[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.CongrExclamation).

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

```lean
def Lean.MVarId.userCongr? (config : Congr!.Config) (mvarId : MVarId) :
    MetaM (Option (List MVarId))
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Tactic.CongrExclamation.Congr (L188-L194)
def Congr!.Config.unfoldSameFun : Congr!.Config where
  partialApp
```

-- snippet 2
```lean
-- Mathlib.Tactic.CongrExclamation.Congr (L195-L198)
def Congr!.Config.numArgsOk (config : Config) (numArgs : Nat) : Bool
```

-- snippet 3
```lean
-- Mathlib.Tactic.CongrExclamation.Congr (L199-L234)
def Congr!.Config.maxArgsFor (config : Config) (numArgs : Nat) : Nat
```

-- snippet 4
```lean
-- Mathlib.Tactic.CongrExclamation.Congr (L235-L271)
def Congr!.plausiblyEqualTypes (ty1 ty2 : Expr) (maxDepth : Nat
```

-- snippet 5
```lean
-- Mathlib.Tactic.CongrExclamation.for (L500-L503)
instance for both the LHS and the RHS.

If successful, this reduces proving `@HEq α x β y` to proving `α = β`.
-/
```

-- snippet 6
```lean
-- Mathlib.Tactic.CongrExclamation.CongrMetaM (L557-L559)
abbrev CongrMetaM
```

-- snippet 7
```lean
-- Mathlib.Tactic.CongrExclamation.CongrMetaM (L560-L588)
def CongrMetaM.nextPattern : CongrMetaM (Option (TSyntax `rcasesPat))
```

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

```lean
def Lean.MVarId.userCongr? (config : Congr!.Config) (mvarId : MVarId) :
    MetaM (Option (List MVarId))
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Tactic.CongrExclamation.Congr (L188-L194)
def Congr!.Config.unfoldSameFun : Congr!.Config where
  partialApp
```

-- snippet 2
```lean
-- Mathlib.Tactic.CongrExclamation.Congr (L195-L198)
def Congr!.Config.numArgsOk (config : Config) (numArgs : Nat) : Bool
```

-- snippet 3
```lean
-- Mathlib.Tactic.CongrExclamation.Congr (L199-L234)
def Congr!.Config.maxArgsFor (config : Config) (numArgs : Nat) : Nat
```

-- snippet 4
```lean
-- Mathlib.Tactic.CongrExclamation.Congr (L235-L271)
def Congr!.plausiblyEqualTypes (ty1 ty2 : Expr) (maxDepth : Nat
```

-- snippet 5
```lean
-- Mathlib.Tactic.CongrExclamation.for (L500-L503)
instance for both the LHS and the RHS.

If successful, this reduces proving `@HEq α x β y` to proving `α = β`.
-/
```

-- snippet 6
```lean
-- Mathlib.Tactic.CongrExclamation.CongrMetaM (L557-L559)
abbrev CongrMetaM
```

-- snippet 7
```lean
-- Mathlib.Tactic.CongrExclamation.CongrMetaM (L560-L588)
def CongrMetaM.nextPattern : CongrMetaM (Option (TSyntax `rcasesPat))
```
[/INST]


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

[User]
Complete the following proof.

```lean
lemma RightHomotopyClass.mk_surjective [CategoryWithWeakEquivalences C] :
    Function.Surjective (mk : (X ⟶ Y) → _)
```

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

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

```lean
lemma RightHomotopyClass.mk_surjective [CategoryWithWeakEquivalences C] :
    Function.Surjective (mk : (X ⟶ Y) → _)
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.[/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.Logic.Small.Defs).

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

```lean
lemma Shrink.rec_equivShrink {α : Type*} [Small.{w} α] {F : Shrink α → Sort v}
    {f : (a : α) → F (equivShrink α a)} (a : α) : Shrink.rec f (equivShrink _ a) = f a
```

Declaration: `Mathlib.Logic.Small.Defs.Shrink` (module Mathlib.Logic.Small.Defs).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Logic.Small.Defs).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
lemma Shrink.rec_equivShrink {α : Type*} [Small.{w} α] {F : Shrink α → Sort v}
    {f : (a : α) → F (equivShrink α a)} (a : α) : Shrink.rec f (equivShrink _ a) = f a
```

Declaration: `Mathlib.Logic.Small.Defs.Shrink` (module Mathlib.Logic.Small.Defs).[/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.Dynamics.FixedPoints.Basic).

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

```lean
instance fixedPoints.decidable [DecidableEq α] (f : α → α) (x : α) :
    Decidable (x ∈ fixedPoints f)
:= thisIsWrongLemmaNameXYZ
IsFixedPt.decidable
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=157, messages=[Message(data="Function expected at\n  fixedPoints\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `fixedPoints` 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=19), pos_end=Position(line=2, column=32), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), 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.Dynamics.FixedPoints.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
instance fixedPoints.decidable [DecidableEq α] (f : α → α) (x : α) :
    Decidable (x ∈ fixedPoints f)
:= thisIsWrongLemmaNameXYZ
IsFixedPt.decidable
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=157, messages=[Message(data="Function expected at\n  fixedPoints\nbut this term has type\n  ?m.2\n\nNote: Expected a function because this term is being applied to the argument\n  f\n\nHint: The identifier `fixedPoints` 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=19), pos_end=Position(line=2, column=32), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), 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]


[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.DFA).

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

```lean
theorem evalFrom_of_append (start : σ) (x y : List α) :
    M.evalFrom start (x ++ y) = M.evalFrom (M.evalFrom start x) y
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Computability.DFA.evalFrom (L74-L76)
def evalFrom (s : σ) : List α → σ :=
List.foldl M.step s
```

-- snippet 2
```lean
-- Mathlib.Computability.DFA.evalFrom_nil (L78-L80)
theorem evalFrom_nil (s : σ) : M.evalFrom s [] = s :=
rfl
```

-- snippet 3
```lean
-- Mathlib.Computability.DFA.evalFrom_cons (L82-L85)
theorem evalFrom_cons (s : σ) (a : α) (x : List α) :
    M.evalFrom s (a :: x) = M.evalFrom (M.step s a) x :=
rfl
```

-- snippet 4
```lean
-- Mathlib.Computability.DFA.evalFrom_singleton (L87-L89)
theorem evalFrom_singleton (s : σ) (a : α) : M.evalFrom s [a] = M.step s a :=
rfl
```

-- snippet 5
```lean
-- Mathlib.Computability.DFA.evalFrom_append_singleton (L91-L95)
theorem evalFrom_append_singleton (s : σ) (x : List α) (a : α) :
    M.evalFrom s (x ++ [a]) = M.step (M.evalFrom s x) a := by
simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]

/-- `M.eval x` evaluates `M` with input `x` starting from the state `M.start`. -/
```

-- snippet 6
```lean
-- Mathlib.Computability.DFA.eval (L96-L98)
def eval : List α → σ :=
M.evalFrom M.start
```

-- snippet 7
```lean
-- Mathlib.Computability.DFA.eval_nil (L100-L102)
theorem eval_nil : M.eval [] = M.start :=
rfl
```

-- snippet 8
```lean
-- Mathlib.Computability.DFA.eval_singleton (L104-L106)
theorem eval_singleton (a : α) : M.eval [a] = M.step M.start a :=
rfl
```

-- snippet 9
```lean
-- Mathlib.Computability.DFA.eval_append_singleton (L108-L110)
theorem eval_append_singleton (x : List α) (a : α) : M.eval (x ++ [a]) = M.step (M.eval x) a :=
evalFrom_append_singleton _ _ _ _
```

-- snippet 10
```lean
-- Mathlib.Computability.DFA.acceptsFrom (L118-L119)
def acceptsFrom (s : σ) : Language α := {x | M.evalFrom s x ∈ M.accept}
```

-- snippet 11
```lean
-- Mathlib.Computability.DFA.mem_acceptsFrom (L120-L123)
theorem mem_acceptsFrom {s : σ} {x : List α} :
    x ∈ M.acceptsFrom s ↔ M.evalFrom s x ∈ M.accept := by rfl
/-- `M.accepts` is the language of `x` such that `M.eval x` is an accept state. -/
```

-- snippet 12
```lean
-- Mathlib.Computability.DFA.accepts (L124-L125)
def accepts : Language α := M.acceptsFrom M.start
```

-- snippet 13
```lean
-- Mathlib.Computability.DFA.mem_accepts (L126-L127)
theorem mem_accepts {x : List α} : x ∈ M.accepts ↔ M.eval x ∈ M.accept := by rfl
```

-- snippet 14
```lean
-- Mathlib.Computability.DFA.evalFrom_split (L128-L157)
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by simp)
  wlog hle : (n : ℕ) ≤ m generalizing n m
  · exact this m n hneq.symm heq.symm (le_of_not_ge hle)
  refine
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n,
                    x.drop m, ?_, ?_, ?_, by rfl, ?_⟩
  · rw [List.take_append_drop, List.take_append_drop]
  · simp only [List.length_drop, List.length_take]
    omega
  · intro h
    have hlen' := congr_arg List.length h
    simp only [List.length_drop, List.length, List.length_take] at hlen'
    omega
  have hq : M.evalFrom (M.evalFrom s ((x.take m).take n)) ((x.take m).drop n) =
      M.evalFrom s ((x.take m).take n) := by
    rw [List.take_take, min_eq_left hle, ← evalFrom_of_append, heq, ← min_eq_left hle, ←
      List.take_take, min_eq_left hle, List.take_append_drop]
  use hq
  rwa [← hq, ← evalFrom_of_append, ← evalFrom_of_append, ← List.append_assoc,
    List.take_append_drop, List.take_append_drop]

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 15
```lean
-- Mathlib.Computability.DFA.evalFrom_of_pow (L158-L172)
theorem evalFrom_of_pow {x y : List α} {s : σ} (hx : M.evalFrom s x = s)
    (hy : y ∈ ({x} : Language α)∗) : M.evalFrom s y = s := by
rw [Language.mem_kstar] at hy
  rcases hy with ⟨S, rfl, hS⟩
  induction S with
  | nil => rfl
  | cons a S ih =>
    have ha := hS a List.mem_cons_self
    rw [Set.mem_singleton_iff] at ha
    rw [List.flatten_cons, evalFrom_of_append, ha, hx]
    apply ih
    intro z hz
    exact hS z (List.mem_cons_of_mem a hz)

set_option backward.isDefEq.respectTransparency false in
```

