s ⊆ t) (hs : Unbounded r s) : Unbounded r t
```

-- snippet 3
```lean
-- Mathlib.Order.Bounded.unbounded_le_of_forall_exists_lt (L39-L43)
theorem unbounded_le_of_forall_exists_lt [Preorder α] (h : ∀ a, ∃ b ∈ s, a < b) :
    Unbounded (· ≤ ·) s
```

-- snippet 4
```lean
-- Mathlib.Order.Bounded.unbounded_le_iff (L44-L46)
theorem unbounded_le_iff [LinearOrder α] : Unbounded (· ≤ ·) s ↔ ∀ a, ∃ b ∈ s, a < b
```

-- snippet 5
```lean
-- Mathlib.Order.Bounded.unbounded_lt_of_forall_exists_le (L47-L51)
theorem unbounded_lt_of_forall_exists_le [Preorder α] (h : ∀ a, ∃ b ∈ s, a ≤ b) :
    Unbounded (· < ·) s
```

-- snippet 6
```lean
-- Mathlib.Order.Bounded.unbounded_lt_iff (L52-L54)
theorem unbounded_lt_iff [LinearOrder α] : Unbounded (· < ·) s ↔ ∀ a, ∃ b ∈ s, a ≤ b
```

-- snippet 7
```lean
-- Mathlib.Order.Bounded.unbounded_ge_of_forall_exists_gt (L55-L58)
theorem unbounded_ge_of_forall_exists_gt [Preorder α] (h : ∀ a, ∃ b ∈ s, b < a) :
    Unbounded (· ≥ ·) s
```

-- snippet 8
```lean
-- Mathlib.Order.Bounded.unbounded_ge_iff (L59-L64)
theorem unbounded_ge_iff [LinearOrder α] : Unbounded (· ≥ ·) s ↔ ∀ a, ∃ b ∈ s, b < a
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Bounded).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem bounded_le_Iic [Preorder α] (a : α) : Bounded (· ≤ ·) (Iic a)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Order.Bounded.Bounded (L29-L31)
theorem Bounded.mono (hst : s ⊆ t) (hs : Bounded r t) : Bounded r s
```

-- snippet 2
```lean
-- Mathlib.Order.Bounded.Unbounded (L32-L38)
theorem Unbounded.mono (hst : s ⊆ t) (hs : Unbounded r s) : Unbounded r t
```

-- snippet 3
```lean
-- Mathlib.Order.Bounded.unbounded_le_of_forall_exists_lt (L39-L43)
theorem unbounded_le_of_forall_exists_lt [Preorder α] (h : ∀ a, ∃ b ∈ s, a < b) :
    Unbounded (· ≤ ·) s
```

-- snippet 4
```lean
-- Mathlib.Order.Bounded.unbounded_le_iff (L44-L46)
theorem unbounded_le_iff [LinearOrder α] : Unbounded (· ≤ ·) s ↔ ∀ a, ∃ b ∈ s, a < b
```

-- snippet 5
```lean
-- Mathlib.Order.Bounded.unbounded_lt_of_forall_exists_le (L47-L51)
theorem unbounded_lt_of_forall_exists_le [Preorder α] (h : ∀ a, ∃ b ∈ s, a ≤ b) :
    Unbounded (· < ·) s
```

-- snippet 6
```lean
-- Mathlib.Order.Bounded.unbounded_lt_iff (L52-L54)
theorem unbounded_lt_iff [LinearOrder α] : Unbounded (· < ·) s ↔ ∀ a, ∃ b ∈ s, a ≤ b
```

-- snippet 7
```lean
-- Mathlib.Order.Bounded.unbounded_ge_of_forall_exists_gt (L55-L58)
theorem unbounded_ge_of_forall_exists_gt [Preorder α] (h : ∀ a, ∃ b ∈ s, b < a) :
    Unbounded (· ≥ ·) s
```

-- snippet 8
```lean
-- Mathlib.Order.Bounded.unbounded_ge_iff (L59-L64)
theorem unbounded_ge_iff [LinearOrder α] : Unbounded (· ≥ ·) s ↔ ∀ a, ∃ b ∈ s, b < a
```
[/INST]


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

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

```text
Proof-state transcript for `Mathlib.NumberTheory.RamificationInertia.HilbertTheory.IsInertiaField`:
State 0: ⊢ theorem IsInertiaField.rank_left (hp : p ≠ ⊥) :
    Module.finrank E L = p.ramificationIdxIn B
Tactic: have : p.IsMaximal := over_def P p ▸ Ideal.IsMaximal.under A P
State 1: ⊢ <goal after have : p.IsMaximal := over_def P p ▸ Ideal.IsMaximal.under A P>
Tactic: have : Finite (A ⧸ p) := Ring.HasFiniteQuotients.finiteQuotient hp
State 2: ⊢ <goal after have : Finite (A ⧸ p) := Ring.HasFiniteQuotients.finiteQuotient hp>
Tactic: rw [← IsGaloisGroup.card_eq_finrank (inertia Gal(L/K) P) E L,  card_inertia_eq_ramificationIdxIn p hp]  /--  The degree `[E : K]` of the inertia field `E` over `K` equals the product of the number of  prime ideals of `B` lying over `p` and the inertia degree of `p` in `B`.  -/
```

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

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

```text
Proof-state transcript for `Mathlib.NumberTheory.RamificationInertia.HilbertTheory.IsInertiaField`:
State 0: ⊢ theorem IsInertiaField.rank_left (hp : p ≠ ⊥) :
    Module.finrank E L = p.ramificationIdxIn B
Tactic: have : p.IsMaximal := over_def P p ▸ Ideal.IsMaximal.under A P
State 1: ⊢ <goal after have : p.IsMaximal := over_def P p ▸ Ideal.IsMaximal.under A P>
Tactic: have : Finite (A ⧸ p) := Ring.HasFiniteQuotients.finiteQuotient hp
State 2: ⊢ <goal after have : Finite (A ⧸ p) := Ring.HasFiniteQuotients.finiteQuotient hp>
Tactic: rw [← IsGaloisGroup.card_eq_finrank (inertia Gal(L/K) P) E L,  card_inertia_eq_ramificationIdxIn p hp]  /--  The degree `[E : K]` of the inertia field `E` over `K` equals the product of the number of  prime ideals of `B` lying over `p` and the inertia degree of `p` in `B`.  -/
```

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


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

[User]
The following Lean code fails to compile. Fix it.

```lean
def fintypeArrow (α : Type*) (β : Type*) [DecidableEq α] [Fintype α] [Encodable β] :
    Trunc (Encodable (α → β))
:= .someMissingNamespaceXYZ
(Fintype.truncEquivFin α).map fun f =>
    Encodable.ofEquiv (Fin (Fintype.card α) → β) <| Equiv.arrowCongr f (Equiv.refl _)

/-- When `α` is finite and all `π a` are encodable, `Π a, π a` is encodable too. Because the
encoding is not unique, we wrap it in `Trunc` to preserve computability. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=283, messages=[Message(data='Unknown constant `Trunc.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  Trunc (Encodable (α → β))', pos=Position(line=3, column=3), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown constant `Quotient.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  Trunc (Encodable (α → β))', pos=Position(line=3, column=3), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown constant `Quot.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  Trunc (Encodable (α → β))', pos=Position(line=3, column=3), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=283, i_end=455, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=8, column=75), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Logic.Encodable.Pi).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def fintypeArrow (α : Type*) (β : Type*) [DecidableEq α] [Fintype α] [Encodable β] :
    Trunc (Encodable (α → β))
:= .someMissingNamespaceXYZ
(Fintype.truncEquivFin α).map fun f =>
    Encodable.ofEquiv (Fin (Fintype.card α) → β) <| Equiv.arrowCongr f (Equiv.refl _)

/-- When `α` is finite and all `π a` are encodable, `Π a, π a` is encodable too. Because the
encoding is not unique, we wrap it in `Trunc` to preserve computability. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=283, messages=[Message(data='Unknown constant `Trunc.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  Trunc (Encodable (α → β))', pos=Position(line=3, column=3), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown constant `Quotient.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  Trunc (Encodable (α → β))', pos=Position(line=3, column=3), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError'), Message(data='Unknown constant `Quot.someMissingNamespaceXYZ`\n\nNote: Inferred this name from the expected resulting type of `.someMissingNamespaceXYZ`:\n  Trunc (Encodable (α → β))', pos=Position(line=3, column=3), pos_end=Position(line=3, column=27), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=283, i_end=455, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=8, column=75), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.[/INST]


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

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

```text
Proof-state transcript for `Mathlib.Geometry.Euclidean.Incenter.excenterWeightsUnnorm_ne_zero`:
State 0: ⊢ lemma excenterWeightsUnnorm_ne_zero (signs : Finset (Fin (n + 1))) (i : Fin (n + 1)) :
    s.excenterWeightsUnnorm signs i ≠ 0
Tactic: rw [excenterWeightsUnnorm]
State 1: ⊢ <goal after rw [excenterWeightsUnnorm]>
Tactic: refine mul_ne_zero ?_ ?_  · grind  · simp [(s.height_pos i).ne']  /-- Whether an excenter exists with a given choice of signs. -/
```

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

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

```text
Proof-state transcript for `Mathlib.Geometry.Euclidean.Incenter.excenterWeightsUnnorm_ne_zero`:
State 0: ⊢ lemma excenterWeightsUnnorm_ne_zero (signs : Finset (Fin (n + 1))) (i : Fin (n + 1)) :
    s.excenterWeightsUnnorm signs i ≠ 0
Tactic: rw [excenterWeightsUnnorm]
State 1: ⊢ <goal after rw [excenterWeightsUnnorm]>
Tactic: refine mul_ne_zero ?_ ?_  · grind  · simp [(s.height_pos i).ne']  /-- Whether an excenter exists with a given choice of signs. -/
```

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


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

[User]
Prove the following theorem. The retrieved context below may help.

```lean
def toOrderTopSubOnePos {x : R⟦Γ⟧} (h : 0 < (x - 1).orderTop) :
    orderTopSubOnePos Γ R where
  val
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coe_mk (L85-L88)
theorem coe_mk (toFun : α → R⟦Γ⟧) (h1 h2) :
    (⟨toFun, h1, h2⟩ : SummableFamily Γ R α) = toFun :=
rfl
```

-- snippet 2
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.isPWO_iUnion_support (L89-L91)
theorem isPWO_iUnion_support (s : SummableFamily Γ R α) : Set.IsPWO (⋃ a : α, (s a).support) :=
s.isPWO_iUnion_support'
```

-- snippet 3
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.finite_co_support (L92-L95)
theorem finite_co_support (s : SummableFamily Γ R α) (g : Γ) :
    (fun a => (s a).coeff g).HasFiniteSupport :=
s.finite_co_support' g
```

-- snippet 4
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coe_injective (L96-L98)
theorem coe_injective : @Function.Injective (SummableFamily Γ R α) (α → R⟦Γ⟧) (⇑) :=
DFunLike.coe_injective
```

-- snippet 5
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.ext (L100-L102)
theorem ext {s t : SummableFamily Γ R α} (h : ∀ a : α, s a = t a) : s = t :=
DFunLike.ext s t h
```

