[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.Computability.Language).

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

```lean
instance instSemiring : Semiring (Language α) where
  add_assoc
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Computability.Language.Language (L65-L70)
def Language (α) :=
Set (List α)
deriving CompleteAtomicBooleanAlgebra

namespace Language
```

-- snippet 2
```lean
-- Mathlib.Computability.Language.zero_def (L100-L102)
theorem zero_def : (0 : Language α) = (∅ : Set _) :=
rfl
```

-- snippet 3
```lean
-- Mathlib.Computability.Language.one_def (L103-L105)
theorem one_def : (1 : Language α) = ({[]} : Set (List α)) :=
rfl
```

-- snippet 4
```lean
-- Mathlib.Computability.Language.add_def (L106-L108)
theorem add_def (l m : Language α) : l + m = (l ∪ m : Set (List α)) :=
rfl
```

-- snippet 5
```lean
-- Mathlib.Computability.Language.sub_def (L109-L111)
theorem sub_def (l m : Language α) : l - m = (l \ m : Set (List α)) :=
rfl
```

-- snippet 6
```lean
-- Mathlib.Computability.Language.mul_def (L112-L116)
theorem mul_def (l m : Language α) : l * m = image2 (· ++ ·) l m :=
rfl

/-- The Kleene star of a language `L` is the set of all strings which can be written by
concatenating strings from `L`. -/
```

-- snippet 7
```lean
-- Mathlib.Computability.Language.kstar_def (L119-L121)
lemma kstar_def (l : Language α) : l∗ = {x | ∃ L : List (List α), x = L.flatten ∧ ∀ y ∈ L, y ∈ l} :=
rfl
```

-- snippet 8
```lean
-- Mathlib.Computability.Language.ext (L123-L125)
theorem ext {l m : Language α} (h : ∀ (x : List α), x ∈ l ↔ x ∈ m) : l = m :=
Set.ext h
```

-- snippet 9
```lean
-- Mathlib.Computability.Language.notMem_zero (L127-L129)
theorem notMem_zero (x : List α) : x ∉ (0 : Language α) :=
id
```

-- snippet 10
```lean
-- Mathlib.Computability.Language.mem_one (L131-L132)
theorem mem_one (x : List α) : x ∈ (1 : Language α) ↔ x = [] := by rfl
```

-- snippet 11
```lean
-- Mathlib.Computability.Language.nil_mem_one (L133-L135)
theorem nil_mem_one : [] ∈ (1 : Language α) :=
Set.mem_singleton _
```

-- snippet 12
```lean
-- Mathlib.Computability.Language.mem_add (L136-L138)
theorem mem_add (l m : Language α) (x : List α) : x ∈ l + m ↔ x ∈ l ∨ x ∈ m :=
Iff.rfl
```

-- snippet 13
```lean
-- Mathlib.Computability.Language.mem_sub (L139-L141)
theorem mem_sub (l m : Language α) (x : List α) : x ∈ l - m ↔ x ∈ l ∧ x ∉ m :=
Iff.rfl
```

-- snippet 14
```lean
-- Mathlib.Computability.Language.mem_mul (L142-L144)
theorem mem_mul : x ∈ l * m ↔ ∃ a ∈ l, ∃ b ∈ m, a ++ b = x :=
mem_image2
```

-- snippet 15
```lean
-- Mathlib.Computability.Language.append_mem_mul (L145-L147)
theorem append_mem_mul : a ∈ l → b ∈ m → a ++ b ∈ l * m :=
mem_image2_of_mem
```

<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.Computability.Language).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
instance instSemiring : Semiring (Language α) where
  add_assoc
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Computability.Language.Language (L65-L70)
def Language (α) :=
Set (List α)
deriving CompleteAtomicBooleanAlgebra

namespace Language
```

-- snippet 2
```lean
-- Mathlib.Computability.Language.zero_def (L100-L102)
theorem zero_def : (0 : Language α) = (∅ : Set _) :=
rfl
```

-- snippet 3
```lean
-- Mathlib.Computability.Language.one_def (L103-L105)
theorem one_def : (1 : Language α) = ({[]} : Set (List α)) :=
rfl
```

-- snippet 4
```lean
-- Mathlib.Computability.Language.add_def (L106-L108)
theorem add_def (l m : Language α) : l + m = (l ∪ m : Set (List α)) :=
rfl
```

-- snippet 5
```lean
-- Mathlib.Computability.Language.sub_def (L109-L111)
theorem sub_def (l m : Language α) : l - m = (l \ m : Set (List α)) :=
rfl
```

-- snippet 6
```lean
-- Mathlib.Computability.Language.mul_def (L112-L116)
theorem mul_def (l m : Language α) : l * m = image2 (· ++ ·) l m :=
rfl

/-- The Kleene star of a language `L` is the set of all strings which can be written by
concatenating strings from `L`. -/
```

-- snippet 7
```lean
-- Mathlib.Computability.Language.kstar_def (L119-L121)
lemma kstar_def (l : Language α) : l∗ = {x | ∃ L : List (List α), x = L.flatten ∧ ∀ y ∈ L, y ∈ l} :=
rfl
```

-- snippet 8
```lean
-- Mathlib.Computability.Language.ext (L123-L125)
theorem ext {l m : Language α} (h : ∀ (x : List α), x ∈ l ↔ x ∈ m) : l = m :=
Set.ext h
```

-- snippet 9
```lean
-- Mathlib.Computability.Language.notMem_zero (L127-L129)
theorem notMem_zero (x : List α) : x ∉ (0 : Language α) :=
id
```

-- snippet 10
```lean
-- Mathlib.Computability.Language.mem_one (L131-L132)
theorem mem_one (x : List α) : x ∈ (1 : Language α) ↔ x = [] := by rfl
```

-- snippet 11
```lean
-- Mathlib.Computability.Language.nil_mem_one (L133-L135)
theorem nil_mem_one : [] ∈ (1 : Language α) :=
Set.mem_singleton _
```

-- snippet 12
```lean
-- Mathlib.Computability.Language.mem_add (L136-L138)
theorem mem_add (l m : Language α) (x : List α) : x ∈ l + m ↔ x ∈ l ∨ x ∈ m :=
Iff.rfl
```

-- snippet 13
```lean
-- Mathlib.Computability.Language.mem_sub (L139-L141)
theorem mem_sub (l m : Language α) (x : List α) : x ∈ l - m ↔ x ∈ l ∧ x ∉ m :=
Iff.rfl
```

-- snippet 14
```lean
-- Mathlib.Computability.Language.mem_mul (L142-L144)
theorem mem_mul : x ∈ l * m ↔ ∃ a ∈ l, ∃ b ∈ m, a ++ b = x :=
mem_image2
```

-- snippet 15
```lean
-- Mathlib.Computability.Language.append_mem_mul (L145-L147)
theorem append_mem_mul : a ∈ l → b ∈ m → a ++ b ∈ l * m :=
mem_image2_of_mem
```
[/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.Computability.Partrec).

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

```lean
theorem option_some_iff {f : α → σ} : (Computable fun a => Option.some (f a)) ↔ Computable f
:= .someMissingNamespaceXYZ
⟨fun h => encode_iff.1 <| Primrec.pred.to_comp.comp <| encode_iff.2 h, option_some.comp⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=219, messages=[Message(data='failed to synthesize instance of type class\n  Primcodable α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=39), pos_end=Position(line=1, column=76), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Primcodable α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=80), pos_end=Position(line=1, column=92), 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.Computability.Partrec).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem option_some_iff {f : α → σ} : (Computable fun a => Option.some (f a)) ↔ Computable f
:= .someMissingNamespaceXYZ
⟨fun h => encode_iff.1 <| Primrec.pred.to_comp.comp <| encode_iff.2 h, option_some.comp⟩
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=219, messages=[Message(data='failed to synthesize instance of type class\n  Primcodable α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=39), pos_end=Position(line=1, column=76), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='failed to synthesize instance of type class\n  Primcodable α\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=80), pos_end=Position(line=1, column=92), 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. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Control.LawfulFix).