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

```lean
theorem evalFrom_of_append (start : σ) (x y : List α) :
    M.evalFrom start (x ++ y) = M.evalFrom (M.evalFrom start x) y
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Computability.DFA.evalFrom (L74-L76)
def evalFrom (s : σ) : List α → σ :=
List.foldl M.step s
```

-- snippet 2
```lean
-- Mathlib.Computability.DFA.evalFrom_nil (L78-L80)
theorem evalFrom_nil (s : σ) : M.evalFrom s [] = s :=
rfl
```

-- snippet 3
```lean
-- Mathlib.Computability.DFA.evalFrom_cons (L82-L85)
theorem evalFrom_cons (s : σ) (a : α) (x : List α) :
    M.evalFrom s (a :: x) = M.evalFrom (M.step s a) x :=
rfl
```

-- snippet 4
```lean
-- Mathlib.Computability.DFA.evalFrom_singleton (L87-L89)
theorem evalFrom_singleton (s : σ) (a : α) : M.evalFrom s [a] = M.step s a :=
rfl
```

-- snippet 5
```lean
-- Mathlib.Computability.DFA.evalFrom_append_singleton (L91-L95)
theorem evalFrom_append_singleton (s : σ) (x : List α) (a : α) :
    M.evalFrom s (x ++ [a]) = M.step (M.evalFrom s x) a := by
simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]

/-- `M.eval x` evaluates `M` with input `x` starting from the state `M.start`. -/
```

-- snippet 6
```lean
-- Mathlib.Computability.DFA.eval (L96-L98)
def eval : List α → σ :=
M.evalFrom M.start
```

-- snippet 7
```lean
-- Mathlib.Computability.DFA.eval_nil (L100-L102)
theorem eval_nil : M.eval [] = M.start :=
rfl
```

-- snippet 8
```lean
-- Mathlib.Computability.DFA.eval_singleton (L104-L106)
theorem eval_singleton (a : α) : M.eval [a] = M.step M.start a :=
rfl
```

-- snippet 9
```lean
-- Mathlib.Computability.DFA.eval_append_singleton (L108-L110)
theorem eval_append_singleton (x : List α) (a : α) : M.eval (x ++ [a]) = M.step (M.eval x) a :=
evalFrom_append_singleton _ _ _ _
```

-- snippet 10
```lean
-- Mathlib.Computability.DFA.acceptsFrom (L118-L119)
def acceptsFrom (s : σ) : Language α := {x | M.evalFrom s x ∈ M.accept}
```

-- snippet 11
```lean
-- Mathlib.Computability.DFA.mem_acceptsFrom (L120-L123)
theorem mem_acceptsFrom {s : σ} {x : List α} :
    x ∈ M.acceptsFrom s ↔ M.evalFrom s x ∈ M.accept := by rfl
/-- `M.accepts` is the language of `x` such that `M.eval x` is an accept state. -/
```

-- snippet 12
```lean
-- Mathlib.Computability.DFA.accepts (L124-L125)
def accepts : Language α := M.acceptsFrom M.start
```

-- snippet 13
```lean
-- Mathlib.Computability.DFA.mem_accepts (L126-L127)
theorem mem_accepts {x : List α} : x ∈ M.accepts ↔ M.eval x ∈ M.accept := by rfl
```

-- snippet 14
```lean
-- Mathlib.Computability.DFA.evalFrom_split (L128-L157)
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by simp)
  wlog hle : (n : ℕ) ≤ m generalizing n m
  · exact this m n hneq.symm heq.symm (le_of_not_ge hle)
  refine
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n,
                    x.drop m, ?_, ?_, ?_, by rfl, ?_⟩
  · rw [List.take_append_drop, List.take_append_drop]
  · simp only [List.length_drop, List.length_take]
    omega
  · intro h
    have hlen' := congr_arg List.length h
    simp only [List.length_drop, List.length, List.length_take] at hlen'
    omega
  have hq : M.evalFrom (M.evalFrom s ((x.take m).take n)) ((x.take m).drop n) =
      M.evalFrom s ((x.take m).take n) := by
    rw [List.take_take, min_eq_left hle, ← evalFrom_of_append, heq, ← min_eq_left hle, ←
      List.take_take, min_eq_left hle, List.take_append_drop]
  use hq
  rwa [← hq, ← evalFrom_of_append, ← evalFrom_of_append, ← List.append_assoc,
    List.take_append_drop, List.take_append_drop]

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 15
```lean
-- Mathlib.Computability.DFA.evalFrom_of_pow (L158-L172)
theorem evalFrom_of_pow {x y : List α} {s : σ} (hx : M.evalFrom s x = s)
    (hy : y ∈ ({x} : Language α)∗) : M.evalFrom s y = s := by
rw [Language.mem_kstar] at hy
  rcases hy with ⟨S, rfl, hS⟩
  induction S with
  | nil => rfl
  | cons a S ih =>
    have ha := hS a List.mem_cons_self
    rw [Set.mem_singleton_iff] at ha
    rw [List.flatten_cons, evalFrom_of_append, ha, hx]
    apply ih
    intro z hz
    exact hS z (List.mem_cons_of_mem a hz)

set_option backward.isDefEq.respectTransparency false in
```
[/INST]


Instruction: Edit `map_sub_nat` so it still compiles and the proof is simplified where possible. Return the full updated `map_sub_nat` 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.Dynamics.Circle.RotationNumber.TranslationNumber).

[User]
You are editing a file in `Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber.map_sub_nat`, L336-L338)
```lean
theorem map_sub_nat (x : ℝ) (n : ℕ) : f (x - n) = f x - n := f.map_sub_int x n
```

## Edit instruction
Edit `map_sub_nat` so it still compiles and the proof is simplified where possible. Return the full updated `map_sub_nat` 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.Dynamics.Circle.RotationNumber.TranslationNumber).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber.map_sub_nat`, L336-L338)
```lean
theorem map_sub_nat (x : ℝ) (n : ℕ) : f (x - n) = f x - n := f.map_sub_int x n
```

## Edit instruction
Edit `map_sub_nat` so it still compiles and the proof is simplified where possible. Return the full updated `map_sub_nat` 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.AlgebraicTopology.SimplexCategory.Basic).

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

```lean
theorem image_ι_eq {Δ Δ'' : SimplexCategory} {φ : Δ ⟶ Δ''} {e : Δ ⟶ image φ} [Epi e]
    {i : image φ ⟶ Δ''} [Mono i] (fac : e ≫ i = φ) : image.ι φ = i
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.congr_toOrderHom_apply (L42-L45)
lemma congr_toOrderHom_apply {a b : SimplexCategory} {f g : a ⟶ b} (h : f = g)
    (x : Fin (a.len + 1)) : f.toOrderHom x = g.toOrderHom x
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.const (L46-L48)
def const (x y : SimplexCategory) (i : Fin (y.len + 1)) : x ⟶ y
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.const_eq_id (L50-L51)
lemma const_eq_id : const ⦋0⦌ ⦋0⦌ 0 = 𝟙 _
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.const_apply (L53-L55)
lemma const_apply (x y : SimplexCategory) (i : Fin (y.len + 1)) (a : Fin (x.len + 1)) :
    (const x y i).toOrderHom a = i
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.const_comp (L57-L61)
theorem const_comp (x : SimplexCategory) {y z : SimplexCategory}
    (f : y ⟶ z) (i : Fin (y.len + 1)) :
    const x y i ≫ f = const x z (f.toOrderHom i)
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.const_fac_thru_zero (L62-L65)
theorem const_fac_thru_zero (n m : SimplexCategory) (i : Fin (m.len + 1)) :
    const n m i = const n ⦋0⦌ 0 ≫ SimplexCategory.const ⦋0⦌ m i
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.Hom (L66-L69)
theorem Hom.ext_zero_left {n : SimplexCategory} (f g : ⦋0⦌ ⟶ n)
    (h0 : f.toOrderHom 0 = g.toOrderHom 0
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.eq_const_of_zero (L70-L73)
theorem eq_const_of_zero {n : SimplexCategory} (f : ⦋0⦌ ⟶ n) :
    f = const _ n (f.toOrderHom 0)
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.AugmentedSimplexCategory (L34-L40)
abbrev AugmentedSimplexCategory
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.inclusion (L42-L43)
def inclusion : SimplexCategory ⥤ AugmentedSimplexCategory
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObjectFunctorCompDropIso (L61-L67)
def equivAugmentedCosimplicialObjectFunctorCompDropIso :
    equivAugmentedCosimplicialObject.functor ⋙ CosimplicialObject.Augmented.drop ≅
    (Functor.whiskeringLeft _ _ C).obj inclusion
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObjectFunctorCompPointIso (L69-L76)
def equivAugmentedCosimplicialObjectFunctorCompPointIso :
    equivAugmentedCosimplicialObject.functor ⋙ CosimplicialObject.Augmented.point ≅
    ((evaluation _ _).obj .star : (AugmentedSimplexCategory ⥤ C) ⥤ C)
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObject (L53-L59)
def equivAugmentedCosimplicialObject :
    (AugmentedSimplexCategory ⥤ C) ≌ CosimplicialObject.Augmented C
```

-- snippet 14
```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)
```

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