-- snippet 6
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coe_add (L127-L129)
theorem coe_add (s t : SummableFamily Γ R α) : ⇑(s + t) = s + t :=
rfl
```

-- snippet 7
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.add_apply (L130-L132)
theorem add_apply {s t : SummableFamily Γ R α} {a : α} : (s + t) a = s a + t a :=
rfl
```

-- snippet 8
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coe_zero (L134-L136)
theorem coe_zero : ((0 : SummableFamily Γ R α) : α → R⟦Γ⟧) = 0 :=
rfl
```

-- snippet 9
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.zero_apply (L137-L144)
theorem zero_apply {a : α} : (0 : SummableFamily Γ R α) a = 0 :=
rfl

section SMul

variable {M} [SMulZeroClass M R]
```

-- snippet 10
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coe_smul' (L159-L161)
theorem coe_smul' (m : M) (s : SummableFamily Γ R α) : ⇑(m • s) = m • s :=
rfl
```

-- snippet 11
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.smul_apply' (L162-L166)
theorem smul_apply' (m : M) (s : SummableFamily Γ R α) (a : α) : (m • s) a = m • s a :=
rfl

end SMul
```

-- snippet 12
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coeff (L172-L176)
def coeff (s : SummableFamily Γ R α) (g : Γ) : α →₀ R where
  support := (s.finite_co_support g).toFinset
toFun a := (s a).coeff g
  mem_support_toFun a := by simp
```

-- snippet 13
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coeff_def (L178-L181)
theorem coeff_def (s : SummableFamily Γ R α) (a : α) (g : Γ) : s.coeff g a = (s a).coeff g :=
rfl

/-- The infinite sum of a `SummableFamily` of Hahn series. -/
```

-- snippet 14
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.hsum (L182-L191)
def hsum (s : SummableFamily Γ R α) : R⟦Γ⟧ where
  coeff g := ∑ᶠ i, (s i).coeff g
isPWO_support' :=
    s.isPWO_iUnion_support.mono fun g => by
      contrapose
      rw [Set.mem_iUnion, not_exists, Function.mem_support, Classical.not_not]
      simp_rw [mem_support, Classical.not_not]
      intro h
      rw [finsum_congr h, finsum_zero]
```

-- snippet 15
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coeff_hsum (L193-L195)
theorem coeff_hsum {s : SummableFamily Γ R α} {g : Γ} : s.hsum.coeff g = ∑ᶠ i, (s i).coeff g :=
rfl
```

-- snippet 16
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.coeff_add (L114-L116)
theorem coeff_add {x y : R⟦Γ⟧} {a : Γ} : (x + y).coeff a = x.coeff a + y.coeff a :=
rfl
```

-- snippet 17
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv (L137-L148)
def addOppositeEquiv : Rᵃᵒᵖ⟦Γ⟧ ≃+ R⟦Γ⟧ᵃᵒᵖ where
  toFun x := .op ⟨fun a ↦ (x.coeff a).unop, by convert x.isPWO_support; ext; simp⟩
invFun x := ⟨fun a ↦ .op (x.unop.coeff a), by convert x.unop.isPWO_support; ext; simp⟩
  left_inv x := by simp
  right_inv x := by
    apply AddOpposite.unop_injective
    simp
  map_add' x y := by
    apply AddOpposite.unop_injective
    ext
    simp
```

-- snippet 18
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.support_add_subset (L107-L109)
theorem support_add_subset (x y : R⟦Γ⟧) : (x + y).support ⊆ x.support ∪ y.support :=
Function.support_add ..
```

-- snippet 19
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.orderTop_add_eq_right (L228-L233)
theorem orderTop_add_eq_right {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧}
    (hxy : y.orderTop < x.orderTop) : (x + y).orderTop = y.orderTop := by
simpa [← map_add, ← AddOpposite.op_add, hxy] using orderTop_add_eq_left
    (x := addOppositeEquiv.symm (.op y))
    (y := addOppositeEquiv.symm (.op x))
```

-- snippet 20
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.coeff_add' (L111-L113)
theorem coeff_add' (x y : R⟦Γ⟧) : (x + y).coeff = x.coeff + y.coeff :=
rfl
```

-- snippet 21
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.leadingCoeff_add_eq_left (L234-L243)
theorem leadingCoeff_add_eq_left {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧}
    (hxy : x.orderTop < y.orderTop) : (x + y).leadingCoeff = x.leadingCoeff := by
have hx : x ≠ 0 := orderTop_ne_top.1 hxy.ne_top
  have ho : (x + y).orderTop = x.orderTop := orderTop_add_eq_left hxy
  by_cases h : x + y = 0
  · rw [h, orderTop_zero] at ho
    rw [h, orderTop_eq_top.mp ho.symm]
  · simp_rw [leadingCoeff_of_ne_zero h, leadingCoeff_of_ne_zero hx, ho, coeff_add]
    rw [coeff_eq_zero_of_lt_orderTop (x := y) (by simpa using hxy), add_zero]
```

-- snippet 22
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_leadingCoeff (L176-L184)
lemma addOppositeEquiv_leadingCoeff (x : Rᵃᵒᵖ⟦Γ⟧) :
    (addOppositeEquiv x).unop.leadingCoeff = x.leadingCoeff.unop := by
classical
  obtain rfl | hx := eq_or_ne x 0
  · simp
  simp only [ne_eq, AddOpposite.unop_eq_zero_iff, EmbeddingLike.map_eq_zero_iff, hx,
    not_false_eq_true, leadingCoeff_of_ne_zero, addOppositeEquiv_orderTop]
  simp [addOppositeEquiv]
```

-- snippet 23
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.support_smul_subset (L51-L53)
theorem support_smul_subset (r : R) (x : HahnSeries Γ V) : (r • x).support ⊆ x.support :=
Function.support_const_smul_subset ..
```

-- snippet 24
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.le_order_smul (L85-L88)
theorem le_order_smul {Γ} [Zero Γ] [LinearOrder Γ] (r : R) (x : V⟦Γ⟧) (h : r • x ≠ 0) :
    x.order ≤ (r • x).order :=
le_of_not_gt (order_smul_not_lt r x h)
```

-- snippet 25
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.coeff_smul (L59-L61)
theorem coeff_smul {r : R} {x : V⟦Γ⟧} {a : Γ} : (r • x).coeff a = r • x.coeff a :=
rfl
```

-- snippet 26
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.min_order_le_order_add (L208-L214)
theorem min_order_le_order_add {Γ} [Zero Γ] [LinearOrder Γ] {x y : R⟦Γ⟧}
    (hxy : x + y ≠ 0) : min x.order y.order ≤ (x + y).order := by
by_cases hx : x = 0; · simp [hx]
  by_cases hy : y = 0; · simp [hy]
  rw [order_of_ne hx, order_of_ne hy, order_of_ne hxy]
  exact HahnSeries.min_le_min_add hx hy hxy
```

-- snippet 27
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.orderTop_smul_not_lt (L67-L74)
theorem orderTop_smul_not_lt (r : R) (x : V⟦Γ⟧) : ¬ (r • x).orderTop < x.orderTop := by
by_cases hrx : r • x = 0
  · rw [hrx, orderTop_zero]
    exact not_top_lt
  · simp only [orderTop_of_ne_zero hrx, orderTop_of_ne_zero <| right_ne_zero_of_smul hrx,
      WithTop.coe_lt_coe]
    exact Set.IsWF.min_of_subset_not_lt_min (Function.support_smul_subset_right ..)
```

-- snippet 28
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.orderTop_add_eq_left (L215-L227)
theorem orderTop_add_eq_left {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧}
    (hxy : x.orderTop < y.orderTop) : (x + y).orderTop = x.orderTop := by
have hx : x ≠ 0 := orderTop_ne_top.1 hxy.ne_top
  let g : Γ := Set.IsWF.min x.isWF_support (support_nonempty_iff.2 hx)
  have hcxyne : (x + y).coeff g ≠ 0 := by
    rw [coeff_add, coeff_eq_zero_of_lt_orderTop (lt_of_eq_of_lt (orderTop_of_ne_zero hx).symm hxy),
      add_zero]
    exact coeff_orderTop_ne (orderTop_of_ne_zero hx)
  have hxyx : (x + y).orderTop ≤ x.orderTop := by
    rw [orderTop_of_ne_zero hx]
    exact orderTop_le_of_coeff_ne_zero hcxyne
  exact le_antisymm hxyx (le_of_eq_of_le (min_eq_left_of_lt hxy).symm min_orderTop_le_orderTop_add)
```

-- snippet 29
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.orderTop_le_orderTop_smul (L75-L78)
theorem orderTop_le_orderTop_smul {Γ} [LinearOrder Γ] (r : R) (x : V⟦Γ⟧) :
    x.orderTop ≤ (r • x).orderTop :=
le_of_not_gt <| orderTop_smul_not_lt r x
```

-- snippet 30
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_orderTop (L161-L169)
lemma addOppositeEquiv_orderTop (x : Rᵃᵒᵖ⟦Γ⟧) :
    (addOppositeEquiv x).unop.orderTop = x.orderTop := by
classical
  simp only [orderTop,
    addOppositeEquiv_support]
  simp only [addOppositeEquiv_apply, AddOpposite.unop_op, mk_eq_zero]
  simp_rw [HahnSeries.ext_iff, funext_iff]
  simp only [Pi.zero_apply, AddOpposite.unop_eq_zero_iff, coeff_zero]
```

-- snippet 31
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.leadingCoeff_add_eq_right (L244-L249)
theorem leadingCoeff_add_eq_right {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧}
    (hxy : y.orderTop < x.orderTop) : (x + y).leadingCoeff = y.leadingCoeff := by
simpa [← map_add, ← AddOpposite.op_add, hxy] using leadingCoeff_add_eq_left
    (x := addOppositeEquiv.symm (.op y))
    (y := addOppositeEquiv.symm (.op x))
```

-- snippet 32
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_symm_leadingCoeff (L186-L198)
lemma addOppositeEquiv_symm_leadingCoeff (x : R⟦Γ⟧ᵃᵒᵖ) :
    (addOppositeEquiv.symm x).leadingCoeff = .op x.unop.leadingCoeff := by
apply AddOpposite.unop_injective
  rw [← addOppositeEquiv_leadingCoeff, AddEquiv.apply_symm_apply, AddOpposite.unop_op]

protected theorem min_le_min_add {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧} (hx : x ≠ 0)
    (hy : y ≠ 0) (hxy : x + y ≠ 0) :
    min (Set.IsWF.min x.isWF_support (support_nonempty_iff.2 hx))
      (Set.IsWF.min y.isWF_support (support_nonempty_iff.2 hy)) ≤
      Set.IsWF.min (x + y).isWF_support (support_nonempty_iff.2 hxy) := by
  rw [← Set.IsWF.min_union]
  exact Set.IsWF.min_le_min_of_subset (support_add_subset (x := x) (y := y))
```

-- snippet 33
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.coeff_nsmul (L125-L126)
theorem coeff_nsmul {x : R⟦Γ⟧} {n : ℕ} : (n • x).coeff = n • x.coeff := coeff_smul' _ _
```