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

```lean
def monotoneCurry [(x y : _) → Preorder <| γ x y] :
    (∀ x : Σ a, β a, γ x.1 x.2) →o ∀ (a) (b : β a), γ a b where
  toFun
:= curry .someMissingNamespaceXYZ
monotone' _x _y h a b := h ⟨a, b⟩

/-- `Sigma.uncurry` as a monotone function. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=172, messages=[Message(data="Function expected at\n  γ\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `γ` 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=43), pos_end=Position(line=1, column=48), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  β\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `β` 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=2, column=16), pos_end=Position(line=2, column=19), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  γ\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  x.1\n\nHint: The identifier `γ` 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=2, column=21), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  β\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `β` 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=2, column=46), pos_end=Position(line=2, column=49), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  γ\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `γ` 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=2, column=52), pos_end=Position(line=2, column=57), severity=<Severity.ERROR: 3>, kind=None), Message(data='stuck at solving universe constraint\n  imax ?u.74 ?u.77 ?u.80 =?= ?u.57+1\nwhile trying to unify\n  Sort (imax ?u.74 ?u.77 ?u.80) : Type (imax ?u.74 ?u.77 ?u.80)\nwith\n  Type ?u.57 : Type (?u.57 + 1)', pos=Position(line=1, column=0), pos_end=Position(line=2, column=57), severity=<Severity.ERROR: 3>, kind=None), Message(data='stuck at solving universe constraint\n  imax (max (?u.60+1) (?u.61+1)) ?u.70 =?= ?u.58+1\nwhile trying to unify\n  Sort (imax (max (?u.60 + 1) (?u.61 + 1)) ?u.70) : Type (imax (max (?u.60 + 1) (?u.61 + 1)) ?u.70)\nwith\n  Type ?u.58 : Type (?u.58 + 1)', pos=Position(line=1, column=0), pos_end=Position(line=2, column=57), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=172, i_end=257, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=5, column=0), pos_end=Position(line=5, column=9), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, column=46), 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.Control.LawfulFix).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def monotoneCurry [(x y : _) → Preorder <| γ x y] :
    (∀ x : Σ a, β a, γ x.1 x.2) →o ∀ (a) (b : β a), γ a b where
  toFun
:= curry .someMissingNamespaceXYZ
monotone' _x _y h a b := h ⟨a, b⟩

/-- `Sigma.uncurry` as a monotone function. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=172, messages=[Message(data="Function expected at\n  γ\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  x\n\nHint: The identifier `γ` 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=43), pos_end=Position(line=1, column=48), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  β\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `β` 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=2, column=16), pos_end=Position(line=2, column=19), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  γ\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  x.1\n\nHint: The identifier `γ` 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=2, column=21), pos_end=Position(line=2, column=30), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  β\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `β` 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=2, column=46), pos_end=Position(line=2, column=49), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  γ\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  a\n\nHint: The identifier `γ` 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=2, column=52), pos_end=Position(line=2, column=57), severity=<Severity.ERROR: 3>, kind=None), Message(data='stuck at solving universe constraint\n  imax ?u.74 ?u.77 ?u.80 =?= ?u.57+1\nwhile trying to unify\n  Sort (imax ?u.74 ?u.77 ?u.80) : Type (imax ?u.74 ?u.77 ?u.80)\nwith\n  Type ?u.57 : Type (?u.57 + 1)', pos=Position(line=1, column=0), pos_end=Position(line=2, column=57), severity=<Severity.ERROR: 3>, kind=None), Message(data='stuck at solving universe constraint\n  imax (max (?u.60+1) (?u.61+1)) ?u.70 =?= ?u.58+1\nwhile trying to unify\n  Sort (imax (max (?u.60 + 1) (?u.61 + 1)) ?u.70) : Type (imax (max (?u.60 + 1) (?u.61 + 1)) ?u.70)\nwith\n  Type ?u.58 : Type (?u.58 + 1)', pos=Position(line=1, column=0), pos_end=Position(line=2, column=57), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=172, i_end=257, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=5, column=0), pos_end=Position(line=5, column=9), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=7, column=46), 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.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic).

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