```lean
theorem image_ι_eq {Δ Δ'' : SimplexCategory} {φ : Δ ⟶ Δ''} {e : Δ ⟶ image φ} [Epi e]
    {i : image φ ⟶ Δ''} [Mono i] (fac : e ≫ i = φ) : image.ι φ = i
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.congr_toOrderHom_apply (L42-L45)
lemma congr_toOrderHom_apply {a b : SimplexCategory} {f g : a ⟶ b} (h : f = g)
    (x : Fin (a.len + 1)) : f.toOrderHom x = g.toOrderHom x
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.const (L46-L48)
def const (x y : SimplexCategory) (i : Fin (y.len + 1)) : x ⟶ y
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.const_eq_id (L50-L51)
lemma const_eq_id : const ⦋0⦌ ⦋0⦌ 0 = 𝟙 _
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.const_apply (L53-L55)
lemma const_apply (x y : SimplexCategory) (i : Fin (y.len + 1)) (a : Fin (x.len + 1)) :
    (const x y i).toOrderHom a = i
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.const_comp (L57-L61)
theorem const_comp (x : SimplexCategory) {y z : SimplexCategory}
    (f : y ⟶ z) (i : Fin (y.len + 1)) :
    const x y i ≫ f = const x z (f.toOrderHom i)
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.const_fac_thru_zero (L62-L65)
theorem const_fac_thru_zero (n m : SimplexCategory) (i : Fin (m.len + 1)) :
    const n m i = const n ⦋0⦌ 0 ≫ SimplexCategory.const ⦋0⦌ m i
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.Hom (L66-L69)
theorem Hom.ext_zero_left {n : SimplexCategory} (f g : ⦋0⦌ ⟶ n)
    (h0 : f.toOrderHom 0 = g.toOrderHom 0
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Basic.eq_const_of_zero (L70-L73)
theorem eq_const_of_zero {n : SimplexCategory} (f : ⦋0⦌ ⟶ n) :
    f = const _ n (f.toOrderHom 0)
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.AugmentedSimplexCategory (L34-L40)
abbrev AugmentedSimplexCategory
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.inclusion (L42-L43)
def inclusion : SimplexCategory ⥤ AugmentedSimplexCategory
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObjectFunctorCompDropIso (L61-L67)
def equivAugmentedCosimplicialObjectFunctorCompDropIso :
    equivAugmentedCosimplicialObject.functor ⋙ CosimplicialObject.Augmented.drop ≅
    (Functor.whiskeringLeft _ _ C).obj inclusion
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObjectFunctorCompPointIso (L69-L76)
def equivAugmentedCosimplicialObjectFunctorCompPointIso :
    equivAugmentedCosimplicialObject.functor ⋙ CosimplicialObject.Augmented.point ≅
    ((evaluation _ _).obj .star : (AugmentedSimplexCategory ⥤ C) ⥤ C)
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Basic.equivAugmentedCosimplicialObject (L53-L59)
def equivAugmentedCosimplicialObject :
    (AugmentedSimplexCategory ⥤ C) ≌ CosimplicialObject.Augmented C
```

-- snippet 14
```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)
```
[/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.Characterization).

[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.Characterization.isDiscrete_tfae`:
State 0: ⊢ theorem isDiscrete_tfae (X : LightCondSet.{u}) :
    TFAE
    [ X.IsDiscrete
    , IsIso ((LightCondensed.discreteUnderlyingAdj _).counit.app X)
    , (LightCondensed.discrete _).essImage X
    , LightCondSet.LocallyConstant.functor.essImage X
    , IsIso (LightCondSet.LocallyConstant.adjunction.counit.app X)
    , ∀ S : LightProfinite.{u}, Nonempty
        (IsColimit <| X.obj.mapCocone (coconeRightOpOfCone S.asLimitCone))
    ]
Tactic: have : functor.Faithful := inferInstance
State 1: ⊢ <goal after have : functor.Faithful := inferInstance>
Tactic: have : functor.Full := inferInstance  -- These `have` statements above shouldn't be needed, but they are.  Sheaf.isConstant_iff_isIso_counit_app' _ LightProfinite.isTerminalPUnit adjunction X  tfae_have 6 → 4 := fun h ↦  mem_locallyConstant_essImage_of_isColimit_mapCocone X (fun S ↦ (h S).some)  tfae_have 4 → 6 := fun ⟨Y, ⟨i⟩⟩ S ↦  ⟨IsColimit.mapCoconeEquiv ((sheafToPresheaf _ _).mapIso i)  (LightCondensed.isColimitLocallyConstantPresheafDiagram Y S)⟩  tfae_finish  end LightCondSet  namespace LightCondMod  variable (R : Type u) [Ring R]
```

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.Characterization).[/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.Characterization.isDiscrete_tfae`:
State 0: ⊢ theorem isDiscrete_tfae (X : LightCondSet.{u}) :
    TFAE
    [ X.IsDiscrete
    , IsIso ((LightCondensed.discreteUnderlyingAdj _).counit.app X)
    , (LightCondensed.discrete _).essImage X
    , LightCondSet.LocallyConstant.functor.essImage X
    , IsIso (LightCondSet.LocallyConstant.adjunction.counit.app X)
    , ∀ S : LightProfinite.{u}, Nonempty
        (IsColimit <| X.obj.mapCocone (coconeRightOpOfCone S.asLimitCone))
    ]
Tactic: have : functor.Faithful := inferInstance
State 1: ⊢ <goal after have : functor.Faithful := inferInstance>
Tactic: have : functor.Full := inferInstance  -- These `have` statements above shouldn't be needed, but they are.  Sheaf.isConstant_iff_isIso_counit_app' _ LightProfinite.isTerminalPUnit adjunction X  tfae_have 6 → 4 := fun h ↦  mem_locallyConstant_essImage_of_isColimit_mapCocone X (fun S ↦ (h S).some)  tfae_have 4 → 6 := fun ⟨Y, ⟨i⟩⟩ S ↦  ⟨IsColimit.mapCoconeEquiv ((sheafToPresheaf _ _).mapIso i)  (LightCondensed.isColimitLocallyConstantPresheafDiagram Y S)⟩  tfae_finish  end LightCondSet  namespace LightCondMod  variable (R : Type u) [Ring R]
```

Question: what is the next tactic after the last recorded state, and why?[/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
theorem symm_symm_apply (f : α ≃ β) (b : α) : f.symm.symm b = f b
```

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

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

```lean
theorem symm_symm_apply (f : α ≃ β) (b : α) : f.symm.symm b = f b
```

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


Instruction: Edit `union_apply_left` so it still compiles and the proof is simplified where possible. Return the full updated `union_apply_left` 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.Logic.Equiv.Set).

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

## Current excerpt (around `Mathlib.Logic.Equiv.Set.union_apply_left`, L236-L239)
```lean
theorem union_apply_left {α} {s t : Set α} [DecidablePred fun x => x ∈ s] (H : Disjoint s t)
    {a : (s ∪ t : Set α)} (ha : ↑a ∈ s) : Equiv.Set.union H a = Sum.inl ⟨a, ha⟩ := dif_pos ha
```

## Edit instruction
Edit `union_apply_left` so it still compiles and the proof is simplified where possible. Return the full updated `union_apply_left` 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.Logic.Equiv.Set).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Logic/Equiv/Set.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Logic.Equiv.Set.union_apply_left`, L236-L239)
```lean
theorem union_apply_left {α} {s t : Set α} [DecidablePred fun x => x ∈ s] (H : Disjoint s t)
    {a : (s ∪ t : Set α)} (ha : ↑a ∈ s) : Equiv.Set.union H a = Sum.inl ⟨a, ha⟩ := dif_pos ha
```

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


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

[User]
Complete the following proof.

```lean
def mapCechConerve {f g : Arrow C}
    [∀ n : ℕ, HasWidePushout f.left (fun _ : Fin (n + 1) => f.right) fun _ => f.hom]
    [∀ n : ℕ, HasWidePushout g.left (fun _ : Fin (n + 1) => g.right) fun _ => g.hom] (F : f ⟶ g) :
    f.cechConerve ⟶ g.cechConerve where
  app n
```

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

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

```lean
def mapCechConerve {f g : Arrow C}
    [∀ n : ℕ, HasWidePushout f.left (fun _ : Fin (n + 1) => f.right) fun _ => f.hom]
    [∀ n : ℕ, HasWidePushout g.left (fun _ : Fin (n + 1) => g.right) fun _ => g.hom] (F : f ⟶ g) :
    f.cechConerve ⟶ g.cechConerve where
  app n
```

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


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

[User]
Complete the following proof.

```lean
def decodeBool : List Bool → Bool
  | b :: _ => b
  | _ => Inhabited.default
```

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

```lean
def decodeBool : List Bool → Bool
  | b :: _ => b
  | _ => Inhabited.default
```

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.Control.ULiftable).

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

```lean
theorem up_down {f : Type u₀ → Type u₁} {g : Type max u₀ v₀ → Type v₁} [ULiftable f g] {α}
    (x : g (ULift.{v₀} α)) : up (down x : f α) = x
:= thisIsWrongLemmaNameXYZ
(ULiftable.congr Equiv.ulift.symm).right_inv _
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=234, messages=[Message(data="Function expected at\n  up\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  (down x : f α)\n\nHint: The identifier `up` 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=46), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), 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.Control.ULiftable).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem up_down {f : Type u₀ → Type u₁} {g : Type max u₀ v₀ → Type v₁} [ULiftable f g] {α}
    (x : g (ULift.{v₀} α)) : up (down x : f α) = x
:= thisIsWrongLemmaNameXYZ
(ULiftable.congr Equiv.ulift.symm).right_inv _
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=234, messages=[Message(data="Function expected at\n  up\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  (down x : f α)\n\nHint: The identifier `up` 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=46), severity=<Severity.ERROR: 3>, kind=None), Message(data='Unknown identifier `thisIsWrongLemmaNameXYZ`', pos=Position(line=3, column=3), pos_end=Position(line=3, column=26), 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]


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

[User]
Complete the following proof.

```lean
theorem map_birkhoffSum {F N : Type*} [AddCommMonoid N] [FunLike F M N] [AddMonoidHomClass F M N]
    (g' : F) (f : α → α) (g : α → M) (n : ℕ) (x : α) :
    g' (birkhoffSum f g n x) = birkhoffSum f (g' ∘ g) n x
```

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

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

```lean
theorem map_birkhoffSum {F N : Type*} [AddCommMonoid N] [FunLike F M N] [AddMonoidHomClass F M N]
    (g' : F) (f : α → α) (g : α → M) (n : ℕ) (x : α) :
    g' (birkhoffSum f g n x) = birkhoffSum f (g' ∘ g) n x
```

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


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

[User]
Complete the following proof.

```lean
theorem jacobiSymNat.zero_right (a : ℕ) : jacobiSymNat a 0 = 1
```

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

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

```lean
theorem jacobiSymNat.zero_right (a : ℕ) : jacobiSymNat a 0 = 1
```

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


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

[User]
Complete the following proof.

```lean
theorem eval_append_singleton (x : List α) (a : α) : M.eval (x ++ [a]) = M.stepSet (M.eval x) 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.Computability.EpsilonNFA).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
theorem eval_append_singleton (x : List α) (a : α) : M.eval (x ++ [a]) = M.stepSet (M.eval x) 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.Quasicategory.StrictSegal).