-- snippet 34
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.min_orderTop_le_orderTop_add (L199-L207)
theorem min_orderTop_le_orderTop_add {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧} :
    min x.orderTop y.orderTop ≤ (x + y).orderTop := by
by_cases hx : x = 0; · simp [hx]
  by_cases hy : y = 0; · simp [hy]
  by_cases hxy : x + y = 0; · simp [hxy]
  rw [orderTop_of_ne_zero hx, orderTop_of_ne_zero hy, orderTop_of_ne_zero hxy, ← WithTop.coe_min,
    WithTop.coe_le_coe]
  exact HahnSeries.min_le_min_add hx hy hxy
```

-- snippet 35
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.coeff_smul' (L55-L57)
theorem coeff_smul' (r : R) (x : V⟦Γ⟧) : (r • x).coeff = r • x.coeff :=
rfl
```

-- snippet 36
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.truncLT_smul (L89-L101)
theorem truncLT_smul [DecidableLT Γ] (c : Γ) (r : R) (x : V⟦Γ⟧) :
    truncLT c (r • x) = r • truncLT c x := by ext; simp
end SMulZeroClass

section Addition

variable [PartialOrder Γ]

section AddMonoid

variable [AddMonoid R]
```

-- snippet 37
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_symm_orderTop (L171-L174)
lemma addOppositeEquiv_symm_orderTop (x : R⟦Γ⟧ᵃᵒᵖ) :
    (addOppositeEquiv.symm x).orderTop = x.unop.orderTop := by
rw [← addOppositeEquiv_orderTop, AddEquiv.apply_symm_apply]
```

-- snippet 38
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_symm_support (L156-L159)
lemma addOppositeEquiv_symm_support (x : R⟦Γ⟧ᵃᵒᵖ) :
    (addOppositeEquiv.symm x).support = x.unop.support := by
rw [← addOppositeEquiv_support, AddEquiv.apply_symm_apply]
```

-- snippet 39
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.order_smul_not_lt (L79-L84)
theorem order_smul_not_lt [Zero Γ] (r : R) (x : V⟦Γ⟧) (h : r • x ≠ 0) :
    ¬ (r • x).order < x.order := by
have hx : x ≠ 0 := right_ne_zero_of_smul h
  simp_all only [order, dite_false]
  exact Set.IsWF.min_of_subset_not_lt_min (Function.support_smul_subset_right ..)
```

-- snippet 40
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_support (L150-L154)
lemma addOppositeEquiv_support (x : Rᵃᵒᵖ⟦Γ⟧) :
    (addOppositeEquiv x).unop.support = x.support := by
ext
  simp [addOppositeEquiv_apply]
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.HahnSeries.Summable).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
def toOrderTopSubOnePos {x : R⟦Γ⟧} (h : 0 < (x - 1).orderTop) :
    orderTopSubOnePos Γ R where
  val
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coe_mk (L85-L88)
theorem coe_mk (toFun : α → R⟦Γ⟧) (h1 h2) :
    (⟨toFun, h1, h2⟩ : SummableFamily Γ R α) = toFun :=
rfl
```

-- snippet 2
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.isPWO_iUnion_support (L89-L91)
theorem isPWO_iUnion_support (s : SummableFamily Γ R α) : Set.IsPWO (⋃ a : α, (s a).support) :=
s.isPWO_iUnion_support'
```

-- snippet 3
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.finite_co_support (L92-L95)
theorem finite_co_support (s : SummableFamily Γ R α) (g : Γ) :
    (fun a => (s a).coeff g).HasFiniteSupport :=
s.finite_co_support' g
```

-- snippet 4
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coe_injective (L96-L98)
theorem coe_injective : @Function.Injective (SummableFamily Γ R α) (α → R⟦Γ⟧) (⇑) :=
DFunLike.coe_injective
```

-- snippet 5
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.ext (L100-L102)
theorem ext {s t : SummableFamily Γ R α} (h : ∀ a : α, s a = t a) : s = t :=
DFunLike.ext s t h
```

-- snippet 6
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coe_add (L127-L129)
theorem coe_add (s t : SummableFamily Γ R α) : ⇑(s + t) = s + t :=
rfl
```

-- snippet 7
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.add_apply (L130-L132)
theorem add_apply {s t : SummableFamily Γ R α} {a : α} : (s + t) a = s a + t a :=
rfl
```

-- snippet 8
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coe_zero (L134-L136)
theorem coe_zero : ((0 : SummableFamily Γ R α) : α → R⟦Γ⟧) = 0 :=
rfl
```

-- snippet 9
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.zero_apply (L137-L144)
theorem zero_apply {a : α} : (0 : SummableFamily Γ R α) a = 0 :=
rfl

section SMul

variable {M} [SMulZeroClass M R]
```

-- snippet 10
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coe_smul' (L159-L161)
theorem coe_smul' (m : M) (s : SummableFamily Γ R α) : ⇑(m • s) = m • s :=
rfl
```

-- snippet 11
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.smul_apply' (L162-L166)
theorem smul_apply' (m : M) (s : SummableFamily Γ R α) (a : α) : (m • s) a = m • s a :=
rfl

end SMul
```

-- snippet 12
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coeff (L172-L176)
def coeff (s : SummableFamily Γ R α) (g : Γ) : α →₀ R where
  support := (s.finite_co_support g).toFinset
toFun a := (s a).coeff g
  mem_support_toFun a := by simp
```

-- snippet 13
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coeff_def (L178-L181)
theorem coeff_def (s : SummableFamily Γ R α) (a : α) (g : Γ) : s.coeff g a = (s a).coeff g :=
rfl

/-- The infinite sum of a `SummableFamily` of Hahn series. -/
```

-- snippet 14
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.hsum (L182-L191)
def hsum (s : SummableFamily Γ R α) : R⟦Γ⟧ where
  coeff g := ∑ᶠ i, (s i).coeff g
isPWO_support' :=
    s.isPWO_iUnion_support.mono fun g => by
      contrapose
      rw [Set.mem_iUnion, not_exists, Function.mem_support, Classical.not_not]
      simp_rw [mem_support, Classical.not_not]
      intro h
      rw [finsum_congr h, finsum_zero]
```

-- snippet 15
```lean
-- Mathlib.RingTheory.HahnSeries.Summable.coeff_hsum (L193-L195)
theorem coeff_hsum {s : SummableFamily Γ R α} {g : Γ} : s.hsum.coeff g = ∑ᶠ i, (s i).coeff g :=
rfl
```

-- snippet 16
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.coeff_add (L114-L116)
theorem coeff_add {x y : R⟦Γ⟧} {a : Γ} : (x + y).coeff a = x.coeff a + y.coeff a :=
rfl
```

-- snippet 17
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv (L137-L148)
def addOppositeEquiv : Rᵃᵒᵖ⟦Γ⟧ ≃+ R⟦Γ⟧ᵃᵒᵖ where
  toFun x := .op ⟨fun a ↦ (x.coeff a).unop, by convert x.isPWO_support; ext; simp⟩
invFun x := ⟨fun a ↦ .op (x.unop.coeff a), by convert x.unop.isPWO_support; ext; simp⟩
  left_inv x := by simp
  right_inv x := by
    apply AddOpposite.unop_injective
    simp
  map_add' x y := by
    apply AddOpposite.unop_injective
    ext
    simp
```

-- snippet 18
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.support_add_subset (L107-L109)
theorem support_add_subset (x y : R⟦Γ⟧) : (x + y).support ⊆ x.support ∪ y.support :=
Function.support_add ..
```

-- snippet 19
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.orderTop_add_eq_right (L228-L233)
theorem orderTop_add_eq_right {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧}
    (hxy : y.orderTop < x.orderTop) : (x + y).orderTop = y.orderTop := by
simpa [← map_add, ← AddOpposite.op_add, hxy] using orderTop_add_eq_left
    (x := addOppositeEquiv.symm (.op y))
    (y := addOppositeEquiv.symm (.op x))
```

-- snippet 20
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.coeff_add' (L111-L113)
theorem coeff_add' (x y : R⟦Γ⟧) : (x + y).coeff = x.coeff + y.coeff :=
rfl
```

-- snippet 21
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.leadingCoeff_add_eq_left (L234-L243)
theorem leadingCoeff_add_eq_left {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧}
    (hxy : x.orderTop < y.orderTop) : (x + y).leadingCoeff = x.leadingCoeff := by
have hx : x ≠ 0 := orderTop_ne_top.1 hxy.ne_top
  have ho : (x + y).orderTop = x.orderTop := orderTop_add_eq_left hxy
  by_cases h : x + y = 0
  · rw [h, orderTop_zero] at ho
    rw [h, orderTop_eq_top.mp ho.symm]
  · simp_rw [leadingCoeff_of_ne_zero h, leadingCoeff_of_ne_zero hx, ho, coeff_add]
    rw [coeff_eq_zero_of_lt_orderTop (x := y) (by simpa using hxy), add_zero]
```

-- snippet 22
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_leadingCoeff (L176-L184)
lemma addOppositeEquiv_leadingCoeff (x : Rᵃᵒᵖ⟦Γ⟧) :
    (addOppositeEquiv x).unop.leadingCoeff = x.leadingCoeff.unop := by
classical
  obtain rfl | hx := eq_or_ne x 0
  · simp
  simp only [ne_eq, AddOpposite.unop_eq_zero_iff, EmbeddingLike.map_eq_zero_iff, hx,
    not_false_eq_true, leadingCoeff_of_ne_zero, addOppositeEquiv_orderTop]
  simp [addOppositeEquiv]
```

-- snippet 23
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.support_smul_subset (L51-L53)
theorem support_smul_subset (r : R) (x : HahnSeries Γ V) : (r • x).support ⊆ x.support :=
Function.support_const_smul_subset ..
```

-- snippet 24
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.le_order_smul (L85-L88)
theorem le_order_smul {Γ} [Zero Γ] [LinearOrder Γ] (r : R) (x : V⟦Γ⟧) (h : r • x ≠ 0) :
    x.order ≤ (r • x).order :=
le_of_not_gt (order_smul_not_lt r x h)
```

-- snippet 25
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.coeff_smul (L59-L61)
theorem coeff_smul {r : R} {x : V⟦Γ⟧} {a : Γ} : (r • x).coeff a = r • x.coeff a :=
rfl
```

-- snippet 26
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.min_order_le_order_add (L208-L214)
theorem min_order_le_order_add {Γ} [Zero Γ] [LinearOrder Γ] {x y : R⟦Γ⟧}
    (hxy : x + y ≠ 0) : min x.order y.order ≤ (x + y).order := by
by_cases hx : x = 0; · simp [hx]
  by_cases hy : y = 0; · simp [hy]
  rw [order_of_ne hx, order_of_ne hy, order_of_ne hxy]
  exact HahnSeries.min_le_min_add hx hy hxy
```

-- snippet 27
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.orderTop_smul_not_lt (L67-L74)
theorem orderTop_smul_not_lt (r : R) (x : V⟦Γ⟧) : ¬ (r • x).orderTop < x.orderTop := by
by_cases hrx : r • x = 0
  · rw [hrx, orderTop_zero]
    exact not_top_lt
  · simp only [orderTop_of_ne_zero hrx, orderTop_of_ne_zero <| right_ne_zero_of_smul hrx,
      WithTop.coe_lt_coe]
    exact Set.IsWF.min_of_subset_not_lt_min (Function.support_smul_subset_right ..)
```