```lean
lemma ext {x y : SimplexCategoryGenRel} (h : x.len = y.len) : x = y
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.FreeSimplexQuiver (L25-L27)
def FreeSimplexQuiver
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.FreeSimplexQuiver (L28-L30)
def FreeSimplexQuiver.mk (n : ℕ) : FreeSimplexQuiver
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.FreeSimplexQuiver (L31-L39)
def FreeSimplexQuiver.len (x : FreeSimplexQuiver) : ℕ
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.quiv (L40-L43)
instance quiv : Quiver FreeSimplexQuiver where
  Hom
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.δ (L44-L47)
abbrev δ {n : ℕ} (i : Fin (n + 2)) : FreeSimplexQuiver.mk n ⟶ .mk (n + 1)
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.σ (L48-L74)
abbrev σ {n : ℕ} (i : Fin (n + 1)) : FreeSimplexQuiver.mk (n + 1) ⟶ .mk n
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.SimplexCategoryGenRel (L75-L78)
def SimplexCategoryGenRel
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.SimplexCategoryGenRel (L79-L84)
def SimplexCategoryGenRel.mk (n : ℕ) : SimplexCategoryGenRel where
  as
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.inclusion (L42-L43)
def inclusion : SimplexCategory ⥤ AugmentedSimplexCategory
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObjectFunctorCompToArrowIso (L78-L85)
def equivAugmentedCosimplicialObjectFunctorCompToArrowIso :
    equivAugmentedCosimplicialObject.functor ⋙ CosimplicialObject.Augmented.toArrow ≅
    Functor.mapArrowFunctor _ C ⋙
      (evaluation _ _ |>.obj <| .mk <| WithInitial.homTo <| .mk 0)
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObject (L53-L59)
def equivAugmentedCosimplicialObject :
    (AugmentedSimplexCategory ⥤ C) ≌ CosimplicialObject.Augmented C
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObjectFunctorCompDropIso (L61-L67)
def equivAugmentedCosimplicialObjectFunctorCompDropIso :
    equivAugmentedCosimplicialObject.functor ⋙ CosimplicialObject.Augmented.drop ≅
    (Functor.whiskeringLeft _ _ C).obj inclusion
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.AugmentedSimplexCategory (L34-L40)
abbrev AugmentedSimplexCategory
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObjectFunctorCompPointIso (L69-L76)
def equivAugmentedCosimplicialObjectFunctorCompPointIso :
    equivAugmentedCosimplicialObject.functor ⋙ CosimplicialObject.Augmented.point ≅
    ((evaluation _ _).obj .star : (AugmentedSimplexCategory ⥤ C) ⥤ C)
```

<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.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
lemma ext {x y : SimplexCategoryGenRel} (h : x.len = y.len) : x = y
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.FreeSimplexQuiver (L25-L27)
def FreeSimplexQuiver
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.FreeSimplexQuiver (L28-L30)
def FreeSimplexQuiver.mk (n : ℕ) : FreeSimplexQuiver
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.FreeSimplexQuiver (L31-L39)
def FreeSimplexQuiver.len (x : FreeSimplexQuiver) : ℕ
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.quiv (L40-L43)
instance quiv : Quiver FreeSimplexQuiver where
  Hom
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.δ (L44-L47)
abbrev δ {n : ℕ} (i : Fin (n + 2)) : FreeSimplexQuiver.mk n ⟶ .mk (n + 1)
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.σ (L48-L74)
abbrev σ {n : ℕ} (i : Fin (n + 1)) : FreeSimplexQuiver.mk (n + 1) ⟶ .mk n
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.SimplexCategoryGenRel (L75-L78)
def SimplexCategoryGenRel
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic.SimplexCategoryGenRel (L79-L84)
def SimplexCategoryGenRel.mk (n : ℕ) : SimplexCategoryGenRel where
  as
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.inclusion (L42-L43)
def inclusion : SimplexCategory ⥤ AugmentedSimplexCategory
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObjectFunctorCompToArrowIso (L78-L85)
def equivAugmentedCosimplicialObjectFunctorCompToArrowIso :
    equivAugmentedCosimplicialObject.functor ⋙ CosimplicialObject.Augmented.toArrow ≅
    Functor.mapArrowFunctor _ C ⋙
      (evaluation _ _ |>.obj <| .mk <| WithInitial.homTo <| .mk 0)
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObject (L53-L59)
def equivAugmentedCosimplicialObject :
    (AugmentedSimplexCategory ⥤ C) ≌ CosimplicialObject.Augmented C
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObjectFunctorCompDropIso (L61-L67)
def equivAugmentedCosimplicialObjectFunctorCompDropIso :
    equivAugmentedCosimplicialObject.functor ⋙ CosimplicialObject.Augmented.drop ≅
    (Functor.whiskeringLeft _ _ C).obj inclusion
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.AugmentedSimplexCategory (L34-L40)
abbrev AugmentedSimplexCategory
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObjectFunctorCompPointIso (L69-L76)
def equivAugmentedCosimplicialObjectFunctorCompPointIso :
    equivAugmentedCosimplicialObject.functor ⋙ CosimplicialObject.Augmented.point ≅
    ((evaluation _ _).obj .star : (AugmentedSimplexCategory ⥤ C) ⥤ C)
```
[/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.OpClass).

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

```lean
theorem IsSymmOp.flip_eq (op : α → α → β) [IsSymmOp op] : flip op = op
:= ⟨⟩⟨⟩
funext fun a ↦ funext fun b ↦ (IsSymmOp.symm_op a b).symm
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=87, messages=[Message(data='`IsSymmOp.flip_eq` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=24), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=87, i_end=155, 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)], 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.OpClass).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem IsSymmOp.flip_eq (op : α → α → β) [IsSymmOp op] : flip op = op
:= ⟨⟩⟨⟩
funext fun a ↦ funext fun b ↦ (IsSymmOp.symm_op a b).symm
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=87, messages=[Message(data='`IsSymmOp.flip_eq` has already been declared', pos=Position(line=1, column=8), pos_end=Position(line=1, column=24), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=87, i_end=155, 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)], 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.Logic.Equiv.Defs).

[User]
Complete the following proof.

```lean
def sigmaCongrRight {α} {β₁ β₂ : α → Type*} (F : ∀ a, β₁ a ≃ β₂ a) : (Σ a, β₁ a) ≃ Σ a, β₂ a where
  toFun a
```

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

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

```lean
def sigmaCongrRight {α} {β₁ β₂ : α → Type*} (F : ∀ a, β₁ a ≃ β₂ a) : (Σ a, β₁ a) ≃ Σ a, β₂ a where
  toFun a
```

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


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

[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.AlgebraicTopology.SimplexCategory.Augmented.Monoidal.inr_comp_inl_comp_associator`:
State 0: ⊢ lemma inr_comp_inl_comp_associator (x y z : AugmentedSimplexCategory) :
    inr _ _ ≫ inl _ _ ≫ (α_ x y z).hom = inl _ _ ≫ inr _ _
Tactic: ext i : 3  dsimp [MonoidalCategoryStruct.associator, associator]
State 1: ⊢ <goal after ext i : 3>
Tactic: have e₁ := inl'_eval y z i
State 2: ⊢ <goal after have e₁ := inl'_eval y z i>
Tactic: have e₂ := inr'_eval x y i
State 3: ⊢ <goal after have e₂ := inr'_eval x y i>
Tactic: have e₃ := inl'_eval (tensorObjOf x y) z <| Fin.cast (by simp +arith) <| i.natAdd (x.len + 1)
State 4: ⊢ <goal after have e₃ := inl'_eval (tensorObjOf x y) z <| Fin.cast (by simp +arith) <| i.natAdd (x.len + 1)>
Tactic: have e₄ := inr'_eval x (tensorObjOf y z) <| Fin.cast (by simp +arith) <| i.castAdd (z.len + 1)
State 5: ⊢ <goal after have e₄ := inr'_eval x (tensorObjOf y z) <| Fin.cast (by simp +arith) <| i.castAdd (z.len + 1)>
Tactic: simp only [SimplexCategory.len_mk] at e₁ e₂ e₃ e₄
State 6: ⊢ <goal after simp only [SimplexCategory.len_mk] at e₁ e₂ e₃ e₄>
Tactic: rw [e₁, e₂, e₃, e₄]
State 7: ⊢ <goal after rw [e₁, e₂, e₃, e₄]>
Tactic: ext; simp +arith  | .star, _, _ => by cat_disch  | _, .star, _ => by cat_disch  | _, _, .star => by cat_disch
```