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

```lean
instance quasicategory' (X : SSet.{u}) [IsStrictSegal X] : Quasicategory X
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.StrictSegal.quasicategory (L33-L97)
theorem quasicategory {X : SSet.{u}} (sx : StrictSegal X) : Quasicategory X := by
apply quasicategory_of_filler X
  intro n i σ₀ h₀ hₙ
  use sx.spineToSimplex <| Path.map (horn.spineId i h₀ hₙ) σ₀
  intro j hj
  apply sx.spineInjective
  ext k
  dsimp only [spineEquiv, spine_arrow, Function.comp_apply, Equiv.coe_fn_mk]
  rw [← types_comp_apply (σ₀.app _) (X.map _), ← σ₀.naturality]
  let ksucc := k.succ.castSucc
  obtain hlt | hgt | heq : ksucc < j ∨ j < ksucc ∨ j = ksucc := by lia
  · rw [← spine_arrow, spine_δ_arrow_lt sx _ hlt]
    dsimp only [Path.map_arrow, spine_arrow, Fin.coe_eq_castSucc]
    apply congr_arg
    apply Subtype.ext
    dsimp [horn.face, CosimplicialObject.δ]
    rw [Subcomplex.yonedaEquiv_coe, Subfunctor.lift_ι, stdSimplex.map_apply,
      Quiver.Hom.unop_op, stdSimplex.yonedaEquiv_map, Equiv.apply_symm_apply,
      mkOfSucc_δ_lt hlt]
    rfl
  · rw [← spine_arrow, spine_δ_arrow_gt sx _ hgt]
    dsimp only [Path.map_arrow, spine_arrow, Fin.coe_eq_castSucc]
    apply congr_arg
    apply Subtype.ext
    dsimp [horn.face, CosimplicialObject.δ]
    rw [Subcomplex.yonedaEquiv_coe, Subfunctor.lift_ι, stdSimplex.map_apply,
      Quiver.Hom.unop_op, stdSimplex.yonedaEquiv_map, Equiv.apply_symm_apply,
      mkOfSucc_δ_gt hgt]
    rfl
  · obtain _ | n := n
    · /- The only inner horn of `Δ[2]` does not contain the diagonal edge. -/
      obtain rfl : k = 0 := by omega
      fin_cases i <;> contradiction
    · /- We construct the triangle in the standard simplex as a 2-simplex in
      the horn. While the triangle is not contained in the inner horn `Λ[2, 1]`,
      it suffices to inhabit `Λ[n + 3, i] _⦋2⦌`. -/
      let triangle : (Λ[n + 3, i] : SSet.{u}) _⦋2⦌ :=
        horn.primitiveTriangle i h₀ hₙ k (by lia)
      /- The interval spanning from `k` to `k + 2` is equivalently the spine
      of the triangle with vertices `k`, `k + 1`, and `k + 2`. -/
      have hi : ((horn.spineId i h₀ hₙ).map σ₀).interval k 2 (by lia) =
          X.spine 2 (σ₀.app _ triangle) := by
        ext m
        dsimp [spine_arrow, Path.map_interval, Path.map_arrow]
        rw [← types_comp_apply (σ₀.app _) (X.map _), ← σ₀.naturality]
        apply congr_arg
        apply Subtype.ext
        ext a : 1
        fin_cases a <;> fin_cases m <;> rfl
      rw [← spine_arrow, spine_δ_arrow_eq sx _ heq, hi]
      simp only [spineToDiagonal, diagonal, spineToSimplex_spine_apply]
      rw [← types_comp_apply (σ₀.app _) (X.map _), ← σ₀.naturality, types_comp_apply]
      apply congr_arg
      apply Subtype.ext
      ext z : 1
      dsimp [horn.face]
      rw [Subcomplex.yonedaEquiv_coe, Subfunctor.lift_ι, stdSimplex.map_apply,
        Quiver.Hom.unop_op, stdSimplex.map_apply, Quiver.Hom.unop_op]
      dsimp [CosimplicialObject.δ]
      rw [stdSimplex.yonedaEquiv_map]
      simp only [Equiv.apply_symm_apply, triangle]
      rw [mkOfSucc_δ_eq heq]
      fin_cases z <;> rfl

/-- Any simplicial set satisfying `IsStrictSegal` is a `Quasicategory`. -/
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.homotopicL_iff_homotopicR (L155-L168)
theorem homotopicL_iff_homotopicR [Quasicategory₂ X] {x y : X _⦋0⦌₂} {f g : Edge x y} :
    HomotopicL f g ↔ HomotopicR f g :=
⟨HomotopicL.homotopicR, HomotopicR.homotopicL⟩

end homotopy_eqrel

section homotopy_category

variable {A : Truncated 2} [Quasicategory₂ A] {x y z : A _⦋0⦌₂}

/--
Given `CompStruct f g h` and `CompStruct f' g' h'` with the same vertices and edges such
that `f` ≃ `f'` and `g` ≃ `g'`, then the long diagonal edges `h` and `h'` are also homotopic.
-/
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.homMk_surjective (L245-L250)
lemma homMk_surjective : Function.Surjective (homMk : Edge x y → _) := Quotient.mk_surjective
/--
The trivial (degenerate) edge at a vertex `x` is a representative for the
identity morphism `x ⟶ x`.
-/
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.HomotopicR (L86-L94)
abbrev HomotopicR {X : Truncated 2} {x y : X _⦋0⦌₂} (f g : Edge x y) :=
Nonempty (CompStruct (id x) f g)

section homotopy_eqrel
variable {X : Truncated 2}

/--
The left homotopy relation is reflexive.
-/
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.Basic.quasicategory_of_filler (L61-L74)
lemma quasicategory_of_filler (S : SSet)
    (filler : ∀ ⦃n : ℕ⦄ ⦃i : Fin (n + 3)⦄ (σ₀ : (Λ[n + 2, i] : SSet) ⟶ S)
      (_h0 : 0 < i) (_hn : i < Fin.last (n + 2)),
      ∃ σ : S _⦋n + 2⦌, ∀ (j) (h : j ≠ i), S.δ j σ = σ₀.app _ (horn.face i j h)) :
    Quasicategory S where
  hornFilling' n i σ₀ h₀ hₙ := by
obtain ⟨σ, h⟩ := filler σ₀ h₀ hₙ
    refine ⟨yonedaEquiv.symm σ, ?_⟩
    apply horn.hom_ext
    intro j hj
    rw [← h j hj, NatTrans.comp_app]
    rfl

end SSet
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.Hom (L213-L219)
def Hom (x y : HomotopyCategory₂ A) := Quotient (instSetoidEdge x.pt y.pt)
/--
Composition of morphisms in `HomotopyCategory₂ A` is given by lifting the edge
chosen by `composeEdges`.
-/
noncomputable
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.Basic.Quasicategory (L44-L56)
lemma Quasicategory.hornFilling {S : SSet} [Quasicategory S] ⦃n : ℕ⦄ ⦃i : Fin (n + 1)⦄
    (h0 : 0 < i) (hn : i < Fin.last n)
    (σ₀ : (Λ[n, i] : SSet) ⟶ S) : ∃ σ : Δ[n] ⟶ S, σ₀ = Λ[n, i].ι ≫ σ := by
cases n using Nat.casesAuxOn with
  | zero => simp [Fin.lt_def] at hn
  | succ n =>
  cases n using Nat.casesAuxOn with
  | zero =>
    simp only [Fin.lt_def, Fin.val_zero, Fin.val_last, zero_add, Nat.lt_one_iff] at h0 hn
    simp [hn] at h0
  | succ n => exact Quasicategory.hornFilling' σ₀ h0 hn

/-- Every Kan complex is a quasicategory. -/
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.homMk (L239-L244)
def homMk (f : Edge x y) : mk x ⟶ mk y := ⟦f⟧
/--
Every morphism in the homotopy category `HomotopyCategory₂ A` is the equivalence class of
an edge of `A`.
-/
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.mk_surjective (L232-L238)
lemma mk_surjective : Function.Surjective (mk : A _⦋0⦌₂ → _) :=
fun ⟨x⟩ ↦ ⟨x, rfl⟩

/--
Any edge in the 2-truncated simplicial set `A` defines a morphism in the homotopy category
by taking its equivalence class.
-/
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.HomotopicL (L78-L85)
abbrev HomotopicL {X : Truncated 2} {x y : X _⦋0⦌₂} (f g : Edge x y) :=
Nonempty (CompStruct f (id y) g)

/--
Two edges `f` and `g` are right homotopic if there is a `CompStruct` with
(0, 1)-edge `Edge.id`, (1, 2)-edge `f`, and (0, 2)-edge `g`. We use `Nonempty` to
have a `Prop` valued `HomotopicR`.
-/
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.StrictBicategory.QCat (L51-L54)
abbrev QCat := ObjectProperty.FullSubcategory Quasicategory
/-- `QCat` obtains a `Cat`-enriched ordinary category structure by applying `hoFunctor` to the
hom objects in its `SSet`-enriched ordinary structure. -/
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.Edge (L169-L202)
lemma Edge.CompStruct.comp_unique {f f' : Edge x y} {g g' : Edge y z} {h h' : Edge x z}
    (s : CompStruct f g h) (s' : CompStruct f' g' h')
    (hf : HomotopicL f f') (hg : HomotopicL g g') : HomotopicL h h' := by
rcases hg.homotopicR with ⟨hg⟩
  rcases hf with ⟨hf⟩
  let ⟨s₁⟩ := Quasicategory₂.fill32 hf (idComp g') s'
  let ⟨s₂⟩ := Quasicategory₂.fill31 (compId f) hg s₁
  exact Quasicategory₂.fill31 s (compId g) s₂

/--
Given two consecutive edges `f`, `g`  in a 2-truncated quasicategory, nonconstructively choose
an edge that is the diagonal of a 2-simplex with spine given by `f` and `g`. The `CompStruct`
witnessing this property is given by `Edge.composeStruct`.
-/
noncomputable def Edge.comp (f : Edge x y) (g : Edge y z) : Edge x z :=
  (Quasicategory₂.fill21 f g).some.1

/--
See `Edge.comp`
-/
noncomputable def Edge.compStruct (f : Edge x y) (g : Edge y z) : CompStruct f g (f.comp g) :=
  (Quasicategory₂.fill21 f g).some.2

/--
The homotopy category of a 2-truncated quasicategory `A` has as objects the vertices of `A`
-/
structure HomotopyCategory₂ (A : Truncated 2) where
  /-- An object of the homotopy category is a vertex of `A`. -/
  pt : A _⦋0⦌₂

/--
Left homotopy is an equivalence relation on the edges of `A`.
Remark: We could have equivalently chosen right homotopy, as shown by `homotopicL_iff_homotopicR`.
-/
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.Nerve.quasicategory (L32-L34)
instance quasicategory {C : Type u} [Category.{v} C] : Quasicategory (nerve C) := inferInstance
end CategoryTheory.Nerve
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.homMk_id (L252-L260)
lemma homMk_id (x : HomotopyCategory₂ A) : homMk (Edge.id x.pt) = 𝟙 x := rfl
end HomotopyCategory₂

open HomotopyCategory₂

/--
Left homotopic edges represent the same morphism in the homotopy category.
-/
```