-- snippet 28
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.orderTop_add_eq_left (L215-L227)
theorem orderTop_add_eq_left {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧}
    (hxy : x.orderTop < y.orderTop) : (x + y).orderTop = x.orderTop := by
have hx : x ≠ 0 := orderTop_ne_top.1 hxy.ne_top
  let g : Γ := Set.IsWF.min x.isWF_support (support_nonempty_iff.2 hx)
  have hcxyne : (x + y).coeff g ≠ 0 := by
    rw [coeff_add, coeff_eq_zero_of_lt_orderTop (lt_of_eq_of_lt (orderTop_of_ne_zero hx).symm hxy),
      add_zero]
    exact coeff_orderTop_ne (orderTop_of_ne_zero hx)
  have hxyx : (x + y).orderTop ≤ x.orderTop := by
    rw [orderTop_of_ne_zero hx]
    exact orderTop_le_of_coeff_ne_zero hcxyne
  exact le_antisymm hxyx (le_of_eq_of_le (min_eq_left_of_lt hxy).symm min_orderTop_le_orderTop_add)
```

-- snippet 29
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.orderTop_le_orderTop_smul (L75-L78)
theorem orderTop_le_orderTop_smul {Γ} [LinearOrder Γ] (r : R) (x : V⟦Γ⟧) :
    x.orderTop ≤ (r • x).orderTop :=
le_of_not_gt <| orderTop_smul_not_lt r x
```

-- snippet 30
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_orderTop (L161-L169)
lemma addOppositeEquiv_orderTop (x : Rᵃᵒᵖ⟦Γ⟧) :
    (addOppositeEquiv x).unop.orderTop = x.orderTop := by
classical
  simp only [orderTop,
    addOppositeEquiv_support]
  simp only [addOppositeEquiv_apply, AddOpposite.unop_op, mk_eq_zero]
  simp_rw [HahnSeries.ext_iff, funext_iff]
  simp only [Pi.zero_apply, AddOpposite.unop_eq_zero_iff, coeff_zero]
```

-- snippet 31
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.leadingCoeff_add_eq_right (L244-L249)
theorem leadingCoeff_add_eq_right {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧}
    (hxy : y.orderTop < x.orderTop) : (x + y).leadingCoeff = y.leadingCoeff := by
simpa [← map_add, ← AddOpposite.op_add, hxy] using leadingCoeff_add_eq_left
    (x := addOppositeEquiv.symm (.op y))
    (y := addOppositeEquiv.symm (.op x))
```

-- snippet 32
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_symm_leadingCoeff (L186-L198)
lemma addOppositeEquiv_symm_leadingCoeff (x : R⟦Γ⟧ᵃᵒᵖ) :
    (addOppositeEquiv.symm x).leadingCoeff = .op x.unop.leadingCoeff := by
apply AddOpposite.unop_injective
  rw [← addOppositeEquiv_leadingCoeff, AddEquiv.apply_symm_apply, AddOpposite.unop_op]

protected theorem min_le_min_add {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧} (hx : x ≠ 0)
    (hy : y ≠ 0) (hxy : x + y ≠ 0) :
    min (Set.IsWF.min x.isWF_support (support_nonempty_iff.2 hx))
      (Set.IsWF.min y.isWF_support (support_nonempty_iff.2 hy)) ≤
      Set.IsWF.min (x + y).isWF_support (support_nonempty_iff.2 hxy) := by
  rw [← Set.IsWF.min_union]
  exact Set.IsWF.min_le_min_of_subset (support_add_subset (x := x) (y := y))
```

-- snippet 33
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.coeff_nsmul (L125-L126)
theorem coeff_nsmul {x : R⟦Γ⟧} {n : ℕ} : (n • x).coeff = n • x.coeff := coeff_smul' _ _
```

-- snippet 34
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.min_orderTop_le_orderTop_add (L199-L207)
theorem min_orderTop_le_orderTop_add {Γ} [LinearOrder Γ] {x y : R⟦Γ⟧} :
    min x.orderTop y.orderTop ≤ (x + y).orderTop := by
by_cases hx : x = 0; · simp [hx]
  by_cases hy : y = 0; · simp [hy]
  by_cases hxy : x + y = 0; · simp [hxy]
  rw [orderTop_of_ne_zero hx, orderTop_of_ne_zero hy, orderTop_of_ne_zero hxy, ← WithTop.coe_min,
    WithTop.coe_le_coe]
  exact HahnSeries.min_le_min_add hx hy hxy
```

-- snippet 35
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.coeff_smul' (L55-L57)
theorem coeff_smul' (r : R) (x : V⟦Γ⟧) : (r • x).coeff = r • x.coeff :=
rfl
```

-- snippet 36
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.truncLT_smul (L89-L101)
theorem truncLT_smul [DecidableLT Γ] (c : Γ) (r : R) (x : V⟦Γ⟧) :
    truncLT c (r • x) = r • truncLT c x := by ext; simp
end SMulZeroClass

section Addition

variable [PartialOrder Γ]

section AddMonoid

variable [AddMonoid R]
```

-- snippet 37
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_symm_orderTop (L171-L174)
lemma addOppositeEquiv_symm_orderTop (x : R⟦Γ⟧ᵃᵒᵖ) :
    (addOppositeEquiv.symm x).orderTop = x.unop.orderTop := by
rw [← addOppositeEquiv_orderTop, AddEquiv.apply_symm_apply]
```

-- snippet 38
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_symm_support (L156-L159)
lemma addOppositeEquiv_symm_support (x : R⟦Γ⟧ᵃᵒᵖ) :
    (addOppositeEquiv.symm x).support = x.unop.support := by
rw [← addOppositeEquiv_support, AddEquiv.apply_symm_apply]
```

-- snippet 39
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.order_smul_not_lt (L79-L84)
theorem order_smul_not_lt [Zero Γ] (r : R) (x : V⟦Γ⟧) (h : r • x ≠ 0) :
    ¬ (r • x).order < x.order := by
have hx : x ≠ 0 := right_ne_zero_of_smul h
  simp_all only [order, dite_false]
  exact Set.IsWF.min_of_subset_not_lt_min (Function.support_smul_subset_right ..)
```

-- snippet 40
```lean
-- Mathlib.RingTheory.HahnSeries.Addition.addOppositeEquiv_support (L150-L154)
lemma addOppositeEquiv_support (x : Rᵃᵒᵖ⟦Γ⟧) :
    (addOppositeEquiv x).unop.support = x.support := by
ext
  simp [addOppositeEquiv_apply]
```
[/INST]


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

[User]
Prove the following theorem. The retrieved context below may help.

```lean
lemma le_one (s : stdSimplex S X) (x : X) : s x ≤ 1
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Analysis.Convex.StdSimplex.stdSimplex (L35-L37)
def stdSimplex : Set (ι → 𝕜)
```

-- snippet 2
```lean
-- Mathlib.Analysis.Convex.StdSimplex.stdSimplex_eq_inter (L38-L41)
theorem stdSimplex_eq_inter : stdSimplex 𝕜 ι = (⋂ x, { f | 0 ≤ f x }) ∩ { f | ∑ x, f x = 1 }
```

-- snippet 3
```lean
-- Mathlib.Analysis.Convex.StdSimplex.convex_stdSimplex (L42-L48)
theorem convex_stdSimplex [IsOrderedRing 𝕜] : Convex 𝕜 (stdSimplex 𝕜 ι)
```

-- snippet 4
```lean
-- Mathlib.Analysis.Convex.StdSimplex.stdSimplex_of_isEmpty_index (L53-L55)
lemma stdSimplex_of_isEmpty_index [IsEmpty ι] [Nontrivial 𝕜] : stdSimplex 𝕜 ι = ∅
```

-- snippet 5
```lean
-- Mathlib.Analysis.Convex.StdSimplex.stdSimplex_unique (L56-L67)
lemma stdSimplex_unique [ZeroLEOneClass 𝕜] [Nonempty ι] [Subsingleton ι] :
    stdSimplex 𝕜 ι = {fun _ ↦ 1}
```

-- snippet 6
```lean
-- Mathlib.Analysis.Convex.StdSimplex.mem_Icc_of_mem_stdSimplex (L68-L73)
theorem mem_Icc_of_mem_stdSimplex [IsOrderedAddMonoid 𝕜]
    {f : ι → 𝕜} (hf : f ∈ stdSimplex 𝕜 ι) (x) :
    f x ∈ Icc (0 : 𝕜) 1
```

-- snippet 7
```lean
-- Mathlib.Analysis.Convex.StdSimplex.stdSimplex_subset_Icc (L74-L80)
theorem stdSimplex_subset_Icc [IsOrderedAddMonoid 𝕜] : stdSimplex 𝕜 ι ⊆ Icc 0 1
```

-- snippet 8
```lean
-- Mathlib.Analysis.Convex.StdSimplex.single_mem_stdSimplex (L81-L83)
theorem single_mem_stdSimplex (i : ι) : Pi.single i 1 ∈ stdSimplex 𝕜 ι
```

-- snippet 9
```lean
-- Mathlib.Analysis.Convex.AmpleSet.ampleSet_univ (L58-L63)
theorem ampleSet_univ {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] :
    AmpleSet (univ : Set F)
```

-- snippet 10
```lean
-- Mathlib.Analysis.Convex.AmpleSet.AmpleSet (L53-L56)
def AmpleSet (s : Set F) : Prop
```

-- snippet 11
```lean
-- Mathlib.Analysis.Convex.AmpleSet.ampleSet_empty (L65-L69)
theorem ampleSet_empty : AmpleSet (∅ : Set F)
```

-- snippet 12
```lean
-- Mathlib.Analysis.Convex.AmpleSet.union (L70-L83)
theorem union {s t : Set F} (hs : AmpleSet s) (ht : AmpleSet t) : AmpleSet (s ∪ t)
```

-- snippet 13
```lean
-- Mathlib.Analysis.Convex.AmpleSet.image (L84-L93)
theorem image {s : Set E} (h : AmpleSet s) (L : E ≃ᴬ[ℝ] F) :
    AmpleSet (L '' s)
```

-- snippet 14
```lean
-- Mathlib.Analysis.Convex.AmpleSet.image_iff (L94-L98)
theorem image_iff {s : Set E} (L : E ≃ᴬ[ℝ] F) :
    AmpleSet (L '' s) ↔ AmpleSet s
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Convex.StdSimplex).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
lemma le_one (s : stdSimplex S X) (x : X) : s x ≤ 1
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Analysis.Convex.StdSimplex.stdSimplex (L35-L37)
def stdSimplex : Set (ι → 𝕜)
```

-- snippet 2
```lean
-- Mathlib.Analysis.Convex.StdSimplex.stdSimplex_eq_inter (L38-L41)
theorem stdSimplex_eq_inter : stdSimplex 𝕜 ι = (⋂ x, { f | 0 ≤ f x }) ∩ { f | ∑ x, f x = 1 }
```

-- snippet 3
```lean
-- Mathlib.Analysis.Convex.StdSimplex.convex_stdSimplex (L42-L48)
theorem convex_stdSimplex [IsOrderedRing 𝕜] : Convex 𝕜 (stdSimplex 𝕜 ι)
```

-- snippet 4
```lean
-- Mathlib.Analysis.Convex.StdSimplex.stdSimplex_of_isEmpty_index (L53-L55)
lemma stdSimplex_of_isEmpty_index [IsEmpty ι] [Nontrivial 𝕜] : stdSimplex 𝕜 ι = ∅
```

-- snippet 5
```lean
-- Mathlib.Analysis.Convex.StdSimplex.stdSimplex_unique (L56-L67)
lemma stdSimplex_unique [ZeroLEOneClass 𝕜] [Nonempty ι] [Subsingleton ι] :
    stdSimplex 𝕜 ι = {fun _ ↦ 1}
```