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.AlgebraicTopology.SimplexCategory.Augmented.Monoidal).[/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.AlgebraicTopology.SimplexCategory.Augmented.Monoidal.inr_comp_inl_comp_associator`:
State 0: ⊢ lemma inr_comp_inl_comp_associator (x y z : AugmentedSimplexCategory) :
    inr _ _ ≫ inl _ _ ≫ (α_ x y z).hom = inl _ _ ≫ inr _ _
Tactic: ext i : 3  dsimp [MonoidalCategoryStruct.associator, associator]
State 1: ⊢ <goal after ext i : 3>
Tactic: have e₁ := inl'_eval y z i
State 2: ⊢ <goal after have e₁ := inl'_eval y z i>
Tactic: have e₂ := inr'_eval x y i
State 3: ⊢ <goal after have e₂ := inr'_eval x y i>
Tactic: have e₃ := inl'_eval (tensorObjOf x y) z <| Fin.cast (by simp +arith) <| i.natAdd (x.len + 1)
State 4: ⊢ <goal after have e₃ := inl'_eval (tensorObjOf x y) z <| Fin.cast (by simp +arith) <| i.natAdd (x.len + 1)>
Tactic: have e₄ := inr'_eval x (tensorObjOf y z) <| Fin.cast (by simp +arith) <| i.castAdd (z.len + 1)
State 5: ⊢ <goal after have e₄ := inr'_eval x (tensorObjOf y z) <| Fin.cast (by simp +arith) <| i.castAdd (z.len + 1)>
Tactic: simp only [SimplexCategory.len_mk] at e₁ e₂ e₃ e₄
State 6: ⊢ <goal after simp only [SimplexCategory.len_mk] at e₁ e₂ e₃ e₄>
Tactic: rw [e₁, e₂, e₃, e₄]
State 7: ⊢ <goal after rw [e₁, e₂, e₃, e₄]>
Tactic: ext; simp +arith  | .star, _, _ => by cat_disch  | _, .star, _ => by cat_disch  | _, _, .star => by cat_disch
```

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.Control.ULiftable).

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

```lean
instance ReaderT.instULiftableULiftULift {m m'} [ULiftable m m'] :
    ULiftable (ReaderT (ULift.{max v₀ u₀} s) m) (ReaderT (ULift.{max v₁ u₀} s) m')
:= thisIsWrongLemmaNameXYZ
ReaderT.uliftable' <| Equiv.ulift.trans Equiv.ulift.symm

/-- for specific continuation passing monads, this function helps to create a uliftable instance -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=243, messages=[Message(data='`ReaderT.instULiftableULiftULift` has already been declared', pos=Position(line=1, column=9), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=243, i_end=342, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=99), 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.Control.ULiftable).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
instance ReaderT.instULiftableULiftULift {m m'} [ULiftable m m'] :
    ULiftable (ReaderT (ULift.{max v₀ u₀} s) m) (ReaderT (ULift.{max v₁ u₀} s) m')
:= thisIsWrongLemmaNameXYZ
ReaderT.uliftable' <| Equiv.ulift.trans Equiv.ulift.symm

/-- for specific continuation passing monads, this function helps to create a uliftable instance -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=243, messages=[Message(data='`ReaderT.instULiftableULiftULift` has already been declared', pos=Position(line=1, column=9), pos_end=Position(line=1, column=40), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=243, i_end=342, messages=[Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=6, column=99), 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.Condensed.Discrete.Colimit).

[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.Condensed.Discrete.Colimit.lanPresheafNatIso`:
State 0: ⊢ def lanPresheafNatIso
    (hF : ∀ S : LightProfinite, IsColimit <| F.mapCocone (coconeRightOpOfCone S.asLimitCone)) :
    lanPresheaf F ≅ F
Tactic: refine NatIso.ofComponents  (fun ⟨S⟩ ↦ (lanPresheafIso (hF S))) fun _ ↦ ?_
State 1: ⊢ <goal after refine NatIso.ofComponents>
Tactic: simp only [lanPresheaf, pointwiseLeftKanExtension_obj, pointwiseLeftKanExtension_map,  lanPresheafIso_hom, Opposite.op_unop]
State 2: ⊢ <goal after simp only [lanPresheaf, pointwiseLeftKanExtension_obj, pointwiseLeftKanExtension_map,>
Tactic: exact colimit.hom_ext fun _ ↦ (by simp)
```

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

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Condensed.Discrete.Colimit).[/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.Condensed.Discrete.Colimit.lanPresheafNatIso`:
State 0: ⊢ def lanPresheafNatIso
    (hF : ∀ S : LightProfinite, IsColimit <| F.mapCocone (coconeRightOpOfCone S.asLimitCone)) :
    lanPresheaf F ≅ F
Tactic: refine NatIso.ofComponents  (fun ⟨S⟩ ↦ (lanPresheafIso (hF S))) fun _ ↦ ?_
State 1: ⊢ <goal after refine NatIso.ofComponents>
Tactic: simp only [lanPresheaf, pointwiseLeftKanExtension_obj, pointwiseLeftKanExtension_map,  lanPresheafIso_hom, Opposite.op_unop]
State 2: ⊢ <goal after simp only [lanPresheaf, pointwiseLeftKanExtension_obj, pointwiseLeftKanExtension_map,>
Tactic: exact colimit.hom_ext fun _ ↦ (by simp)
```

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


Instruction: Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

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

[User]
Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

## Imports
```lean
import Mathlib.Tactic.Attr.Register
```

## Surrounding context in Mathlib/Control/Basic.lean
```lean
-- Mathlib.Control.Basic.zipWithM (L37-L41)
def zipWithM {α₁ α₂ φ : Type u} (f : α₁ → α₂ → F φ) : ∀ (_ : List α₁) (_ : List α₂), F (List φ)
  | x :: xs, y :: ys => (· :: ·) <$> f x y <*> zipWithM f xs ys
  | _, _ => pure []

/-- Like `zipWithM` but evaluates the result as it traverses the lists using `*>`. -/

-- Mathlib.Control.Basic.zipWithM' (L42-L48)
def zipWithM' (f : α → β → F γ) : List α → List β → F PUnit
  | x :: xs, y :: ys => f x y *> zipWithM' f xs ys
  | [], _ => pure PUnit.unit
  | _, [] => pure PUnit.unit

variable [LawfulApplicative F]

-- Mathlib.Control.Basic.pure_id'_seq (L50-L52)
theorem pure_id'_seq (x : F α) : (pure fun x => x) <*> x = x
```

```lean
theorem fish_assoc {α β γ φ} (f : α → m β) (g : β → m γ) (h : γ → m φ) :
    (f >=> g) >=> h = f >=> g >=> h
```

Declaration: `Mathlib.Control.Basic.fish_assoc` (module Mathlib.Control.Basic).

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Control.Basic).

Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

## Imports
```lean
import Mathlib.Tactic.Attr.Register
```

## Surrounding context in Mathlib/Control/Basic.lean
```lean
-- Mathlib.Control.Basic.zipWithM (L37-L41)
def zipWithM {α₁ α₂ φ : Type u} (f : α₁ → α₂ → F φ) : ∀ (_ : List α₁) (_ : List α₂), F (List φ)
  | x :: xs, y :: ys => (· :: ·) <$> f x y <*> zipWithM f xs ys
  | _, _ => pure []

/-- Like `zipWithM` but evaluates the result as it traverses the lists using `*>`. -/

-- Mathlib.Control.Basic.zipWithM' (L42-L48)
def zipWithM' (f : α → β → F γ) : List α → List β → F PUnit
  | x :: xs, y :: ys => f x y *> zipWithM' f xs ys
  | [], _ => pure PUnit.unit
  | _, [] => pure PUnit.unit

variable [LawfulApplicative F]

-- Mathlib.Control.Basic.pure_id'_seq (L50-L52)
theorem pure_id'_seq (x : F α) : (pure fun x => x) <*> x = x
```

```lean
theorem fish_assoc {α β γ φ} (f : α → m β) (g : β → m γ) (h : γ → m φ) :
    (f >=> g) >=> h = f >=> g >=> h
```

Declaration: `Mathlib.Control.Basic.fish_assoc` (module Mathlib.Control.Basic).


