Instruction: Edit `fix_eq_of_ωScottContinuous` so it still compiles and the proof is simplified where possible. Return the full updated `fix_eq_of_ωScottContinuous` 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.Control.LawfulFix).

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

## Current excerpt (around `Mathlib.Control.LawfulFix.fix_eq_of_ωScottContinuous`, L156-L172)
Open namespaces: OmegaCompletePartialOrder, Nat, Nat.Upto, Fix

## Surrounding context in Mathlib/Control/LawfulFix.lean
```lean
-- Mathlib.Control.LawfulFix.approx_mono' (L49-L53)
theorem approx_mono' {i : ℕ} : Fix.approx f i ≤ Fix.approx f (succ i)

-- Mathlib.Control.LawfulFix.approx_mono (L54-L60)
theorem approx_mono ⦃i j : ℕ⦄ (hij : i ≤ j) : approx f i ≤ approx f j

-- Mathlib.Control.LawfulFix.mem_iff (L61-L83)
theorem mem_iff (a : α) (b : β a) : b ∈ Part.fix f a ↔ ∃ i, b ∈ approx f i a
```

Relevant lemmas available in scope: `le_antisymm`, `i`, `le_f_of_mem_approx`

```lean
theorem fix_eq_of_ωScottContinuous (hc : ωScottContinuous g) :
    Part.fix g = g (Part.fix g) := by
rw [fix_eq_ωSup_of_ωScottContinuous hc, hc.map_ωSup]
  apply le_antisymm
  · apply ωSup_le_ωSup_of_le _
    intro i
    exists i
    apply le_f_of_mem_approx _ ⟨i, rfl⟩
  · apply ωSup_le_ωSup_of_le _
    intro i
    exists i.succ

end Part

namespace Part

/-- `toUnit` as a monotone function -/
```

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

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).

You are editing a file in `Mathlib/Control/LawfulFix.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Control.LawfulFix.fix_eq_of_ωScottContinuous`, L156-L172)
Open namespaces: OmegaCompletePartialOrder, Nat, Nat.Upto, Fix

## Surrounding context in Mathlib/Control/LawfulFix.lean
```lean
-- Mathlib.Control.LawfulFix.approx_mono' (L49-L53)
theorem approx_mono' {i : ℕ} : Fix.approx f i ≤ Fix.approx f (succ i)

-- Mathlib.Control.LawfulFix.approx_mono (L54-L60)
theorem approx_mono ⦃i j : ℕ⦄ (hij : i ≤ j) : approx f i ≤ approx f j

-- Mathlib.Control.LawfulFix.mem_iff (L61-L83)
theorem mem_iff (a : α) (b : β a) : b ∈ Part.fix f a ↔ ∃ i, b ∈ approx f i a
```

Relevant lemmas available in scope: `le_antisymm`, `i`, `le_f_of_mem_approx`

```lean
theorem fix_eq_of_ωScottContinuous (hc : ωScottContinuous g) :
    Part.fix g = g (Part.fix g) := by
rw [fix_eq_ωSup_of_ωScottContinuous hc, hc.map_ωSup]
  apply le_antisymm
  · apply ωSup_le_ωSup_of_le _
    intro i
    exists i
    apply le_f_of_mem_approx _ ⟨i, rfl⟩
  · apply ωSup_le_ωSup_of_le _
    intro i
    exists i.succ

end Part

namespace Part

/-- `toUnit` as a monotone function -/
```

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


Instruction: Edit `ManyOneEquiv` so it still compiles and the proof is simplified where possible. Return the full updated `ManyOneEquiv` 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.Computability.Reduce).

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

## Current excerpt (around `Mathlib.Computability.Reduce.ManyOneEquiv`, L147-L150)
Open namespaces: Function, Computable

## Surrounding context in Mathlib/Computability/Reduce.lean
```lean
-- Mathlib.Computability.Reduce.ManyOneReducible (L41-L43)
def ManyOneReducible {α β} [Primcodable α] [Primcodable β] (p : α → Prop) (q : β → Prop)

-- Mathlib.Computability.Reduce.ManyOneReducible (L47-L50)
theorem ManyOneReducible.mk {α β} [Primcodable α] [Primcodable β] {f : α → β} (q : β → Prop)
    (h : Computable f) : (fun a => q (f a)) ≤₀ q

-- Mathlib.Computability.Reduce.manyOneReducible_refl (L52-L54)
theorem manyOneReducible_refl {α} [Primcodable α] (p : α → Prop) : p ≤₀ p
```

```lean
theorem ManyOneEquiv.symm {α β} [Primcodable α] [Primcodable β] {p : α → Prop} {q : β → Prop} :
    ManyOneEquiv p q → ManyOneEquiv q p := And.symm
```

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

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.Reduce).

You are editing a file in `Mathlib/Computability/Reduce.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Computability.Reduce.ManyOneEquiv`, L147-L150)
Open namespaces: Function, Computable

## Surrounding context in Mathlib/Computability/Reduce.lean
```lean
-- Mathlib.Computability.Reduce.ManyOneReducible (L41-L43)
def ManyOneReducible {α β} [Primcodable α] [Primcodable β] (p : α → Prop) (q : β → Prop)

-- Mathlib.Computability.Reduce.ManyOneReducible (L47-L50)
theorem ManyOneReducible.mk {α β} [Primcodable α] [Primcodable β] {f : α → β} (q : β → Prop)
    (h : Computable f) : (fun a => q (f a)) ≤₀ q

-- Mathlib.Computability.Reduce.manyOneReducible_refl (L52-L54)
theorem manyOneReducible_refl {α} [Primcodable α] (p : α → Prop) : p ≤₀ p
```

```lean
theorem ManyOneEquiv.symm {α β} [Primcodable α] [Primcodable β] {p : α → Prop} {q : β → Prop} :
    ManyOneEquiv p q → ManyOneEquiv q p := And.symm
```

## Edit instruction
Edit `ManyOneEquiv` so it still compiles and the proof is simplified where possible. Return the full updated `ManyOneEquiv` 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.Computability.Language).

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

```lean
theorem append_mem_mul : a ∈ l → b ∈ m → a ++ b ∈ l * m
  frobnicateTacticThatDoesNotExist :=
mem_image2_of_mem
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=121, messages=[Message(data="Function expected at\n  m\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist\n\nHint: The identifier `m` 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=54), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `mem_image2_of_mem`', pos=Position(line=3, column=0), pos_end=Position(line=3, column=17), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._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.Language).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem append_mem_mul : a ∈ l → b ∈ m → a ++ b ∈ l * m
  frobnicateTacticThatDoesNotExist :=
mem_image2_of_mem
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=121, messages=[Message(data="Function expected at\n  m\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist\n\nHint: The identifier `m` 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=54), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `mem_image2_of_mem`', pos=Position(line=3, column=0), pos_end=Position(line=3, column=17), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

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