-- snippet 6
```lean
-- Mathlib.Analysis.Convex.StdSimplex.mem_Icc_of_mem_stdSimplex (L68-L73)
theorem mem_Icc_of_mem_stdSimplex [IsOrderedAddMonoid 𝕜]
    {f : ι → 𝕜} (hf : f ∈ stdSimplex 𝕜 ι) (x) :
    f x ∈ Icc (0 : 𝕜) 1
```

-- snippet 7
```lean
-- Mathlib.Analysis.Convex.StdSimplex.stdSimplex_subset_Icc (L74-L80)
theorem stdSimplex_subset_Icc [IsOrderedAddMonoid 𝕜] : stdSimplex 𝕜 ι ⊆ Icc 0 1
```

-- snippet 8
```lean
-- Mathlib.Analysis.Convex.StdSimplex.single_mem_stdSimplex (L81-L83)
theorem single_mem_stdSimplex (i : ι) : Pi.single i 1 ∈ stdSimplex 𝕜 ι
```

-- snippet 9
```lean
-- Mathlib.Analysis.Convex.AmpleSet.ampleSet_univ (L58-L63)
theorem ampleSet_univ {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] :
    AmpleSet (univ : Set F)
```

-- snippet 10
```lean
-- Mathlib.Analysis.Convex.AmpleSet.AmpleSet (L53-L56)
def AmpleSet (s : Set F) : Prop
```

-- snippet 11
```lean
-- Mathlib.Analysis.Convex.AmpleSet.ampleSet_empty (L65-L69)
theorem ampleSet_empty : AmpleSet (∅ : Set F)
```

-- snippet 12
```lean
-- Mathlib.Analysis.Convex.AmpleSet.union (L70-L83)
theorem union {s t : Set F} (hs : AmpleSet s) (ht : AmpleSet t) : AmpleSet (s ∪ t)
```

-- snippet 13
```lean
-- Mathlib.Analysis.Convex.AmpleSet.image (L84-L93)
theorem image {s : Set E} (h : AmpleSet s) (L : E ≃ᴬ[ℝ] F) :
    AmpleSet (L '' s)
```

-- snippet 14
```lean
-- Mathlib.Analysis.Convex.AmpleSet.image_iff (L94-L98)
theorem image_iff {s : Set E} (L : E ≃ᴬ[ℝ] F) :
    AmpleSet (L '' s) ↔ AmpleSet s
```
[/INST]


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

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem integrableOn_exp_mul_complex_Iic {a : ℂ} (ha : 0 < a.re) (c : ℝ) :
    IntegrableOn (fun x : ℝ => Complex.exp (a * x)) (Iic c)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_exp_Iic (L37-L44)
theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) := by
refine
    integrableOn_Iic_of_intervalIntegral_norm_bounded (exp c) c
      (fun y => intervalIntegrable_exp.1) tendsto_id
      (eventually_of_mem (Iic_mem_atBot 0) fun y _ => ?_)
  simp_rw [norm_of_nonneg (exp_pos _).le, integral_exp, sub_le_self_iff]
  exact (exp_pos _).le
```

-- snippet 2
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_exp_neg_Ioi (L45-L47)
theorem integrableOn_exp_neg_Ioi (c : ℝ) : IntegrableOn (fun (x : ℝ) => exp (-x)) (Ioi c) :=
Iff.mp integrableOn_Ici_iff_integrableOn_Ioi (integrableOn_exp_Iic (-c)).comp_neg_Ici
```

-- snippet 3
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_Iic (L48-L54)
theorem integral_exp_Iic (c : ℝ) : ∫ x : ℝ in Iic c, exp x = exp c := by
refine
    tendsto_nhds_unique
      (intervalIntegral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) ?_
  simp_rw [integral_exp, show 𝓝 (exp c) = 𝓝 (exp c - 0) by rw [sub_zero]]
  exact tendsto_exp_atBot.const_sub _
```

-- snippet 4
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_Iic_zero (L55-L57)
theorem integral_exp_Iic_zero : ∫ x : ℝ in Iic 0, exp x = 1 :=
exp_zero ▸ integral_exp_Iic 0
```

-- snippet 5
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_neg_Ioi (L58-L60)
theorem integral_exp_neg_Ioi (c : ℝ) : (∫ x : ℝ in Ioi c, exp (-x)) = exp (-c) := by
simpa only [integral_comp_neg_Ioi] using integral_exp_Iic (-c)
```

-- snippet 6
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_neg_Ioi_zero (L61-L63)
theorem integral_exp_neg_Ioi_zero : (∫ x : ℝ in Ioi 0, exp (-x)) = 1 := by
simpa only [neg_zero, exp_zero] using integral_exp_neg_Ioi 0
```

-- snippet 7
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_exp_mul_complex_Ioi (L64-L72)
theorem integrableOn_exp_mul_complex_Ioi {a : ℂ} (ha : a.re < 0) (c : ℝ) :
    IntegrableOn (fun x : ℝ => Complex.exp (a * x)) (Ioi c) := by
refine (integrable_norm_iff ?_).mp ?_
  · apply Continuous.aestronglyMeasurable
    fun_prop
  · simpa [Complex.norm_exp] using
      (integrableOn_Ioi_comp_mul_left_iff (fun x => exp (-x)) c (a := -a.re) (by simpa)).mpr <|
        integrableOn_exp_neg_Ioi _
```

-- snippet 8
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_exp_mul_Ioi (L78-L82)
theorem integrableOn_exp_mul_Ioi {a : ℝ} (ha : a < 0) (c : ℝ) :
    IntegrableOn (fun x : ℝ => Real.exp (a * x)) (Ioi c) := by
have := Integrable.norm <| integrableOn_exp_mul_complex_Ioi (a := a) (by simpa using ha) c
  simpa [Complex.norm_exp] using this
```

-- snippet 9
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_exp_mul_Iic (L83-L87)
theorem integrableOn_exp_mul_Iic {a : ℝ} (ha : 0 < a) (c : ℝ) :
    IntegrableOn (fun x : ℝ => Real.exp (a * x)) (Iic c) := by
have := Integrable.norm <| integrableOn_exp_mul_complex_Iic (a := a) (by simpa using ha) c
  simpa [Complex.norm_exp] using this
```

-- snippet 10
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_mul_complex_Ioi (L88-L96)
theorem integral_exp_mul_complex_Ioi {a : ℂ} (ha : a.re < 0) (c : ℝ) :
    ∫ x : ℝ in Set.Ioi c, Complex.exp (a * x) = - Complex.exp (a * c) / a := by
refine tendsto_nhds_unique (intervalIntegral_tendsto_integral_Ioi c
    (integrableOn_exp_mul_complex_Ioi ha c) tendsto_id) ?_
  simp_rw [integral_exp_mul_complex (c := a) (by aesop), id_eq]
  suffices Tendsto (fun x : ℝ ↦ Complex.exp (a * x)) atTop (𝓝 0) by
    simpa using this.sub_const _ |>.div_const _
  simpa [Complex.tendsto_exp_nhds_zero_iff] using tendsto_const_nhds.neg_mul_atTop ha tendsto_id
```

-- snippet 11
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_mul_complex_Iic (L97-L103)
theorem integral_exp_mul_complex_Iic {a : ℂ} (ha : 0 < a.re) (c : ℝ) :
    ∫ x : ℝ in Set.Iic c, Complex.exp (a * x) = Complex.exp (a * c) / a := by
simpa [neg_mul, ← mul_neg, ← Complex.ofReal_neg,
    integral_comp_neg_Ioi (f := fun x : ℝ ↦ Complex.exp (a * x))]
    using integral_exp_mul_complex_Ioi (a := -a) (by simpa) (-c)

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 12
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_mul_Ioi (L104-L110)
theorem integral_exp_mul_Ioi {a : ℝ} (ha : a < 0) (c : ℝ) :
    ∫ x : ℝ in Set.Ioi c, Real.exp (a * x) = - Real.exp (a * c) / a := by
simp_rw [Real.exp, ← RCLike.re_to_complex, Complex.ofReal_mul]
  rw [integral_re, integral_exp_mul_complex_Ioi (by simpa using ha), RCLike.re_to_complex,
    RCLike.re_to_complex, Complex.div_ofReal_re, Complex.neg_re]
  exact integrableOn_exp_mul_complex_Ioi (by simpa using ha) _
```

-- snippet 13
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_mul_Iic (L111-L116)
theorem integral_exp_mul_Iic {a : ℝ} (ha : 0 < a) (c : ℝ) :
    ∫ x : ℝ in Set.Iic c, Real.exp (a * x) = Real.exp (a * c) / a := by
simpa [neg_mul, ← mul_neg, integral_comp_neg_Ioi (f := fun x : ℝ ↦ Real.exp (a * x))]
    using integral_exp_mul_Ioi (a := -a) (by simpa) (-c)

/-- If `-m < c`, then `(fun t : ℝ ↦ (t + m) ^ a)` is integrable on `(c, ∞)` for all `a < -1`. -/
```

-- snippet 14
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_add_rpow_Ioi_of_lt (L117-L131)
theorem integrableOn_add_rpow_Ioi_of_lt {a c m : ℝ} (ha : a < -1) (hc : -m < c) :
    IntegrableOn (fun (x : ℝ) ↦ (x + m) ^ a) (Ioi c) := by
have hd : ∀ x ∈ Ici c, HasDerivAt (fun t ↦ (t + m) ^ (a + 1) / (a + 1)) ((x + m) ^ a) x := by
    intro x hx
    convert (((hasDerivAt_id _).add_const _).rpow_const _).div_const _ using 1
    · simp [show a + 1 ≠ 0 by linarith]
    left; linarith [mem_Ici.mp hx, id_eq x]
  have ht : Tendsto (fun t ↦ ((t + m) ^ (a + 1)) / (a + 1)) atTop (nhds (0 / (a + 1))) := by
    rw [← neg_neg (a + 1)]
    exact (tendsto_rpow_neg_atTop (by linarith)).comp
      (tendsto_atTop_add_const_right _ m tendsto_id) |>.div_const _
  exact integrableOn_Ioi_deriv_of_nonneg' hd
    (fun t ht ↦ rpow_nonneg (by linarith [mem_Ioi.mp ht]) a) ht

/-- If `0 < c`, then `(fun t : ℝ ↦ t ^ a)` is integrable on `(c, ∞)` for all `a < -1`. -/
```

-- snippet 15
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_Ioi_rpow_of_lt (L132-L135)
theorem integrableOn_Ioi_rpow_of_lt {a c : ℝ} (ha : a < -1) (hc : 0 < c) :
    IntegrableOn (fun t : ℝ ↦ t ^ a) (Ioi c) := by
simpa using integrableOn_add_rpow_Ioi_of_lt ha (by simpa : -0 < c)
```

-- snippet 16
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh (L61-L63)
def arcosh (x : ℝ) :=
log (x + √(x ^ 2 - 1))
```

-- snippet 17
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_zero (L69-L70)
theorem arcosh_zero : arcosh 1 = 0 := by simp [arcosh]
```