Instruction: Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

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

[User]
Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

Open namespaces: Nat, Set, Function, (Commute), Fintype

## Surrounding context in Mathlib/Dynamics/PeriodicPts/Lemmas.lean
```lean
-- Mathlib.Dynamics.PeriodicPts.Lemmas.directed_ptsOfPeriod_pnat (L31-L34)
theorem directed_ptsOfPeriod_pnat (f : α → α) : Directed (· ⊆ ·) fun n : ℕ+ => ptsOfPeriod f n

-- Mathlib.Dynamics.PeriodicPts.Lemmas.bijOn_periodicPts (L35-L38)
theorem bijOn_periodicPts : BijOn f (periodicPts f) (periodicPts f)

-- Mathlib.Dynamics.PeriodicPts.Lemmas.minimalPeriod_eq_prime_iff (L39-L45)
theorem minimalPeriod_eq_prime_iff {p : ℕ} [hp : Fact p.Prime] :
    minimalPeriod f x = p ↔ IsPeriodicPt f p x ∧ ¬IsFixedPt f x
```

```lean
theorem minimalPeriod_piMap :
    minimalPeriod (Pi.map f) x = sInf { n > 0 | ∀ i, minimalPeriod (f i) (x i) ∣ n }
```

Declaration: `Mathlib.Dynamics.PeriodicPts.Lemmas.minimalPeriod_piMap` (module Mathlib.Dynamics.PeriodicPts.Lemmas).

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Dynamics.PeriodicPts.Lemmas).

Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

Open namespaces: Nat, Set, Function, (Commute), Fintype

## Surrounding context in Mathlib/Dynamics/PeriodicPts/Lemmas.lean
```lean
-- Mathlib.Dynamics.PeriodicPts.Lemmas.directed_ptsOfPeriod_pnat (L31-L34)
theorem directed_ptsOfPeriod_pnat (f : α → α) : Directed (· ⊆ ·) fun n : ℕ+ => ptsOfPeriod f n

-- Mathlib.Dynamics.PeriodicPts.Lemmas.bijOn_periodicPts (L35-L38)
theorem bijOn_periodicPts : BijOn f (periodicPts f) (periodicPts f)

-- Mathlib.Dynamics.PeriodicPts.Lemmas.minimalPeriod_eq_prime_iff (L39-L45)
theorem minimalPeriod_eq_prime_iff {p : ℕ} [hp : Fact p.Prime] :
    minimalPeriod f x = p ↔ IsPeriodicPt f p x ∧ ¬IsFixedPt f x
```

```lean
theorem minimalPeriod_piMap :
    minimalPeriod (Pi.map f) x = sInf { n > 0 | ∀ i, minimalPeriod (f i) (x i) ∣ n }
```

Declaration: `Mathlib.Dynamics.PeriodicPts.Lemmas.minimalPeriod_piMap` (module Mathlib.Dynamics.PeriodicPts.Lemmas).


[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.Lean.ContextInfo).

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

```lean
def runTacticCapturingInfoTree (ctx : ContextInfo) (i : TacticInfo) (goal : MVarId)
    (x : MVarId → MetaM α) : CommandElabM (α × PersistentArray InfoTree)
:= do thisIsWrongLemmaNameXYZ
if !i.goalsBefore.contains goal then
    panic!"ContextInfo.runTacticCapturingInfoTree: `goal` must be an element of `i.goalsBefore`"
  let mctx := i.mctxBefore
  let lctx := (mctx.decls.find! goal).2
  ctx.runMetaMCapturingInfoTree lctx do
    let type ← goal.getType
    let goal ← Meta.mkFreshExprSyntheticOpaqueMVar type
    x goal.mvarId!

/-- Run tactic code in the context of an infotree node, capturing InfoTrees for suggestion extraction.

Returns both the resulting goals and the InfoTrees produced during tactic execution.
Use `collectTryThisSuggestions` from `Mathlib.Lean.Elab.InfoTree` to extract suggestions. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=192, messages=[Message(data="Function expected at\n  MetaM\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `MetaM` 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=2, column=18), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  CommandElabM\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  (α × PersistentArray InfoTree)\n\nHint: The identifier `CommandElabM` 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=2, column=29), pos_end=Position(line=2, column=72), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=192, i_end=822, messages=[Message(data="unexpected token 'if'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=2), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=16, column=92), 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.Lean.ContextInfo).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def runTacticCapturingInfoTree (ctx : ContextInfo) (i : TacticInfo) (goal : MVarId)
    (x : MVarId → MetaM α) : CommandElabM (α × PersistentArray InfoTree)
:= do thisIsWrongLemmaNameXYZ
if !i.goalsBefore.contains goal then
    panic!"ContextInfo.runTacticCapturingInfoTree: `goal` must be an element of `i.goalsBefore`"
  let mctx := i.mctxBefore
  let lctx := (mctx.decls.find! goal).2
  ctx.runMetaMCapturingInfoTree lctx do
    let type ← goal.getType
    let goal ← Meta.mkFreshExprSyntheticOpaqueMVar type
    x goal.mvarId!

/-- Run tactic code in the context of an infotree node, capturing InfoTrees for suggestion extraction.

Returns both the resulting goals and the InfoTrees produced during tactic execution.
Use `collectTryThisSuggestions` from `Mathlib.Lean.Elab.InfoTree` to extract suggestions. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=192, messages=[Message(data="Function expected at\n  MetaM\nbut this term has type\n  ?m.4\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `MetaM` 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=2, column=18), pos_end=Position(line=2, column=25), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  CommandElabM\nbut this term has type\n  ?m.5\n\nNote: Expected a function because this term is being applied to the argument\n  (α × PersistentArray InfoTree)\n\nHint: The identifier `CommandElabM` 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=2, column=29), pos_end=Position(line=2, column=72), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=192, i_end=822, messages=[Message(data="unexpected token 'if'; expected command", pos=Position(line=4, column=0), pos_end=Position(line=4, column=2), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=16, column=92), 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.Tactic.FieldSimp.Lemmas).

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