-- snippet 15
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.instSetoidEdge (L203-L212)
instance instSetoidEdge (x y : A _⦋0⦌₂) : Setoid (Edge x y) where
  r := HomotopicL
iseqv := ⟨fun _ ↦ HomotopicL.refl, HomotopicL.symm, HomotopicL.trans⟩

namespace HomotopyCategory₂

/--
The morphisms between two vertices `x`, `y` in `HomotopyCategory₂ A` are homotopy classes
of edges between `x` and `y`.
-/
```

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

```lean
instance quasicategory' (X : SSet.{u}) [IsStrictSegal X] : Quasicategory X
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.StrictSegal.quasicategory (L33-L97)
theorem quasicategory {X : SSet.{u}} (sx : StrictSegal X) : Quasicategory X := by
apply quasicategory_of_filler X
  intro n i σ₀ h₀ hₙ
  use sx.spineToSimplex <| Path.map (horn.spineId i h₀ hₙ) σ₀
  intro j hj
  apply sx.spineInjective
  ext k
  dsimp only [spineEquiv, spine_arrow, Function.comp_apply, Equiv.coe_fn_mk]
  rw [← types_comp_apply (σ₀.app _) (X.map _), ← σ₀.naturality]
  let ksucc := k.succ.castSucc
  obtain hlt | hgt | heq : ksucc < j ∨ j < ksucc ∨ j = ksucc := by lia
  · rw [← spine_arrow, spine_δ_arrow_lt sx _ hlt]
    dsimp only [Path.map_arrow, spine_arrow, Fin.coe_eq_castSucc]
    apply congr_arg
    apply Subtype.ext
    dsimp [horn.face, CosimplicialObject.δ]
    rw [Subcomplex.yonedaEquiv_coe, Subfunctor.lift_ι, stdSimplex.map_apply,
      Quiver.Hom.unop_op, stdSimplex.yonedaEquiv_map, Equiv.apply_symm_apply,
      mkOfSucc_δ_lt hlt]
    rfl
  · rw [← spine_arrow, spine_δ_arrow_gt sx _ hgt]
    dsimp only [Path.map_arrow, spine_arrow, Fin.coe_eq_castSucc]
    apply congr_arg
    apply Subtype.ext
    dsimp [horn.face, CosimplicialObject.δ]
    rw [Subcomplex.yonedaEquiv_coe, Subfunctor.lift_ι, stdSimplex.map_apply,
      Quiver.Hom.unop_op, stdSimplex.yonedaEquiv_map, Equiv.apply_symm_apply,
      mkOfSucc_δ_gt hgt]
    rfl
  · obtain _ | n := n
    · /- The only inner horn of `Δ[2]` does not contain the diagonal edge. -/
      obtain rfl : k = 0 := by omega
      fin_cases i <;> contradiction
    · /- We construct the triangle in the standard simplex as a 2-simplex in
      the horn. While the triangle is not contained in the inner horn `Λ[2, 1]`,
      it suffices to inhabit `Λ[n + 3, i] _⦋2⦌`. -/
      let triangle : (Λ[n + 3, i] : SSet.{u}) _⦋2⦌ :=
        horn.primitiveTriangle i h₀ hₙ k (by lia)
      /- The interval spanning from `k` to `k + 2` is equivalently the spine
      of the triangle with vertices `k`, `k + 1`, and `k + 2`. -/
      have hi : ((horn.spineId i h₀ hₙ).map σ₀).interval k 2 (by lia) =
          X.spine 2 (σ₀.app _ triangle) := by
        ext m
        dsimp [spine_arrow, Path.map_interval, Path.map_arrow]
        rw [← types_comp_apply (σ₀.app _) (X.map _), ← σ₀.naturality]
        apply congr_arg
        apply Subtype.ext
        ext a : 1
        fin_cases a <;> fin_cases m <;> rfl
      rw [← spine_arrow, spine_δ_arrow_eq sx _ heq, hi]
      simp only [spineToDiagonal, diagonal, spineToSimplex_spine_apply]
      rw [← types_comp_apply (σ₀.app _) (X.map _), ← σ₀.naturality, types_comp_apply]
      apply congr_arg
      apply Subtype.ext
      ext z : 1
      dsimp [horn.face]
      rw [Subcomplex.yonedaEquiv_coe, Subfunctor.lift_ι, stdSimplex.map_apply,
        Quiver.Hom.unop_op, stdSimplex.map_apply, Quiver.Hom.unop_op]
      dsimp [CosimplicialObject.δ]
      rw [stdSimplex.yonedaEquiv_map]
      simp only [Equiv.apply_symm_apply, triangle]
      rw [mkOfSucc_δ_eq heq]
      fin_cases z <;> rfl

/-- Any simplicial set satisfying `IsStrictSegal` is a `Quasicategory`. -/
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.homotopicL_iff_homotopicR (L155-L168)
theorem homotopicL_iff_homotopicR [Quasicategory₂ X] {x y : X _⦋0⦌₂} {f g : Edge x y} :
    HomotopicL f g ↔ HomotopicR f g :=
⟨HomotopicL.homotopicR, HomotopicR.homotopicL⟩

end homotopy_eqrel

section homotopy_category

variable {A : Truncated 2} [Quasicategory₂ A] {x y z : A _⦋0⦌₂}

/--
Given `CompStruct f g h` and `CompStruct f' g' h'` with the same vertices and edges such
that `f` ≃ `f'` and `g` ≃ `g'`, then the long diagonal edges `h` and `h'` are also homotopic.
-/
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.homMk_surjective (L245-L250)
lemma homMk_surjective : Function.Surjective (homMk : Edge x y → _) := Quotient.mk_surjective
/--
The trivial (degenerate) edge at a vertex `x` is a representative for the
identity morphism `x ⟶ x`.
-/
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.HomotopicR (L86-L94)
abbrev HomotopicR {X : Truncated 2} {x y : X _⦋0⦌₂} (f g : Edge x y) :=
Nonempty (CompStruct (id x) f g)

section homotopy_eqrel
variable {X : Truncated 2}

/--
The left homotopy relation is reflexive.
-/
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.Basic.quasicategory_of_filler (L61-L74)
lemma quasicategory_of_filler (S : SSet)
    (filler : ∀ ⦃n : ℕ⦄ ⦃i : Fin (n + 3)⦄ (σ₀ : (Λ[n + 2, i] : SSet) ⟶ S)
      (_h0 : 0 < i) (_hn : i < Fin.last (n + 2)),
      ∃ σ : S _⦋n + 2⦌, ∀ (j) (h : j ≠ i), S.δ j σ = σ₀.app _ (horn.face i j h)) :
    Quasicategory S where
  hornFilling' n i σ₀ h₀ hₙ := by
obtain ⟨σ, h⟩ := filler σ₀ h₀ hₙ
    refine ⟨yonedaEquiv.symm σ, ?_⟩
    apply horn.hom_ext
    intro j hj
    rw [← h j hj, NatTrans.comp_app]
    rfl

end SSet
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.Hom (L213-L219)
def Hom (x y : HomotopyCategory₂ A) := Quotient (instSetoidEdge x.pt y.pt)
/--
Composition of morphisms in `HomotopyCategory₂ A` is given by lifting the edge
chosen by `composeEdges`.
-/
noncomputable
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.Basic.Quasicategory (L44-L56)
lemma Quasicategory.hornFilling {S : SSet} [Quasicategory S] ⦃n : ℕ⦄ ⦃i : Fin (n + 1)⦄
    (h0 : 0 < i) (hn : i < Fin.last n)
    (σ₀ : (Λ[n, i] : SSet) ⟶ S) : ∃ σ : Δ[n] ⟶ S, σ₀ = Λ[n, i].ι ≫ σ := by
cases n using Nat.casesAuxOn with
  | zero => simp [Fin.lt_def] at hn
  | succ n =>
  cases n using Nat.casesAuxOn with
  | zero =>
    simp only [Fin.lt_def, Fin.val_zero, Fin.val_last, zero_add, Nat.lt_one_iff] at h0 hn
    simp [hn] at h0
  | succ n => exact Quasicategory.hornFilling' σ₀ h0 hn

/-- Every Kan complex is a quasicategory. -/
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.homMk (L239-L244)
def homMk (f : Edge x y) : mk x ⟶ mk y := ⟦f⟧
/--
Every morphism in the homotopy category `HomotopyCategory₂ A` is the equivalence class of
an edge of `A`.
-/
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.mk_surjective (L232-L238)
lemma mk_surjective : Function.Surjective (mk : A _⦋0⦌₂ → _) :=
fun ⟨x⟩ ↦ ⟨x, rfl⟩

/--
Any edge in the 2-truncated simplicial set `A` defines a morphism in the homotopy category
by taking its equivalence class.
-/
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.HomotopicL (L78-L85)
abbrev HomotopicL {X : Truncated 2} {x y : X _⦋0⦌₂} (f g : Edge x y) :=
Nonempty (CompStruct f (id y) g)

/--
Two edges `f` and `g` are right homotopic if there is a `CompStruct` with
(0, 1)-edge `Edge.id`, (1, 2)-edge `f`, and (0, 2)-edge `g`. We use `Nonempty` to
have a `Prop` valued `HomotopicR`.
-/
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.StrictBicategory.QCat (L51-L54)
abbrev QCat := ObjectProperty.FullSubcategory Quasicategory
/-- `QCat` obtains a `Cat`-enriched ordinary category structure by applying `hoFunctor` to the
hom objects in its `SSet`-enriched ordinary structure. -/
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.Edge (L169-L202)
lemma Edge.CompStruct.comp_unique {f f' : Edge x y} {g g' : Edge y z} {h h' : Edge x z}
    (s : CompStruct f g h) (s' : CompStruct f' g' h')
    (hf : HomotopicL f f') (hg : HomotopicL g g') : HomotopicL h h' := by
rcases hg.homotopicR with ⟨hg⟩
  rcases hf with ⟨hf⟩
  let ⟨s₁⟩ := Quasicategory₂.fill32 hf (idComp g') s'
  let ⟨s₂⟩ := Quasicategory₂.fill31 (compId f) hg s₁
  exact Quasicategory₂.fill31 s (compId g) s₂