-- snippet 18
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.coshOpenPartialHomeomorph (L139-L152)
def coshOpenPartialHomeomorph : OpenPartialHomeomorph ℝ ℝ where
  toFun := cosh
invFun := arcosh
  source := Ioi 0
  target := Ioi 1
  map_source' _ hr := one_lt_cosh.mpr (ne_of_lt hr).symm
  map_target' _ hr := arcosh_pos hr
  left_inv' _ hr := arcosh_cosh (le_of_lt hr)
  right_inv' _ hr := cosh_arcosh (le_of_lt hr)
  open_source := isOpen_Ioi
  open_target := isOpen_Ioi
  continuousOn_toFun := by fun_prop
  continuousOn_invFun := continuousOn_arcosh.mono Ioi_subset_Ici_self
```

-- snippet 19
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.analyticAt_arcosh (L180-L183)
lemma analyticAt_arcosh {x : ℝ} (hx : x ∈ Ioi 1) : AnalyticAt ℝ arcosh x :=
(contDiffAt_arcosh hx).analyticAt

/-- The function `Real.arcosh` is real analytic. -/
```

-- snippet 20
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.continuousOn_arcosh (L133-L138)
theorem continuousOn_arcosh : ContinuousOn arcosh (Ici 1) :=
have {x : ℝ} (hx : x ∈ Ici 1) : 0 < x + √(x ^ 2 - 1) :=
    add_pos_of_pos_of_nonneg (show 0 < x by grind) (sqrt_nonneg _)
  continuousOn_log.comp (by fun_prop) (by grind [MapsTo])

/-- `Real.cosh` as an `OpenPartialHomeomorph` from $(0, ∞)$ to $(1, ∞)$. -/
```

-- snippet 21
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.exp_arcosh (L64-L67)
theorem exp_arcosh {x : ℝ} (hx : 1 ≤ x) : exp (arcosh x) = x + √(x ^ 2 - 1) := by
apply exp_log
  positivity
```

-- snippet 22
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.coshPartialEquiv (L123-L132)
def coshPartialEquiv : PartialEquiv ℝ ℝ where
  toFun := cosh
invFun := arcosh
  source := Ici 0
  target := Ici 1
  map_source' r _ := one_le_cosh r
  map_target' _ hr := arcosh_nonneg hr
  left_inv' _ hr := arcosh_cosh hr
  right_inv' _ hr := cosh_arcosh hr
```

-- snippet 23
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.contDiffAt_arcosh (L169-L174)
theorem contDiffAt_arcosh {n : WithTop ℕ∞} {x : ℝ} (hx : x ∈ Ioi 1) : ContDiffAt ℝ n arcosh x := by
refine coshOpenPartialHomeomorph.contDiffAt_symm_deriv ?_ hx (hasDerivAt_cosh _)
    contDiff_cosh.contDiffAt
  rw [ne_eq, sinh_eq_zero]
  exact (arcosh_pos hx).ne'
```

-- snippet 24
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.differentiableOn_arcosh (L166-L168)
theorem differentiableOn_arcosh : DifferentiableOn ℝ arcosh (Ioi 1) := fun _ hx =>
(differentiableAt_arcosh hx).differentiableWithinAt
```

-- snippet 25
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.analyticWithinAt_arcosh (L184-L188)
lemma analyticWithinAt_arcosh {s : Set ℝ} {x : ℝ} (hx : x ∈ Ioi 1) :
    AnalyticWithinAt ℝ arcosh s x :=
(contDiffAt_arcosh hx).contDiffWithinAt.analyticWithinAt

/-- The function `Real.arcosh` is real analytic. -/
```

-- snippet 26
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_pos (L103-L109)
theorem arcosh_pos {x : ℝ} (hx : 1 < x) : 0 < arcosh x := by
apply log_pos
  calc
    1 < x + 0 := by simpa
    _ ≤ x + √(x ^ 2 - 1) := by gcongr; positivity

/-- This holds for `Ioi 0` instead of only `Ici 1` due to junk values. -/
```

-- snippet 27
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.hasDerivAt_arcosh (L160-L162)
theorem hasDerivAt_arcosh {x : ℝ} (hx : x ∈ Ioi 1) : HasDerivAt arcosh (√(x ^ 2 - 1))⁻¹ x :=
(hasStrictDerivAt_arcosh hx).hasDerivAt
```

-- snippet 28
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_lt_arcosh (L119-L122)
theorem arcosh_lt_arcosh {x y : ℝ} (hx : 0 < x) (hy : 0 < y) : arcosh x < arcosh y ↔ x < y :=
strictMonoOn_arcosh.lt_iff_lt hx hy

/-- `Real.cosh` as a `PartialEquiv` from $[0, ∞)$ to $[1, ∞)$. -/
```

-- snippet 29
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_cosh (L93-L96)
theorem arcosh_cosh {x : ℝ} (hx : 0 ≤ x) : arcosh (cosh x) = x := by
rw [arcosh, ← exp_eq_exp, exp_log (by positivity), ← eq_sub_iff_add_eq', exp_sub_cosh,
    ← sq_eq_sq₀ (sqrt_nonneg _) (sinh_nonneg_iff.mpr hx), ← sinh_sq, sq_sqrt (pow_two_nonneg _)]
```

-- snippet 30
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_eq_zero_iff (L81-L84)
theorem arcosh_eq_zero_iff {x : ℝ} (hx : 1 ≤ x) : arcosh x = 0 ↔ x = 1 := by
rw [← exp_injective.eq_iff, exp_arcosh hx, exp_zero]
  grind
```

-- snippet 31
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_nonneg (L97-L102)
theorem arcosh_nonneg {x : ℝ} (hx : 1 ≤ x) : 0 ≤ arcosh x := by
apply log_nonneg
  calc
    1 ≤ x + 0 := by simpa
    _ ≤ x + √(x ^ 2 - 1) := by gcongr; positivity
```

-- snippet 32
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.tanh_arcosh (L89-L92)
theorem tanh_arcosh {x : ℝ} (hx : 1 ≤ x) : tanh (arcosh x) = √(x ^ 2 - 1) / x := by
rw [tanh_eq_sinh_div_cosh, sinh_arcosh hx, cosh_arcosh hx]

/-- `arcosh` is the left inverse of `cosh` over $[0, ∞)$. -/
```

-- snippet 33
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_le_arcosh (L115-L118)
theorem arcosh_le_arcosh {x y : ℝ} (hx : 0 < x) (hy : 0 < y) : arcosh x ≤ arcosh y ↔ x ≤ y :=
strictMonoOn_arcosh.le_iff_le hx hy

/-- This holds for `0 < x, y ≤ 1` due to junk values. -/
```

-- snippet 34
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.hasStrictDerivAt_arcosh (L153-L159)
theorem hasStrictDerivAt_arcosh {x : ℝ} (hx : x ∈ Ioi 1) :
    HasStrictDerivAt arcosh (√(x ^ 2 - 1))⁻¹ x := by
rw [← sinh_arcosh (le_of_lt hx)]
  refine coshOpenPartialHomeomorph.hasStrictDerivAt_symm hx ?_ (hasStrictDerivAt_cosh _)
  rw [ne_eq, sinh_eq_zero]
  exact ne_of_gt (arcosh_pos hx)
```

-- snippet 35
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.sinh_arcosh (L85-L88)
theorem sinh_arcosh {x : ℝ} (hx : 1 ≤ x) : sinh (arcosh x) = √(x ^ 2 - 1) := by
rw [arcosh, sinh_eq, exp_neg, exp_log (by positivity), add_sqrt_self_sq_sub_one_inv hx]
  ring
```

-- snippet 36
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.cosh_arcosh (L77-L80)
theorem cosh_arcosh {x : ℝ} (hx : 1 ≤ x) : cosh (arcosh x) = x := by
rw [arcosh, cosh_eq, exp_neg, exp_log (by positivity), add_sqrt_self_sq_sub_one_inv hx]
  ring
```

-- snippet 37
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.contDiffOn_arcosh (L175-L178)
theorem contDiffOn_arcosh {n : WithTop ℕ∞} : ContDiffOn ℝ n arcosh (Ioi 1) := fun _ hx =>
(contDiffAt_arcosh hx).contDiffWithinAt

/-- The function `Real.arcosh` is real analytic. -/
```

-- snippet 38
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.add_sqrt_self_sq_sub_one_inv (L71-L76)
lemma add_sqrt_self_sq_sub_one_inv {x : ℝ} (hx : 1 ≤ x) :
    (x + √(x ^ 2 - 1))⁻¹ = x - √(x ^ 2 - 1) := by
apply inv_eq_of_mul_eq_one_right
  rw [← pow_two_sub_pow_two, sq_sqrt (sub_nonneg_of_le (one_le_pow₀ hx)), sub_sub_cancel]

/-- `arcosh` is the right inverse of `cosh` over $[1, ∞)$. -/
```

-- snippet 39
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.differentiableAt_arcosh (L163-L165)
theorem differentiableAt_arcosh {x : ℝ} (hx : x ∈ Ioi 1) : DifferentiableAt ℝ arcosh x :=
(hasDerivAt_arcosh hx).differentiableAt
```

-- snippet 40
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.strictMonoOn_arcosh (L110-L114)
theorem strictMonoOn_arcosh : StrictMonoOn arcosh (Ioi 0) := by
refine strictMonoOn_log.comp ?_ fun x (hx : 0 < x) ↦ show 0 < x + √(x ^ 2 - 1) by positivity
  exact strictMonoOn_id.add_monotone fun x (hx : 0 < x) y (hy : 0 < y) hxy ↦ by gcongr

/-- This holds for `0 < x, y ≤ 1` due to junk values. -/
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.SpecialFunctions.ImproperIntegrals).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem integrableOn_exp_mul_complex_Iic {a : ℂ} (ha : 0 < a.re) (c : ℝ) :
    IntegrableOn (fun x : ℝ => Complex.exp (a * x)) (Iic c)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_exp_Iic (L37-L44)
theorem integrableOn_exp_Iic (c : ℝ) : IntegrableOn exp (Iic c) := by
refine
    integrableOn_Iic_of_intervalIntegral_norm_bounded (exp c) c
      (fun y => intervalIntegrable_exp.1) tendsto_id
      (eventually_of_mem (Iic_mem_atBot 0) fun y _ => ?_)
  simp_rw [norm_of_nonneg (exp_pos _).le, integral_exp, sub_le_self_iff]
  exact (exp_pos _).le
```

-- snippet 2
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_exp_neg_Ioi (L45-L47)
theorem integrableOn_exp_neg_Ioi (c : ℝ) : IntegrableOn (fun (x : ℝ) => exp (-x)) (Ioi c) :=
Iff.mp integrableOn_Ici_iff_integrableOn_Ioi (integrableOn_exp_Iic (-c)).comp_neg_Ici
```

-- snippet 3
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_Iic (L48-L54)
theorem integral_exp_Iic (c : ℝ) : ∫ x : ℝ in Iic c, exp x = exp c := by
refine
    tendsto_nhds_unique
      (intervalIntegral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) ?_
  simp_rw [integral_exp, show 𝓝 (exp c) = 𝓝 (exp c - 0) by rw [sub_zero]]
  exact tendsto_exp_atBot.const_sub _
```

-- snippet 4
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_Iic_zero (L55-L57)
theorem integral_exp_Iic_zero : ∫ x : ℝ in Iic 0, exp x = 1 :=
exp_zero ▸ integral_exp_Iic 0
```