```lean
theorem lt_eq_cancel_lt {M : Type*} [MonoidWithZero M] [PartialOrder M] [PosMulStrictMono M]
    [PosMulReflectLT M] {e₁ e₂ f₁ f₂ L : M}
    (H₁ : e₁ = L * f₁) (H₂ : e₂ = L * f₂) (HL : 0 < L) :
    (e₁ < e₂) = (f₁ < f₂)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_add (L39-L47)
theorem zpow'_add (a : α) (m n : ℤ) :
    zpow' a (m + n) = zpow' a m * zpow' a n := by
by_cases ha : a = 0
  · simp [zpow', ha]
    by_cases hn : n = 0
    · simp +contextual [hn, zero_zpow]
    · simp +contextual [hn, zero_zpow]
  · simp [zpow', ha, zpow_add₀]
```

-- snippet 2
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_of_ne_zero_right (L48-L50)
theorem zpow'_of_ne_zero_right (a : α) (n : ℤ) (hn : n ≠ 0) : zpow' a n = a ^ n := by
simp [zpow', hn]
```

-- snippet 3
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_of_ne_zero_left (L51-L53)
theorem zpow'_of_ne_zero_left (a : α) (n : ℤ) (ha : a ≠ 0) : zpow' a n = a ^ n := by
simp [zpow', ha]
```

-- snippet 4
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zero_zpow' (L55-L59)
lemma zero_zpow' (n : ℤ) : zpow' (0 : α) n = 0 := by
simp +contextual only [zpow', true_and, ite_eq_left_iff]
  intro hn
  exact zero_zpow n hn
```

-- snippet 5
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_eq_zero_iff (L60-L65)
lemma zpow'_eq_zero_iff (a : α) (n : ℤ) : zpow' a n = 0 ↔ a = 0 := by
obtain rfl | hn := eq_or_ne n 0
  · simp [zpow']
  · simp [zpow', zpow_eq_zero_iff hn]
    tauto
```

-- snippet 6
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.one_zpow' (L67-L69)
lemma one_zpow' (n : ℤ) : zpow' (1 : α) n = 1 := by
simp [zpow']
```

-- snippet 7
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_one (L71-L73)
lemma zpow'_one (a : α) : zpow' a 1 = a := by
simp [zpow']
```

-- snippet 8
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_zero_eq_div (L74-L79)
lemma zpow'_zero_eq_div (a : α) : zpow' a 0 = a / a := by
simp [zpow']
  by_cases h : a = 0
  · simp [h]
  · simp [h]
```

-- snippet 9
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_zero_of_ne_zero (L80-L81)
lemma zpow'_zero_of_ne_zero {a : α} (ha : a ≠ 0) : zpow' a 0 = 1 := by simp [zpow', ha]
```

-- snippet 10
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_neg (L82-L87)
lemma zpow'_neg (a : α) (n : ℤ) : zpow' a (-n) = (zpow' a n)⁻¹ := by
simp +contextual [zpow', apply_ite]
  split_ifs with h
  · tauto
  · tauto
```

-- snippet 11
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_mul (L88-L98)
lemma zpow'_mul (a : α) (m n : ℤ) : zpow' a (m * n) = zpow' (zpow' a m) n := by
by_cases ha : a = 0
  · simp [ha]
  by_cases hn : n = 0
  · rw [hn]
    simp [zpow', ha, zpow_ne_zero ]
  by_cases hm : m = 0
  · rw [hm]
    simp [zpow', ha]
  simpa [zpow', ha, hm, hn] using zpow_mul a m n
```

-- snippet 12
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_ofNat (L99-L103)
lemma zpow'_ofNat (a : α) {n : ℕ} (hn : n ≠ 0) : zpow' a n = a ^ n := by
rw [zpow'_of_ne_zero_right]
  · simp
  exact_mod_cast hn
```

-- snippet 13
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.mul_zpow' (L106-L113)
lemma mul_zpow' [CommGroupWithZero α] (n : ℤ) (a b : α) :
    zpow' (a * b) n = zpow' a n * zpow' b n := by
by_cases ha : a = 0
  · simp [ha]
  by_cases hb : b = 0
  · simp [hb]
  simpa [zpow', ha, hb] using mul_zpow a b n
```

-- snippet 14
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.list_prod_zpow' (L114-L120)
theorem list_prod_zpow' [CommGroupWithZero α] {r : ℤ} {l : List α} :
    zpow' (prod l) r = prod (map (fun x ↦ zpow' x r) l) :=
let fr : α →* α := ⟨⟨fun b ↦ zpow' b r, one_zpow' r⟩, (mul_zpow' r)⟩
  map_list_prod fr l

end zpow'
```

-- snippet 15
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.subst_add (L121-L126)
theorem subst_add {M : Type*} [Semiring M] {x₁ x₂ X₁ X₂ Y y a : M}
    (h₁ : x₁ = a * X₁) (h₂ : x₂ = a * X₂) (H_atom : X₁ + X₂ = Y) (hy : a * Y = y) :
    x₁ + x₂ = y := by
subst h₁ h₂ H_atom hy
  simp [mul_add]
```

-- snippet 16
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.subst_sub (L127-L132)
theorem subst_sub {M : Type*} [Ring M] {x₁ x₂ X₁ X₂ Y y a : M}
    (h₁ : x₁ = a * X₁) (h₂ : x₂ = a * X₂) (H_atom : X₁ - X₂ = Y) (hy : a * Y = y) :
    x₁ - x₂ = y := by
subst h₁ h₂ H_atom hy
  simp [mul_sub]
```

-- snippet 17
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.eq_div_of_eq_one_of_subst (L133-L137)
theorem eq_div_of_eq_one_of_subst {M : Type*} [DivInvOneMonoid M] {l l_n n : M} (h : l = l_n / 1)
    (hn : l_n = n) :
    l = n := by
rw [h, hn, div_one]
```

-- snippet 18
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.eq_div_of_eq_one_of_subst' (L138-L142)
theorem eq_div_of_eq_one_of_subst' {M : Type*} [DivInvOneMonoid M] {l l_d d : M} (h : l = 1 / l_d)
    (hn : l_d = d) :
    l = d⁻¹ := by
rw [h, hn, one_div]
```

-- snippet 19
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.eq_div_of_subst (L143-L147)
theorem eq_div_of_subst {M : Type*} [Div M] {l l_n l_d n d : M} (h : l = l_n / l_d) (hn : l_n = n)
    (hd : l_d = d) :
    l = n / d := by
rw [h, hn, hd]
```

-- snippet 20
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.eq_mul_of_eq_eq_eq_mul (L148-L152)
theorem eq_mul_of_eq_eq_eq_mul {M : Type*} [Mul M] {a b c D e f : M}
    (h₁ : a = b) (h₂ : b = c) (h₃ : c = D * e) (h₄ : e = f) :
    a = D * f := by
rw [h₁, h₂, h₃, h₄]
```

-- snippet 21
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.eq_eq_cancel_eq (L153-L158)
theorem eq_eq_cancel_eq {M : Type*} [MonoidWithZero M] [IsLeftCancelMulZero M] {e₁ e₂ f₁ f₂ L : M}
    (H₁ : e₁ = L * f₁) (H₂ : e₂ = L * f₂) (HL : L ≠ 0) :
    (e₁ = e₂) = (f₁ = f₂) := by
subst H₁ H₂
  rw [mul_right_inj' HL]
```

-- snippet 22
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.le_eq_cancel_le (L159-L166)
theorem le_eq_cancel_le {M : Type*} [MonoidWithZero M] [PartialOrder M] [PosMulMono M]
    [PosMulReflectLE M] {e₁ e₂ f₁ f₂ L : M}
    (H₁ : e₁ = L * f₁) (H₂ : e₂ = L * f₂) (HL : 0 < L) :
    (e₁ ≤ e₂) = (f₁ ≤ f₂) := by
subst H₁ H₂
  apply Iff.eq
  exact mul_le_mul_iff_right₀ HL
```