/--
Given two consecutive edges `f`, `g`  in a 2-truncated quasicategory, nonconstructively choose
an edge that is the diagonal of a 2-simplex with spine given by `f` and `g`. The `CompStruct`
witnessing this property is given by `Edge.composeStruct`.
-/
noncomputable def Edge.comp (f : Edge x y) (g : Edge y z) : Edge x z :=
  (Quasicategory₂.fill21 f g).some.1

/--
See `Edge.comp`
-/
noncomputable def Edge.compStruct (f : Edge x y) (g : Edge y z) : CompStruct f g (f.comp g) :=
  (Quasicategory₂.fill21 f g).some.2

/--
The homotopy category of a 2-truncated quasicategory `A` has as objects the vertices of `A`
-/
structure HomotopyCategory₂ (A : Truncated 2) where
  /-- An object of the homotopy category is a vertex of `A`. -/
  pt : A _⦋0⦌₂

/--
Left homotopy is an equivalence relation on the edges of `A`.
Remark: We could have equivalently chosen right homotopy, as shown by `homotopicL_iff_homotopicR`.
-/
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.Nerve.quasicategory (L32-L34)
instance quasicategory {C : Type u} [Category.{v} C] : Quasicategory (nerve C) := inferInstance
end CategoryTheory.Nerve
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.homMk_id (L252-L260)
lemma homMk_id (x : HomotopyCategory₂ A) : homMk (Edge.id x.pt) = 𝟙 x := rfl
end HomotopyCategory₂

open HomotopyCategory₂

/--
Left homotopic edges represent the same morphism in the homotopy category.
-/
```

-- snippet 15
```lean
-- Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated.instSetoidEdge (L203-L212)
instance instSetoidEdge (x y : A _⦋0⦌₂) : Setoid (Edge x y) where
  r := HomotopicL
iseqv := ⟨fun _ ↦ HomotopicL.refl, HomotopicL.symm, HomotopicL.trans⟩

namespace HomotopyCategory₂

/--
The morphisms between two vertices `x`, `y` in `HomotopyCategory₂ A` are homotopy classes
of edges between `x` and `y`.
-/
```
[/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.DoldKan.Faces).

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

```lean
theorem comp_Hσ_eq_zero {Y : C} {n q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} (v : HigherFacesVanish q φ)
    (hqn : n < q) : φ ≫ (Hσ q).f (n + 1) = 0
:= by .someMissingNamespaceXYZ
simp only [Hσ, Homotopy.nullHomotopicMap'_f (c_mk (n + 2) (n + 1) rfl) (c_mk (n + 1) n rfl)]
  rw [hσ'_eq_zero hqn (c_mk (n + 1) n rfl), comp_zero, zero_add]
  by_cases hqn' : n + 1 < q
  · rw [hσ'_eq_zero hqn' (c_mk (n + 2) (n + 1) rfl), zero_comp, comp_zero]
  · simp only [hσ'_eq (show n + 1 = 0 + q by lia) (c_mk (n + 2) (n + 1) rfl), pow_zero,
      Fin.mk_zero, one_zsmul, eqToHom_refl, comp_id, comp_sum,
      AlternatingFaceMapComplex.obj_d_eq]
    -- All terms of the sum but the first two are zeros
    rw [Fin.sum_univ_succ, Fin.sum_univ_succ, Fintype.sum_eq_zero, add_zero]
    · simp only [Fin.val_zero, Fin.val_succ, Fin.val_castSucc, zero_add, pow_zero, one_smul,
        pow_one, neg_smul, comp_neg, ← Fin.castSucc_zero (n := n + 2), δ_comp_σ_self, δ_comp_σ_succ,
        add_neg_cancel]
    · intro j
      rw [comp_zsmul, comp_zsmul, δ_comp_σ_of_gt', v.comp_δ_eq_zero_assoc, zero_comp, zsmul_zero]
      · simp [Fin.succ_ne_zero]
      · dsimp
        lia
      · simp only [Fin.succ_lt_succ_iff, j.succ_pos]

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=60, messages=[Message(data='expected token', pos=Position(line=1, column=54), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=60, i_end=1287, messages=[Message(data='unexpected end of input', pos=Position(line=23, column=56), 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.AlgebraicTopology.DoldKan.Faces).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem comp_Hσ_eq_zero {Y : C} {n q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} (v : HigherFacesVanish q φ)
    (hqn : n < q) : φ ≫ (Hσ q).f (n + 1) = 0
:= by .someMissingNamespaceXYZ
simp only [Hσ, Homotopy.nullHomotopicMap'_f (c_mk (n + 2) (n + 1) rfl) (c_mk (n + 1) n rfl)]
  rw [hσ'_eq_zero hqn (c_mk (n + 1) n rfl), comp_zero, zero_add]
  by_cases hqn' : n + 1 < q
  · rw [hσ'_eq_zero hqn' (c_mk (n + 2) (n + 1) rfl), zero_comp, comp_zero]
  · simp only [hσ'_eq (show n + 1 = 0 + q by lia) (c_mk (n + 2) (n + 1) rfl), pow_zero,
      Fin.mk_zero, one_zsmul, eqToHom_refl, comp_id, comp_sum,
      AlternatingFaceMapComplex.obj_d_eq]
    -- All terms of the sum but the first two are zeros
    rw [Fin.sum_univ_succ, Fin.sum_univ_succ, Fintype.sum_eq_zero, add_zero]
    · simp only [Fin.val_zero, Fin.val_succ, Fin.val_castSucc, zero_add, pow_zero, one_smul,
        pow_one, neg_smul, comp_neg, ← Fin.castSucc_zero (n := n + 2), δ_comp_σ_self, δ_comp_σ_succ,
        add_neg_cancel]
    · intro j
      rw [comp_zsmul, comp_zsmul, δ_comp_σ_of_gt', v.comp_δ_eq_zero_assoc, zero_comp, zsmul_zero]
      · simp [Fin.succ_ne_zero]
      · dsimp
        lia
      · simp only [Fin.succ_lt_succ_iff, j.succ_pos]

set_option backward.isDefEq.respectTransparency false in
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=60, messages=[Message(data='expected token', pos=Position(line=1, column=54), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=60, i_end=1287, messages=[Message(data='unexpected end of input', pos=Position(line=23, column=56), 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. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne).

[User]
Complete the following proof.

```lean
lemma toMk₁_of_castSucc_lt {n : ℕ} (i : Fin (n + 2)) (j : Fin (n + 1)) (h : j.castSucc < i) :
    dsimp% toMk₁ i j = 0
```

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

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

```lean
lemma toMk₁_of_castSucc_lt {n : ℕ} (i : Fin (n + 2)) (j : Fin (n + 1)) (h : j.castSucc < i) :
    dsimp% toMk₁ i j = 0
```

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


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

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

```lean
def Nat.divModEquiv (n : ℕ) [NeZero n] : ℕ ≃ ℕ × Fin n where
  toFun a
:= (a / n, Fin.ofNat n a) thisIsWrongLemmaNameXYZ
invFun p := p.1 * n + ↑p.2
  -- TODO: is there a canonical order of `*` and `+` here?
  left_inv _ := Nat.div_add_mod' _ _
  right_inv p := by
    refine Prod.ext ?_ (Fin.ext <| Nat.mul_add_mod_of_lt p.2.is_lt)
    dsimp only
    rw [Nat.add_comm, Nat.add_mul_div_right _ _ n.pos_of_neZero, Nat.div_eq_of_lt p.2.is_lt,
      Nat.zero_add]

/-- The equivalence induced by `a ↦ (a / n, a % n)` for nonzero `n`.
See `Int.ediv_emod_unique` for a similar propositional statement. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=130, messages=[Message(data='`Nat.divModEquiv` has already been declared', pos=Position(line=1, column=4), pos_end=Position(line=1, column=19), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=130, i_end=611, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=0), pos_end=Position(line=4, column=6), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=14, column=68), 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.Equiv.Fin.Basic).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
def Nat.divModEquiv (n : ℕ) [NeZero n] : ℕ ≃ ℕ × Fin n where
  toFun a
:= (a / n, Fin.ofNat n a) thisIsWrongLemmaNameXYZ
invFun p := p.1 * n + ↑p.2
  -- TODO: is there a canonical order of `*` and `+` here?
  left_inv _ := Nat.div_add_mod' _ _
  right_inv p := by
    refine Prod.ext ?_ (Fin.ext <| Nat.mul_add_mod_of_lt p.2.is_lt)
    dsimp only
    rw [Nat.add_comm, Nat.add_mul_div_right _ _ n.pos_of_neZero, Nat.div_eq_of_lt p.2.is_lt,
      Nat.zero_add]

/-- The equivalence induced by `a ↦ (a / n, a % n)` for nonzero `n`.
See `Int.ediv_emod_unique` for a similar propositional statement. -/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=130, messages=[Message(data='`Nat.divModEquiv` has already been declared', pos=Position(line=1, column=4), pos_end=Position(line=1, column=19), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=130, i_end=611, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=0), pos_end=Position(line=4, column=6), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=14, column=68), 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. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Logic.Function.OfArity).

[User]
Complete the following proof.

```lean
def fromTypes_fin_const_equiv (α β : Type u) (n : ℕ) :
    FromTypes (fun (_ : Fin n) => α) β ≃ OfArity α β n
```

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

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

```lean
def fromTypes_fin_const_equiv (α β : Type u) (n : ℕ) :
    FromTypes (fun (_ : Fin n) => α) β ≃ OfArity α β n
```

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


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

[User]
Complete the following proof.