-- snippet 5
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_neg_Ioi (L58-L60)
theorem integral_exp_neg_Ioi (c : ℝ) : (∫ x : ℝ in Ioi c, exp (-x)) = exp (-c) := by
simpa only [integral_comp_neg_Ioi] using integral_exp_Iic (-c)
```

-- snippet 6
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_neg_Ioi_zero (L61-L63)
theorem integral_exp_neg_Ioi_zero : (∫ x : ℝ in Ioi 0, exp (-x)) = 1 := by
simpa only [neg_zero, exp_zero] using integral_exp_neg_Ioi 0
```

-- snippet 7
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_exp_mul_complex_Ioi (L64-L72)
theorem integrableOn_exp_mul_complex_Ioi {a : ℂ} (ha : a.re < 0) (c : ℝ) :
    IntegrableOn (fun x : ℝ => Complex.exp (a * x)) (Ioi c) := by
refine (integrable_norm_iff ?_).mp ?_
  · apply Continuous.aestronglyMeasurable
    fun_prop
  · simpa [Complex.norm_exp] using
      (integrableOn_Ioi_comp_mul_left_iff (fun x => exp (-x)) c (a := -a.re) (by simpa)).mpr <|
        integrableOn_exp_neg_Ioi _
```

-- snippet 8
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_exp_mul_Ioi (L78-L82)
theorem integrableOn_exp_mul_Ioi {a : ℝ} (ha : a < 0) (c : ℝ) :
    IntegrableOn (fun x : ℝ => Real.exp (a * x)) (Ioi c) := by
have := Integrable.norm <| integrableOn_exp_mul_complex_Ioi (a := a) (by simpa using ha) c
  simpa [Complex.norm_exp] using this
```

-- snippet 9
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_exp_mul_Iic (L83-L87)
theorem integrableOn_exp_mul_Iic {a : ℝ} (ha : 0 < a) (c : ℝ) :
    IntegrableOn (fun x : ℝ => Real.exp (a * x)) (Iic c) := by
have := Integrable.norm <| integrableOn_exp_mul_complex_Iic (a := a) (by simpa using ha) c
  simpa [Complex.norm_exp] using this
```

-- snippet 10
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_mul_complex_Ioi (L88-L96)
theorem integral_exp_mul_complex_Ioi {a : ℂ} (ha : a.re < 0) (c : ℝ) :
    ∫ x : ℝ in Set.Ioi c, Complex.exp (a * x) = - Complex.exp (a * c) / a := by
refine tendsto_nhds_unique (intervalIntegral_tendsto_integral_Ioi c
    (integrableOn_exp_mul_complex_Ioi ha c) tendsto_id) ?_
  simp_rw [integral_exp_mul_complex (c := a) (by aesop), id_eq]
  suffices Tendsto (fun x : ℝ ↦ Complex.exp (a * x)) atTop (𝓝 0) by
    simpa using this.sub_const _ |>.div_const _
  simpa [Complex.tendsto_exp_nhds_zero_iff] using tendsto_const_nhds.neg_mul_atTop ha tendsto_id
```

-- snippet 11
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_mul_complex_Iic (L97-L103)
theorem integral_exp_mul_complex_Iic {a : ℂ} (ha : 0 < a.re) (c : ℝ) :
    ∫ x : ℝ in Set.Iic c, Complex.exp (a * x) = Complex.exp (a * c) / a := by
simpa [neg_mul, ← mul_neg, ← Complex.ofReal_neg,
    integral_comp_neg_Ioi (f := fun x : ℝ ↦ Complex.exp (a * x))]
    using integral_exp_mul_complex_Ioi (a := -a) (by simpa) (-c)

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 12
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_mul_Ioi (L104-L110)
theorem integral_exp_mul_Ioi {a : ℝ} (ha : a < 0) (c : ℝ) :
    ∫ x : ℝ in Set.Ioi c, Real.exp (a * x) = - Real.exp (a * c) / a := by
simp_rw [Real.exp, ← RCLike.re_to_complex, Complex.ofReal_mul]
  rw [integral_re, integral_exp_mul_complex_Ioi (by simpa using ha), RCLike.re_to_complex,
    RCLike.re_to_complex, Complex.div_ofReal_re, Complex.neg_re]
  exact integrableOn_exp_mul_complex_Ioi (by simpa using ha) _
```

-- snippet 13
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integral_exp_mul_Iic (L111-L116)
theorem integral_exp_mul_Iic {a : ℝ} (ha : 0 < a) (c : ℝ) :
    ∫ x : ℝ in Set.Iic c, Real.exp (a * x) = Real.exp (a * c) / a := by
simpa [neg_mul, ← mul_neg, integral_comp_neg_Ioi (f := fun x : ℝ ↦ Real.exp (a * x))]
    using integral_exp_mul_Ioi (a := -a) (by simpa) (-c)

/-- If `-m < c`, then `(fun t : ℝ ↦ (t + m) ^ a)` is integrable on `(c, ∞)` for all `a < -1`. -/
```

-- snippet 14
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_add_rpow_Ioi_of_lt (L117-L131)
theorem integrableOn_add_rpow_Ioi_of_lt {a c m : ℝ} (ha : a < -1) (hc : -m < c) :
    IntegrableOn (fun (x : ℝ) ↦ (x + m) ^ a) (Ioi c) := by
have hd : ∀ x ∈ Ici c, HasDerivAt (fun t ↦ (t + m) ^ (a + 1) / (a + 1)) ((x + m) ^ a) x := by
    intro x hx
    convert (((hasDerivAt_id _).add_const _).rpow_const _).div_const _ using 1
    · simp [show a + 1 ≠ 0 by linarith]
    left; linarith [mem_Ici.mp hx, id_eq x]
  have ht : Tendsto (fun t ↦ ((t + m) ^ (a + 1)) / (a + 1)) atTop (nhds (0 / (a + 1))) := by
    rw [← neg_neg (a + 1)]
    exact (tendsto_rpow_neg_atTop (by linarith)).comp
      (tendsto_atTop_add_const_right _ m tendsto_id) |>.div_const _
  exact integrableOn_Ioi_deriv_of_nonneg' hd
    (fun t ht ↦ rpow_nonneg (by linarith [mem_Ioi.mp ht]) a) ht

/-- If `0 < c`, then `(fun t : ℝ ↦ t ^ a)` is integrable on `(c, ∞)` for all `a < -1`. -/
```

-- snippet 15
```lean
-- Mathlib.Analysis.SpecialFunctions.ImproperIntegrals.integrableOn_Ioi_rpow_of_lt (L132-L135)
theorem integrableOn_Ioi_rpow_of_lt {a c : ℝ} (ha : a < -1) (hc : 0 < c) :
    IntegrableOn (fun t : ℝ ↦ t ^ a) (Ioi c) := by
simpa using integrableOn_add_rpow_Ioi_of_lt ha (by simpa : -0 < c)
```

-- snippet 16
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh (L61-L63)
def arcosh (x : ℝ) :=
log (x + √(x ^ 2 - 1))
```

-- snippet 17
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_zero (L69-L70)
theorem arcosh_zero : arcosh 1 = 0 := by simp [arcosh]
```

-- snippet 18
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.coshOpenPartialHomeomorph (L139-L152)
def coshOpenPartialHomeomorph : OpenPartialHomeomorph ℝ ℝ where
  toFun := cosh
invFun := arcosh
  source := Ioi 0
  target := Ioi 1
  map_source' _ hr := one_lt_cosh.mpr (ne_of_lt hr).symm
  map_target' _ hr := arcosh_pos hr
  left_inv' _ hr := arcosh_cosh (le_of_lt hr)
  right_inv' _ hr := cosh_arcosh (le_of_lt hr)
  open_source := isOpen_Ioi
  open_target := isOpen_Ioi
  continuousOn_toFun := by fun_prop
  continuousOn_invFun := continuousOn_arcosh.mono Ioi_subset_Ici_self
```

-- snippet 19
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.analyticAt_arcosh (L180-L183)
lemma analyticAt_arcosh {x : ℝ} (hx : x ∈ Ioi 1) : AnalyticAt ℝ arcosh x :=
(contDiffAt_arcosh hx).analyticAt

/-- The function `Real.arcosh` is real analytic. -/
```

-- snippet 20
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.continuousOn_arcosh (L133-L138)
theorem continuousOn_arcosh : ContinuousOn arcosh (Ici 1) :=
have {x : ℝ} (hx : x ∈ Ici 1) : 0 < x + √(x ^ 2 - 1) :=
    add_pos_of_pos_of_nonneg (show 0 < x by grind) (sqrt_nonneg _)
  continuousOn_log.comp (by fun_prop) (by grind [MapsTo])

/-- `Real.cosh` as an `OpenPartialHomeomorph` from $(0, ∞)$ to $(1, ∞)$. -/
```

-- snippet 21
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.exp_arcosh (L64-L67)
theorem exp_arcosh {x : ℝ} (hx : 1 ≤ x) : exp (arcosh x) = x + √(x ^ 2 - 1) := by
apply exp_log
  positivity
```

-- snippet 22
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.coshPartialEquiv (L123-L132)
def coshPartialEquiv : PartialEquiv ℝ ℝ where
  toFun := cosh
invFun := arcosh
  source := Ici 0
  target := Ici 1
  map_source' r _ := one_le_cosh r
  map_target' _ hr := arcosh_nonneg hr
  left_inv' _ hr := arcosh_cosh hr
  right_inv' _ hr := cosh_arcosh hr
```

-- snippet 23
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.contDiffAt_arcosh (L169-L174)
theorem contDiffAt_arcosh {n : WithTop ℕ∞} {x : ℝ} (hx : x ∈ Ioi 1) : ContDiffAt ℝ n arcosh x := by
refine coshOpenPartialHomeomorph.contDiffAt_symm_deriv ?_ hx (hasDerivAt_cosh _)
    contDiff_cosh.contDiffAt
  rw [ne_eq, sinh_eq_zero]
  exact (arcosh_pos hx).ne'
```

-- snippet 24
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.differentiableOn_arcosh (L166-L168)
theorem differentiableOn_arcosh : DifferentiableOn ℝ arcosh (Ioi 1) := fun _ hx =>
(differentiableAt_arcosh hx).differentiableWithinAt
```

-- snippet 25
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.analyticWithinAt_arcosh (L184-L188)
lemma analyticWithinAt_arcosh {s : Set ℝ} {x : ℝ} (hx : x ∈ Ioi 1) :
    AnalyticWithinAt ℝ arcosh s x :=
(contDiffAt_arcosh hx).contDiffWithinAt.analyticWithinAt

/-- The function `Real.arcosh` is real analytic. -/
```

-- snippet 26
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_pos (L103-L109)
theorem arcosh_pos {x : ℝ} (hx : 1 < x) : 0 < arcosh x := by
apply log_pos
  calc
    1 < x + 0 := by simpa
    _ ≤ x + √(x ^ 2 - 1) := by gcongr; positivity

/-- This holds for `Ioi 0` instead of only `Ici 1` due to junk values. -/
```