-- snippet 23
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.NF (L185-L191)
def NF (M : Type*) := List (ℤ × M)
namespace NF
variable {M : Type*}

/-- Augment a `FieldSimp.NF M` object `l`, i.e. a list of pairs in `ℤ × M`, by prepending another
pair `p : ℤ × M`. -/
```

-- snippet 24
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.cons (L193-L194)
def cons (p : ℤ × M) (l : NF M) : NF M := p :: l
```

-- snippet 25
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.cons_ne_zero (L208-L213)
theorem cons_ne_zero [GroupWithZero M] (r : ℤ) {x : M} (hx : x ≠ 0) {l : NF M} (hl : l.eval ≠ 0) :
    ((r, x) ::ᵣ l).eval ≠ 0 := by
unfold eval cons
  apply mul_ne_zero ?_ hl
  simp [zpow'_eq_zero_iff, hx]
```

<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.FieldSimp.Lemmas).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem lt_eq_cancel_lt {M : Type*} [MonoidWithZero M] [PartialOrder M] [PosMulStrictMono M]
    [PosMulReflectLT M] {e₁ e₂ f₁ f₂ L : M}
    (H₁ : e₁ = L * f₁) (H₂ : e₂ = L * f₂) (HL : 0 < L) :
    (e₁ < e₂) = (f₁ < f₂)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_add (L39-L47)
theorem zpow'_add (a : α) (m n : ℤ) :
    zpow' a (m + n) = zpow' a m * zpow' a n := by
by_cases ha : a = 0
  · simp [zpow', ha]
    by_cases hn : n = 0
    · simp +contextual [hn, zero_zpow]
    · simp +contextual [hn, zero_zpow]
  · simp [zpow', ha, zpow_add₀]
```

-- snippet 2
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_of_ne_zero_right (L48-L50)
theorem zpow'_of_ne_zero_right (a : α) (n : ℤ) (hn : n ≠ 0) : zpow' a n = a ^ n := by
simp [zpow', hn]
```

-- snippet 3
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_of_ne_zero_left (L51-L53)
theorem zpow'_of_ne_zero_left (a : α) (n : ℤ) (ha : a ≠ 0) : zpow' a n = a ^ n := by
simp [zpow', ha]
```

-- snippet 4
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zero_zpow' (L55-L59)
lemma zero_zpow' (n : ℤ) : zpow' (0 : α) n = 0 := by
simp +contextual only [zpow', true_and, ite_eq_left_iff]
  intro hn
  exact zero_zpow n hn
```

-- snippet 5
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_eq_zero_iff (L60-L65)
lemma zpow'_eq_zero_iff (a : α) (n : ℤ) : zpow' a n = 0 ↔ a = 0 := by
obtain rfl | hn := eq_or_ne n 0
  · simp [zpow']
  · simp [zpow', zpow_eq_zero_iff hn]
    tauto
```

-- snippet 6
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.one_zpow' (L67-L69)
lemma one_zpow' (n : ℤ) : zpow' (1 : α) n = 1 := by
simp [zpow']
```

-- snippet 7
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_one (L71-L73)
lemma zpow'_one (a : α) : zpow' a 1 = a := by
simp [zpow']
```

-- snippet 8
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_zero_eq_div (L74-L79)
lemma zpow'_zero_eq_div (a : α) : zpow' a 0 = a / a := by
simp [zpow']
  by_cases h : a = 0
  · simp [h]
  · simp [h]
```

-- snippet 9
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_zero_of_ne_zero (L80-L81)
lemma zpow'_zero_of_ne_zero {a : α} (ha : a ≠ 0) : zpow' a 0 = 1 := by simp [zpow', ha]
```

-- snippet 10
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_neg (L82-L87)
lemma zpow'_neg (a : α) (n : ℤ) : zpow' a (-n) = (zpow' a n)⁻¹ := by
simp +contextual [zpow', apply_ite]
  split_ifs with h
  · tauto
  · tauto
```

-- snippet 11
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_mul (L88-L98)
lemma zpow'_mul (a : α) (m n : ℤ) : zpow' a (m * n) = zpow' (zpow' a m) n := by
by_cases ha : a = 0
  · simp [ha]
  by_cases hn : n = 0
  · rw [hn]
    simp [zpow', ha, zpow_ne_zero ]
  by_cases hm : m = 0
  · rw [hm]
    simp [zpow', ha]
  simpa [zpow', ha, hm, hn] using zpow_mul a m n
```

-- snippet 12
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.zpow'_ofNat (L99-L103)
lemma zpow'_ofNat (a : α) {n : ℕ} (hn : n ≠ 0) : zpow' a n = a ^ n := by
rw [zpow'_of_ne_zero_right]
  · simp
  exact_mod_cast hn
```

-- snippet 13
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.mul_zpow' (L106-L113)
lemma mul_zpow' [CommGroupWithZero α] (n : ℤ) (a b : α) :
    zpow' (a * b) n = zpow' a n * zpow' b n := by
by_cases ha : a = 0
  · simp [ha]
  by_cases hb : b = 0
  · simp [hb]
  simpa [zpow', ha, hb] using mul_zpow a b n
```

-- snippet 14
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.list_prod_zpow' (L114-L120)
theorem list_prod_zpow' [CommGroupWithZero α] {r : ℤ} {l : List α} :
    zpow' (prod l) r = prod (map (fun x ↦ zpow' x r) l) :=
let fr : α →* α := ⟨⟨fun b ↦ zpow' b r, one_zpow' r⟩, (mul_zpow' r)⟩
  map_list_prod fr l

end zpow'
```

-- snippet 15
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.subst_add (L121-L126)
theorem subst_add {M : Type*} [Semiring M] {x₁ x₂ X₁ X₂ Y y a : M}
    (h₁ : x₁ = a * X₁) (h₂ : x₂ = a * X₂) (H_atom : X₁ + X₂ = Y) (hy : a * Y = y) :
    x₁ + x₂ = y := by
subst h₁ h₂ H_atom hy
  simp [mul_add]
```

-- snippet 16
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.subst_sub (L127-L132)
theorem subst_sub {M : Type*} [Ring M] {x₁ x₂ X₁ X₂ Y y a : M}
    (h₁ : x₁ = a * X₁) (h₂ : x₂ = a * X₂) (H_atom : X₁ - X₂ = Y) (hy : a * Y = y) :
    x₁ - x₂ = y := by
subst h₁ h₂ H_atom hy
  simp [mul_sub]
```

-- snippet 17
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.eq_div_of_eq_one_of_subst (L133-L137)
theorem eq_div_of_eq_one_of_subst {M : Type*} [DivInvOneMonoid M] {l l_n n : M} (h : l = l_n / 1)
    (hn : l_n = n) :
    l = n := by
rw [h, hn, div_one]
```