```lean
theorem floor_sub_le_translationNumber (x : ℝ) : ↑⌊f x - x⌋ ≤ τ f
```

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

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

```lean
theorem floor_sub_le_translationNumber (x : ℝ) : ↑⌊f x - x⌋ ≤ τ f
```

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


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

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

```lean
theorem option_some : Computable (@Option.some α)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Computability.Partrec.rfindX (L65-L90)
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun _ => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        intro n h
        rcases h.lt_or_eq_dec with h | h
        · exact al _ h
        · rw [h]
          exact ⟨_, e⟩
      · exact ⟨m, ⟨_, e⟩, al⟩)

end Rfind

/-- Find the smallest `n` satisfying `p n`, where all `p k` for `k < n` are defined as false.
Returns a `Part`. -/
```

-- snippet 2
```lean
-- Mathlib.Computability.Partrec.rfind (L91-L93)
def rfind (p : ℕ →. Bool) : Part ℕ :=
⟨_, fun h => (rfindX p h).1⟩
```

-- snippet 3
```lean
-- Mathlib.Computability.Partrec.rfind_spec (L94-L96)
theorem rfind_spec {p : ℕ →. Bool} {n : ℕ} (h : n ∈ rfind p) : true ∈ p n :=
h.snd ▸ (rfindX p h.fst).2.1
```

-- snippet 4
```lean
-- Mathlib.Computability.Partrec.rfind_min (L97-L99)
theorem rfind_min {p : ℕ →. Bool} {n : ℕ} (h : n ∈ rfind p) : ∀ {m : ℕ}, m < n → false ∈ p m :=
@(h.snd ▸ @((rfindX p h.fst).2.2))
```

-- snippet 5
```lean
-- Mathlib.Computability.Partrec.rfind_dom (L101-L104)
theorem rfind_dom {p : ℕ →. Bool} :
    (rfind p).Dom ↔ ∃ n, true ∈ p n ∧ ∀ {m : ℕ}, m < n → (p m).Dom :=
Iff.rfl
```

-- snippet 6
```lean
-- Mathlib.Computability.Partrec.rfind_dom' (L105-L111)
theorem rfind_dom' {p : ℕ →. Bool} :
    (rfind p).Dom ↔ ∃ n, true ∈ p n ∧ ∀ {m : ℕ}, m ≤ n → (p m).Dom :=
exists_congr fun _ =>
    and_congr_right fun pn =>
      ⟨fun H _ h => (Decidable.eq_or_lt_of_le h).elim (fun e => e.symm ▸ pn.fst) (H _), fun H _ h =>
        H (le_of_lt h)⟩
```

-- snippet 7
```lean
-- Mathlib.Computability.Partrec.mem_rfind (L113-L121)
theorem mem_rfind {p : ℕ →. Bool} {n : ℕ} :
    n ∈ rfind p ↔ true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m :=
⟨fun h => ⟨rfind_spec h, @rfind_min _ _ h⟩, fun ⟨h₁, h₂⟩ => by
    let ⟨m, hm⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨_, h₁, fun {m} mn => (h₂ mn).fst⟩
    rcases lt_trichotomy m n with (h | h | h)
    · injection mem_unique (h₂ h) (rfind_spec hm)
    · rwa [← h]
    · injection mem_unique h₁ (rfind_min hm h)⟩
```

-- snippet 8
```lean
-- Mathlib.Computability.Partrec.rfind_min' (L122-L126)
theorem rfind_min' {p : ℕ → Bool} {m : ℕ} (pm : p m) : ∃ n ∈ rfind p, n ≤ m :=
have : true ∈ (p : ℕ →. Bool) m := ⟨trivial, pm⟩
  let ⟨n, hn⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨m, this, fun {_} _ => ⟨⟩⟩
  ⟨n, hn, not_lt.1 fun h => by injection mem_unique this (rfind_min hn h)⟩
```

-- snippet 9
```lean
-- Mathlib.Computability.Partrec.rfind_zero_none (L127-L133)
theorem rfind_zero_none (p : ℕ →. Bool) (p0 : p 0 = Part.none) : rfind p = Part.none :=
eq_none_iff.2 fun _ h =>
    let ⟨_, _, h₂⟩ := rfind_dom'.1 h.fst
    (p0 ▸ h₂ (zero_le _) : (@Part.none Bool).Dom)

/-- Find the smallest `n` satisfying `f n`, where all `f k` for `k < n` are defined as false.
Returns a `Part`. -/
```

-- snippet 10
```lean
-- Mathlib.Computability.Partrec.rfindOpt (L134-L136)
def rfindOpt {α} (f : ℕ → Option α) : Part α :=
(rfind fun n => (f n).isSome).bind fun n => f n
```

-- snippet 11
```lean
-- Mathlib.Computability.Partrec.rfindOpt_spec (L137-L140)
theorem rfindOpt_spec {α} {f : ℕ → Option α} {a} (h : a ∈ rfindOpt f) : ∃ n, a ∈ f n :=
let ⟨n, _, h₂⟩ := mem_bind_iff.1 h
  ⟨n, mem_coe.1 h₂⟩
```

-- snippet 12
```lean
-- Mathlib.Computability.Partrec.rfindOpt_dom (L141-L150)
theorem rfindOpt_dom {α} {f : ℕ → Option α} : (rfindOpt f).Dom ↔ ∃ n a, a ∈ f n :=
⟨fun h => (rfindOpt_spec ⟨h, rfl⟩).imp fun _ h => ⟨_, h⟩, fun h => by
    have h' : ∃ n, (f n).isSome := h.imp fun n => Option.isSome_iff_exists.2
    have s := Nat.find_spec h'
    have fd : (rfind fun n => (f n).isSome).Dom :=
      ⟨Nat.find h', by simpa using s.symm, fun _ _ => trivial⟩
    refine ⟨fd, ?_⟩
    have := rfind_spec (get_mem fd)
    simpa using this⟩
```

-- snippet 13
```lean
-- Mathlib.Computability.Partrec.rfindOpt_mono (L151-L173)
theorem rfindOpt_mono {α} {f : ℕ → Option α} (H : ∀ {a m n}, m ≤ n → a ∈ f m → a ∈ f n) {a} :
    a ∈ rfindOpt f ↔ ∃ n, a ∈ f n :=
⟨rfindOpt_spec, fun ⟨n, h⟩ => by
    have h' := rfindOpt_dom.2 ⟨_, _, h⟩
    obtain ⟨k, hk⟩ := rfindOpt_spec ⟨h', rfl⟩
    have := (H (le_max_left _ _) h).symm.trans (H (le_max_right _ _) hk)
    simp at this; simp [this, get_mem]⟩

/-- `Nat.Partrec f` means that the partial function `f : ℕ →. ℕ` is partially recursive. -/
protected inductive Partrec : (ℕ →. ℕ) → Prop
  | zero : Nat.Partrec (pure 0)
  | succ : Nat.Partrec succ
  | left : Nat.Partrec ↑fun n : ℕ => n.unpair.1
  | right : Nat.Partrec ↑fun n : ℕ => n.unpair.2
  | pair {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec fun n => pair <$> f n <*> g n
  | comp {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec fun n => g n >>= f
  | prec {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec (unpaired fun a n =>
      n.rec (f a) fun y IH => do let i ← IH; g (pair a (pair y i)))
  | rfind {f} : Nat.Partrec f →
    Nat.Partrec fun a => rfind fun n => (fun m => m = 0) <$> f (pair a n)

namespace Partrec
```

-- snippet 14
```lean
-- Mathlib.Computability.Partrec.of_eq (L174-L176)
theorem of_eq {f g : ℕ →. ℕ} (hf : Nat.Partrec f) (H : ∀ n, f n = g n) : Nat.Partrec g :=
(funext H : f = g) ▸ hf
```

-- snippet 15
```lean
-- Mathlib.Computability.Partrec.of_eq_tot (L177-L180)
theorem of_eq_tot {f : ℕ →. ℕ} {g : ℕ → ℕ} (hf : Nat.Partrec f) (H : ∀ n, g n ∈ f n) :
    Nat.Partrec g :=
hf.of_eq fun n => eq_some_iff.2 (H n)
```

-- snippet 16
```lean
-- Mathlib.Computability.Partrec.of_primrec (L181-L203)
theorem of_primrec {f : ℕ → ℕ} (hf : Nat.Primrec f) : Nat.Partrec f := by
induction hf with
  | zero => exact zero
  | succ => exact succ
  | left => exact left
  | right => exact right
  | pair _ _ pf pg =>
    refine (pf.pair pg).of_eq_tot fun n => ?_
    simp [Seq.seq]
  | comp _ _ pf pg =>
    refine (pf.comp pg).of_eq_tot fun n => (by simp)
  | prec _ _ pf pg =>
    refine (pf.prec pg).of_eq_tot fun n => ?_
    simp only [unpaired, PFun.coe_val, bind_eq_bind]
    induction n.unpair.2 with
    | zero => simp
    | succ m IH =>
      simp only [mem_bind_iff, mem_some_iff]
      exact ⟨_, IH, rfl⟩

protected theorem some : Nat.Partrec some :=
  of_primrec Primrec.id
```

-- snippet 17
```lean
-- Mathlib.Computability.Partrec.none (L204-L207)
theorem none : Nat.Partrec fun _ => none :=
(of_primrec (Nat.Primrec.const 1)).rfind.of_eq fun _ =>
    eq_none_iff.2 fun _ ⟨h, _⟩ => by simp at h
```

-- snippet 18
```lean
-- Mathlib.Computability.Partrec.prec' (L208-L213)
theorem prec' {f g h} (hf : Nat.Partrec f) (hg : Nat.Partrec g) (hh : Nat.Partrec h) :
    Nat.Partrec fun a => (f a).bind fun n => n.rec (g a)
      fun y IH => do {let i ← IH; h (Nat.pair a (Nat.pair y i))} :=
((prec hg hh).comp (pair Partrec.some hf)).of_eq fun a =>
    ext fun s => by simp [Seq.seq]
```

-- snippet 19
```lean
-- Mathlib.Computability.Partrec.ppred (L214-L229)
theorem ppred : Nat.Partrec fun n => ppred n :=
have : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1 :=
    (Primrec.ite
      (@PrimrecRel.comp _ _ _ _ _ _ _ _ _
        Primrec.eq Primrec.fst (_root_.Primrec.succ.comp Primrec.snd))
      (_root_.Primrec.const 0) (_root_.Primrec.const 1)).to₂
  (of_primrec (Primrec₂.unpaired'.2 this)).rfind.of_eq fun n => by
    cases n
    · exact eq_none_iff.2 (by simp)
    · exact eq_some_iff.2 (by simp; lia)

end Partrec

end Nat

/-- Partially recursive partial functions `α → σ` between `Primcodable` types -/
```

-- snippet 20
```lean
-- Mathlib.Computability.Partrec.Partrec (L230-L233)
def Partrec {α σ} [Primcodable α] [Primcodable σ] (f : α →. σ) :=
Nat.Partrec fun n => Part.bind (decode (α := α) n) fun a => (f a).map encode

/-- Partially recursive partial functions `α → β → σ` between `Primcodable` types -/
```

-- snippet 21
```lean
-- Mathlib.Computability.Partrec.Partrec₂ (L234-L238)
def Partrec₂ {α β σ} [Primcodable α] [Primcodable β] [Primcodable σ] (f : α → β →. σ) :=
Partrec fun p : α × β => f p.1 p.2