-- snippet 27
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.hasDerivAt_arcosh (L160-L162)
theorem hasDerivAt_arcosh {x : ℝ} (hx : x ∈ Ioi 1) : HasDerivAt arcosh (√(x ^ 2 - 1))⁻¹ x :=
(hasStrictDerivAt_arcosh hx).hasDerivAt
```

-- snippet 28
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_lt_arcosh (L119-L122)
theorem arcosh_lt_arcosh {x y : ℝ} (hx : 0 < x) (hy : 0 < y) : arcosh x < arcosh y ↔ x < y :=
strictMonoOn_arcosh.lt_iff_lt hx hy

/-- `Real.cosh` as a `PartialEquiv` from $[0, ∞)$ to $[1, ∞)$. -/
```

-- snippet 29
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_cosh (L93-L96)
theorem arcosh_cosh {x : ℝ} (hx : 0 ≤ x) : arcosh (cosh x) = x := by
rw [arcosh, ← exp_eq_exp, exp_log (by positivity), ← eq_sub_iff_add_eq', exp_sub_cosh,
    ← sq_eq_sq₀ (sqrt_nonneg _) (sinh_nonneg_iff.mpr hx), ← sinh_sq, sq_sqrt (pow_two_nonneg _)]
```

-- snippet 30
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_eq_zero_iff (L81-L84)
theorem arcosh_eq_zero_iff {x : ℝ} (hx : 1 ≤ x) : arcosh x = 0 ↔ x = 1 := by
rw [← exp_injective.eq_iff, exp_arcosh hx, exp_zero]
  grind
```

-- snippet 31
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_nonneg (L97-L102)
theorem arcosh_nonneg {x : ℝ} (hx : 1 ≤ x) : 0 ≤ arcosh x := by
apply log_nonneg
  calc
    1 ≤ x + 0 := by simpa
    _ ≤ x + √(x ^ 2 - 1) := by gcongr; positivity
```

-- snippet 32
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.tanh_arcosh (L89-L92)
theorem tanh_arcosh {x : ℝ} (hx : 1 ≤ x) : tanh (arcosh x) = √(x ^ 2 - 1) / x := by
rw [tanh_eq_sinh_div_cosh, sinh_arcosh hx, cosh_arcosh hx]

/-- `arcosh` is the left inverse of `cosh` over $[0, ∞)$. -/
```

-- snippet 33
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.arcosh_le_arcosh (L115-L118)
theorem arcosh_le_arcosh {x y : ℝ} (hx : 0 < x) (hy : 0 < y) : arcosh x ≤ arcosh y ↔ x ≤ y :=
strictMonoOn_arcosh.le_iff_le hx hy

/-- This holds for `0 < x, y ≤ 1` due to junk values. -/
```

-- snippet 34
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.hasStrictDerivAt_arcosh (L153-L159)
theorem hasStrictDerivAt_arcosh {x : ℝ} (hx : x ∈ Ioi 1) :
    HasStrictDerivAt arcosh (√(x ^ 2 - 1))⁻¹ x := by
rw [← sinh_arcosh (le_of_lt hx)]
  refine coshOpenPartialHomeomorph.hasStrictDerivAt_symm hx ?_ (hasStrictDerivAt_cosh _)
  rw [ne_eq, sinh_eq_zero]
  exact ne_of_gt (arcosh_pos hx)
```

-- snippet 35
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.sinh_arcosh (L85-L88)
theorem sinh_arcosh {x : ℝ} (hx : 1 ≤ x) : sinh (arcosh x) = √(x ^ 2 - 1) := by
rw [arcosh, sinh_eq, exp_neg, exp_log (by positivity), add_sqrt_self_sq_sub_one_inv hx]
  ring
```

-- snippet 36
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.cosh_arcosh (L77-L80)
theorem cosh_arcosh {x : ℝ} (hx : 1 ≤ x) : cosh (arcosh x) = x := by
rw [arcosh, cosh_eq, exp_neg, exp_log (by positivity), add_sqrt_self_sq_sub_one_inv hx]
  ring
```

-- snippet 37
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.contDiffOn_arcosh (L175-L178)
theorem contDiffOn_arcosh {n : WithTop ℕ∞} : ContDiffOn ℝ n arcosh (Ioi 1) := fun _ hx =>
(contDiffAt_arcosh hx).contDiffWithinAt

/-- The function `Real.arcosh` is real analytic. -/
```

-- snippet 38
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.add_sqrt_self_sq_sub_one_inv (L71-L76)
lemma add_sqrt_self_sq_sub_one_inv {x : ℝ} (hx : 1 ≤ x) :
    (x + √(x ^ 2 - 1))⁻¹ = x - √(x ^ 2 - 1) := by
apply inv_eq_of_mul_eq_one_right
  rw [← pow_two_sub_pow_two, sq_sqrt (sub_nonneg_of_le (one_le_pow₀ hx)), sub_sub_cancel]

/-- `arcosh` is the right inverse of `cosh` over $[1, ∞)$. -/
```

-- snippet 39
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.differentiableAt_arcosh (L163-L165)
theorem differentiableAt_arcosh {x : ℝ} (hx : x ∈ Ioi 1) : DifferentiableAt ℝ arcosh x :=
(hasDerivAt_arcosh hx).differentiableAt
```

-- snippet 40
```lean
-- Mathlib.Analysis.SpecialFunctions.Arcosh.strictMonoOn_arcosh (L110-L114)
theorem strictMonoOn_arcosh : StrictMonoOn arcosh (Ioi 0) := by
refine strictMonoOn_log.comp ?_ fun x (hx : 0 < x) ↦ show 0 < x + √(x ^ 2 - 1) by positivity
  exact strictMonoOn_id.add_monotone fun x (hx : 0 < x) y (hy : 0 < y) hxy ↦ by gcongr

/-- This holds for `0 < x, y ≤ 1` due to junk values. -/
```
[/INST]


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

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

```text
Proof-state transcript for `Mathlib.ModelTheory.Definability.TermDefinable₁`:
State 0: ⊢ theorem TermDefinable₁.definable₂_graph {f : M → M} (h : A.TermDefinable₁ L f) :
    A.Definable₂ L f.graph
Tactic: obtain ⟨t, h⟩ := h.termDefinable.definable_tupleGraph A L
State 1: ⊢ <goal after obtain ⟨t, h⟩ := h.termDefinable.definable_tupleGraph A L>
Tactic: use t.relabel (Option.elim · 1 (fun _ ↦ 0))
State 2: ⊢ <goal after use t.relabel (Option.elim · 1 (fun _ ↦ 0))>
Tactic: ext v
State 3: ⊢ <goal after ext v>
Tactic: convert Set.ext_iff.1 h (v ∘ (Option.elim · 1 (fun _ ↦ 0)))
State 4: ⊢ <goal after convert Set.ext_iff.1 h (v ∘ (Option.elim · 1 (fun _ ↦ 0)))>
Tactic: simp  /-- The identity function is `TermDefinable₁` -/
```

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

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

```text
Proof-state transcript for `Mathlib.ModelTheory.Definability.TermDefinable₁`:
State 0: ⊢ theorem TermDefinable₁.definable₂_graph {f : M → M} (h : A.TermDefinable₁ L f) :
    A.Definable₂ L f.graph
Tactic: obtain ⟨t, h⟩ := h.termDefinable.definable_tupleGraph A L
State 1: ⊢ <goal after obtain ⟨t, h⟩ := h.termDefinable.definable_tupleGraph A L>
Tactic: use t.relabel (Option.elim · 1 (fun _ ↦ 0))
State 2: ⊢ <goal after use t.relabel (Option.elim · 1 (fun _ ↦ 0))>
Tactic: ext v
State 3: ⊢ <goal after ext v>
Tactic: convert Set.ext_iff.1 h (v ∘ (Option.elim · 1 (fun _ ↦ 0)))
State 4: ⊢ <goal after convert Set.ext_iff.1 h (v ∘ (Option.elim · 1 (fun _ ↦ 0)))>
Tactic: simp  /-- The identity function is `TermDefinable₁` -/
```

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


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

[User]
The following Lean code fails to compile. Fix it.

```lean
theorem mem_carrier {s : NonUnitalSubring R} {x : R} : x ∈ s.toNonUnitalSubsemiring ↔ x ∈ s
:= .someMissingNamespaceXYZ
Iff.rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=133, messages=[Message(data='failed to synthesize instance of type class\n  NonUnitalNonAssocRing R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=25), pos_end=Position(line=1, column=43), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.NonUnitalSubring.Defs).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem mem_carrier {s : NonUnitalSubring R} {x : R} : x ∈ s.toNonUnitalSubsemiring ↔ x ∈ s
:= .someMissingNamespaceXYZ
Iff.rfl
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=133, messages=[Message(data='failed to synthesize instance of type class\n  NonUnitalNonAssocRing R\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=25), pos_end=Position(line=1, column=43), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.[/INST]


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

[User]
Complete the following proof.

```lean
theorem monic_charpoly (b : B) : (charpoly G b).Monic
```

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.Invariant.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem monic_charpoly (b : B) : (charpoly G b).Monic
```

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. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.Linarith.Oracle.FourierMotzkin).

[User]
The following Lean code fails to compile. Fix it.

```lean
def getPCompSet : LinarithM PCompSet
:= ⟨⟩⟨⟩
LinarithData.comps <$> get

/-- Throws an exception if a contradictory `PComp` is contained in the current state. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=46, messages=[Message(data="Function expected at\n  LinarithM\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  PCompSet\n\nHint: The identifier `LinarithM` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=18), pos_end=Position(line=1, column=36), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=46, i_end=169, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=2, column=5), pos_end=Position(line=2, column=6), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=5, column=88), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.Linarith.Oracle.FourierMotzkin).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def getPCompSet : LinarithM PCompSet
:= ⟨⟩⟨⟩
LinarithData.comps <$> get

/-- Throws an exception if a contradictory `PComp` is contained in the current state. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=46, messages=[Message(data="Function expected at\n  LinarithM\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  PCompSet\n\nHint: The identifier `LinarithM` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=18), pos_end=Position(line=1, column=36), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=46, i_end=169, messages=[Message(data="unexpected token '⟨'; expected command", pos=Position(line=2, column=5), pos_end=Position(line=2, column=6), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=5, column=88), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.[/INST]


Instruction: Edit `isoCycles₁_hom_comp_i` so it still compiles and the proof is simplified where possible. Return the full updated `isoCycles₁_hom_comp_i` block.

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

[User]
You are editing a file in `Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree.isoCycles₁_hom_comp_i`, L755-L759)
```lean
lemma isoCycles₁_hom_comp_i :
    (isoCycles₁ A).hom ≫ (shortComplexH1 A).moduleCatLeftHomologyData.i =
      iCycles A 1 ≫ (chainsIso₁ A).hom := by
simp [isoCycles₁, iCycles, HomologicalComplex.iCycles, ShortComplex.iCycles]
```

## Edit instruction
Edit `isoCycles₁_hom_comp_i` so it still compiles and the proof is simplified where possible. Return the full updated `isoCycles₁_hom_comp_i` block.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree.isoCycles₁_hom_comp_i`, L755-L759)
```lean
lemma isoCycles₁_hom_comp_i :
    (isoCycles₁ A).hom ≫ (shortComplexH1 A).moduleCatLeftHomologyData.i =
      iCycles A 1 ≫ (chainsIso₁ A).hom := by
simp [isoCycles₁, iCycles, HomologicalComplex.iCycles, ShortComplex.iCycles]
```

## Edit instruction
Edit `isoCycles₁_hom_comp_i` so it still compiles and the proof is simplified where possible. Return the full updated `isoCycles₁_hom_comp_i` block.
[/INST]