-- snippet 18
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.eq_div_of_eq_one_of_subst' (L138-L142)
theorem eq_div_of_eq_one_of_subst' {M : Type*} [DivInvOneMonoid M] {l l_d d : M} (h : l = 1 / l_d)
    (hn : l_d = d) :
    l = d⁻¹ := by
rw [h, hn, one_div]
```

-- snippet 19
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.eq_div_of_subst (L143-L147)
theorem eq_div_of_subst {M : Type*} [Div M] {l l_n l_d n d : M} (h : l = l_n / l_d) (hn : l_n = n)
    (hd : l_d = d) :
    l = n / d := by
rw [h, hn, hd]
```

-- snippet 20
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.eq_mul_of_eq_eq_eq_mul (L148-L152)
theorem eq_mul_of_eq_eq_eq_mul {M : Type*} [Mul M] {a b c D e f : M}
    (h₁ : a = b) (h₂ : b = c) (h₃ : c = D * e) (h₄ : e = f) :
    a = D * f := by
rw [h₁, h₂, h₃, h₄]
```

-- snippet 21
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.eq_eq_cancel_eq (L153-L158)
theorem eq_eq_cancel_eq {M : Type*} [MonoidWithZero M] [IsLeftCancelMulZero M] {e₁ e₂ f₁ f₂ L : M}
    (H₁ : e₁ = L * f₁) (H₂ : e₂ = L * f₂) (HL : L ≠ 0) :
    (e₁ = e₂) = (f₁ = f₂) := by
subst H₁ H₂
  rw [mul_right_inj' HL]
```

-- snippet 22
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.le_eq_cancel_le (L159-L166)
theorem le_eq_cancel_le {M : Type*} [MonoidWithZero M] [PartialOrder M] [PosMulMono M]
    [PosMulReflectLE M] {e₁ e₂ f₁ f₂ L : M}
    (H₁ : e₁ = L * f₁) (H₂ : e₂ = L * f₂) (HL : 0 < L) :
    (e₁ ≤ e₂) = (f₁ ≤ f₂) := by
subst H₁ H₂
  apply Iff.eq
  exact mul_le_mul_iff_right₀ HL
```

-- snippet 23
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.NF (L185-L191)
def NF (M : Type*) := List (ℤ × M)
namespace NF
variable {M : Type*}

/-- Augment a `FieldSimp.NF M` object `l`, i.e. a list of pairs in `ℤ × M`, by prepending another
pair `p : ℤ × M`. -/
```

-- snippet 24
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.cons (L193-L194)
def cons (p : ℤ × M) (l : NF M) : NF M := p :: l
```

-- snippet 25
```lean
-- Mathlib.Tactic.FieldSimp.Lemmas.cons_ne_zero (L208-L213)
theorem cons_ne_zero [GroupWithZero M] (r : ℤ) {x : M} (hx : x ≠ 0) {l : NF M} (hl : l.eval ≠ 0) :
    ((r, x) ::ᵣ l).eval ≠ 0 := by
unfold eval cons
  apply mul_ne_zero ?_ hl
  simp [zpow'_eq_zero_iff, hx]
```
[/INST]


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

[User]
Complete the following proof.

Open namespaces: Function, Prod, Set, Uniformity, UniformSpace, scoped, SetRel, ENNReal

## Surrounding context in Mathlib/Dynamics/TopologicalEntropy/Semiconj.lean
```lean
-- Mathlib.Dynamics.TopologicalEntropy.Semiconj.IsDynCoverOf (L65-L71)
lemma IsDynCoverOf.image (h : Semiconj φ S T) (h' : IsDynCoverOf S F (map φ φ ⁻¹' V) n s) :
    IsDynCoverOf T (φ '' F) V n (φ '' s)

-- Mathlib.Dynamics.TopologicalEntropy.Semiconj.IsDynCoverOf (L72-L92)
lemma IsDynCoverOf.preimage (h : Semiconj φ S T) [V.IsSymm] {t : Finset Y}
    (h' : IsDynCoverOf T (φ '' F) V n t) :
    ∃ s : Finset X, IsDynCoverOf S F ((map φ φ) ⁻¹' (V ○ V)) n s ∧ s.card ≤ t.card

-- Mathlib.Dynamics.TopologicalEntropy.Semiconj.le_coverMincard_image (L93-L101)
lemma le_coverMincard_image (h : Semiconj φ S T) (F : Set X) [V.IsSymm] (n : ℕ) :
    coverMincard S F ((map φ φ) ⁻¹' (V ○ V)) n ≤ coverMincard T (φ '' F) V n
```

Relevant lemmas available in scope: `le_antisymm`, `coverEntropyEntourage_le_coverEntropy`

```lean
theorem coverEntropy_image_of_comap (u : UniformSpace Y) {S : X → X} {T : Y → Y} {φ : X → Y}
    (h : Semiconj φ S T) (F : Set X) :
    coverEntropy T (φ '' F) = @coverEntropy X (comap φ u) S F
```

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

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Dynamics.TopologicalEntropy.Semiconj).

Complete the following proof.

Open namespaces: Function, Prod, Set, Uniformity, UniformSpace, scoped, SetRel, ENNReal

## Surrounding context in Mathlib/Dynamics/TopologicalEntropy/Semiconj.lean
```lean
-- Mathlib.Dynamics.TopologicalEntropy.Semiconj.IsDynCoverOf (L65-L71)
lemma IsDynCoverOf.image (h : Semiconj φ S T) (h' : IsDynCoverOf S F (map φ φ ⁻¹' V) n s) :
    IsDynCoverOf T (φ '' F) V n (φ '' s)

-- Mathlib.Dynamics.TopologicalEntropy.Semiconj.IsDynCoverOf (L72-L92)
lemma IsDynCoverOf.preimage (h : Semiconj φ S T) [V.IsSymm] {t : Finset Y}
    (h' : IsDynCoverOf T (φ '' F) V n t) :
    ∃ s : Finset X, IsDynCoverOf S F ((map φ φ) ⁻¹' (V ○ V)) n s ∧ s.card ≤ t.card

-- Mathlib.Dynamics.TopologicalEntropy.Semiconj.le_coverMincard_image (L93-L101)
lemma le_coverMincard_image (h : Semiconj φ S T) (F : Set X) [V.IsSymm] (n : ℕ) :
    coverMincard S F ((map φ φ) ⁻¹' (V ○ V)) n ≤ coverMincard T (φ '' F) V n
```

Relevant lemmas available in scope: `le_antisymm`, `coverEntropyEntourage_le_coverEntropy`

```lean
theorem coverEntropy_image_of_comap (u : UniformSpace Y) {S : X → X} {T : Y → Y} {φ : X → Y}
    (h : Semiconj φ S T) (F : Set X) :
    coverEntropy T (φ '' F) = @coverEntropy X (comap φ u) S F
```

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