/-- Computable functions `α → σ` between `Primcodable` types:
  a function is computable if and only if it is partially recursive (as a partial function) -/
```

-- snippet 22
```lean
-- Mathlib.Computability.Partrec.Computable (L239-L242)
def Computable {α σ} [Primcodable α] [Primcodable σ] (f : α → σ) :=
Partrec (f : α →. σ)

/-- Computable functions `α → β → σ` between `Primcodable` types -/
```

-- snippet 23
```lean
-- Mathlib.Computability.Partrec.Computable₂ (L243-L245)
def Computable₂ {α β σ} [Primcodable α] [Primcodable β] [Primcodable σ] (f : α → β → σ) :=
Computable fun p : α × β => f p.1 p.2
```

-- snippet 24
```lean
-- Mathlib.Computability.Partrec.Primrec (L246-L267)
theorem Primrec.to_comp {α σ} [Primcodable α] [Primcodable σ] {f : α → σ} (hf : Primrec f) :
    Computable f :=
(Nat.Partrec.ppred.comp (Nat.Partrec.of_primrec hf)).of_eq fun n => by
    simp; cases decode (α := α) n <;> simp

nonrec theorem Primrec₂.to_comp {α β σ} [Primcodable α] [Primcodable β] [Primcodable σ]
    {f : α → β → σ} (hf : Primrec₂ f) : Computable₂ f :=
  hf.to_comp

protected theorem Computable.partrec {α σ} [Primcodable α] [Primcodable σ] {f : α → σ}
    (hf : Computable f) : Partrec (f : α →. σ) :=
  hf

protected theorem Computable₂.partrec₂ {α β σ} [Primcodable α] [Primcodable β] [Primcodable σ]
    {f : α → β → σ} (hf : Computable₂ f) : Partrec₂ fun a => (f a : β →. σ) :=
  hf

namespace Computable

variable {α : Type*} {β : Type*} {γ : Type*} {σ : Type*}
variable [Primcodable α] [Primcodable β] [Primcodable γ] [Primcodable σ]
```

-- snippet 25
```lean
-- Mathlib.Computability.Partrec.of_eq (L268-L270)
theorem of_eq {f g : α → σ} (hf : Computable f) (H : ∀ n, f n = g n) : Computable g :=
(funext H : f = g) ▸ hf
```

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

```lean
theorem option_some : Computable (@Option.some α)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Computability.Partrec.rfindX (L65-L90)
def rfindX : { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } :=
suffices ∀ k, (∀ n < k, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m } from
    this 0 fun _ => (Nat.not_lt_zero _).elim
  @WellFounded.fix _ _ (lbp p) (wf_lbp p H)
    (by
      intro m IH al
      have pm : (p m).Dom := by
        rcases H with ⟨n, h₁, h₂⟩
        rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
        · exact h₂ _ h₃
        · rw [h₃]
          exact h₁.fst
        · injection mem_unique h₁ (al _ h₃)
      cases e : (p m).get pm
      · suffices ∀ᵉ k ≤ m, false ∈ p k from IH _ ⟨rfl, this⟩ fun n h => this _ (le_of_lt_succ h)
        intro n h
        rcases h.lt_or_eq_dec with h | h
        · exact al _ h
        · rw [h]
          exact ⟨_, e⟩
      · exact ⟨m, ⟨_, e⟩, al⟩)

end Rfind

/-- Find the smallest `n` satisfying `p n`, where all `p k` for `k < n` are defined as false.
Returns a `Part`. -/
```

-- snippet 2
```lean
-- Mathlib.Computability.Partrec.rfind (L91-L93)
def rfind (p : ℕ →. Bool) : Part ℕ :=
⟨_, fun h => (rfindX p h).1⟩
```

-- snippet 3
```lean
-- Mathlib.Computability.Partrec.rfind_spec (L94-L96)
theorem rfind_spec {p : ℕ →. Bool} {n : ℕ} (h : n ∈ rfind p) : true ∈ p n :=
h.snd ▸ (rfindX p h.fst).2.1
```

-- snippet 4
```lean
-- Mathlib.Computability.Partrec.rfind_min (L97-L99)
theorem rfind_min {p : ℕ →. Bool} {n : ℕ} (h : n ∈ rfind p) : ∀ {m : ℕ}, m < n → false ∈ p m :=
@(h.snd ▸ @((rfindX p h.fst).2.2))
```

-- snippet 5
```lean
-- Mathlib.Computability.Partrec.rfind_dom (L101-L104)
theorem rfind_dom {p : ℕ →. Bool} :
    (rfind p).Dom ↔ ∃ n, true ∈ p n ∧ ∀ {m : ℕ}, m < n → (p m).Dom :=
Iff.rfl
```

-- snippet 6
```lean
-- Mathlib.Computability.Partrec.rfind_dom' (L105-L111)
theorem rfind_dom' {p : ℕ →. Bool} :
    (rfind p).Dom ↔ ∃ n, true ∈ p n ∧ ∀ {m : ℕ}, m ≤ n → (p m).Dom :=
exists_congr fun _ =>
    and_congr_right fun pn =>
      ⟨fun H _ h => (Decidable.eq_or_lt_of_le h).elim (fun e => e.symm ▸ pn.fst) (H _), fun H _ h =>
        H (le_of_lt h)⟩
```

-- snippet 7
```lean
-- Mathlib.Computability.Partrec.mem_rfind (L113-L121)
theorem mem_rfind {p : ℕ →. Bool} {n : ℕ} :
    n ∈ rfind p ↔ true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m :=
⟨fun h => ⟨rfind_spec h, @rfind_min _ _ h⟩, fun ⟨h₁, h₂⟩ => by
    let ⟨m, hm⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨_, h₁, fun {m} mn => (h₂ mn).fst⟩
    rcases lt_trichotomy m n with (h | h | h)
    · injection mem_unique (h₂ h) (rfind_spec hm)
    · rwa [← h]
    · injection mem_unique h₁ (rfind_min hm h)⟩
```

-- snippet 8
```lean
-- Mathlib.Computability.Partrec.rfind_min' (L122-L126)
theorem rfind_min' {p : ℕ → Bool} {m : ℕ} (pm : p m) : ∃ n ∈ rfind p, n ≤ m :=
have : true ∈ (p : ℕ →. Bool) m := ⟨trivial, pm⟩
  let ⟨n, hn⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨m, this, fun {_} _ => ⟨⟩⟩
  ⟨n, hn, not_lt.1 fun h => by injection mem_unique this (rfind_min hn h)⟩
```

-- snippet 9
```lean
-- Mathlib.Computability.Partrec.rfind_zero_none (L127-L133)
theorem rfind_zero_none (p : ℕ →. Bool) (p0 : p 0 = Part.none) : rfind p = Part.none :=
eq_none_iff.2 fun _ h =>
    let ⟨_, _, h₂⟩ := rfind_dom'.1 h.fst
    (p0 ▸ h₂ (zero_le _) : (@Part.none Bool).Dom)

/-- Find the smallest `n` satisfying `f n`, where all `f k` for `k < n` are defined as false.
Returns a `Part`. -/
```

-- snippet 10
```lean
-- Mathlib.Computability.Partrec.rfindOpt (L134-L136)
def rfindOpt {α} (f : ℕ → Option α) : Part α :=
(rfind fun n => (f n).isSome).bind fun n => f n
```

-- snippet 11
```lean
-- Mathlib.Computability.Partrec.rfindOpt_spec (L137-L140)
theorem rfindOpt_spec {α} {f : ℕ → Option α} {a} (h : a ∈ rfindOpt f) : ∃ n, a ∈ f n :=
let ⟨n, _, h₂⟩ := mem_bind_iff.1 h
  ⟨n, mem_coe.1 h₂⟩
```

-- snippet 12
```lean
-- Mathlib.Computability.Partrec.rfindOpt_dom (L141-L150)
theorem rfindOpt_dom {α} {f : ℕ → Option α} : (rfindOpt f).Dom ↔ ∃ n a, a ∈ f n :=
⟨fun h => (rfindOpt_spec ⟨h, rfl⟩).imp fun _ h => ⟨_, h⟩, fun h => by
    have h' : ∃ n, (f n).isSome := h.imp fun n => Option.isSome_iff_exists.2
    have s := Nat.find_spec h'
    have fd : (rfind fun n => (f n).isSome).Dom :=
      ⟨Nat.find h', by simpa using s.symm, fun _ _ => trivial⟩
    refine ⟨fd, ?_⟩
    have := rfind_spec (get_mem fd)
    simpa using this⟩
```

-- snippet 13
```lean
-- Mathlib.Computability.Partrec.rfindOpt_mono (L151-L173)
theorem rfindOpt_mono {α} {f : ℕ → Option α} (H : ∀ {a m n}, m ≤ n → a ∈ f m → a ∈ f n) {a} :
    a ∈ rfindOpt f ↔ ∃ n, a ∈ f n :=
⟨rfindOpt_spec, fun ⟨n, h⟩ => by
    have h' := rfindOpt_dom.2 ⟨_, _, h⟩
    obtain ⟨k, hk⟩ := rfindOpt_spec ⟨h', rfl⟩
    have := (H (le_max_left _ _) h).symm.trans (H (le_max_right _ _) hk)
    simp at this; simp [this, get_mem]⟩

/-- `Nat.Partrec f` means that the partial function `f : ℕ →. ℕ` is partially recursive. -/
protected inductive Partrec : (ℕ →. ℕ) → Prop
  | zero : Nat.Partrec (pure 0)
  | succ : Nat.Partrec succ
  | left : Nat.Partrec ↑fun n : ℕ => n.unpair.1
  | right : Nat.Partrec ↑fun n : ℕ => n.unpair.2
  | pair {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec fun n => pair <$> f n <*> g n
  | comp {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec fun n => g n >>= f
  | prec {f g} : Nat.Partrec f → Nat.Partrec g → Nat.Partrec (unpaired fun a n =>
      n.rec (f a) fun y IH => do let i ← IH; g (pair a (pair y i)))
  | rfind {f} : Nat.Partrec f →
    Nat.Partrec fun a => rfind fun n => (fun m => m = 0) <$> f (pair a n)

namespace Partrec
```

-- snippet 14
```lean
-- Mathlib.Computability.Partrec.of_eq (L174-L176)
theorem of_eq {f g : ℕ →. ℕ} (hf : Nat.Partrec f) (H : ∀ n, f n = g n) : Nat.Partrec g :=
(funext H : f = g) ▸ hf
```

-- snippet 15
```lean
-- Mathlib.Computability.Partrec.of_eq_tot (L177-L180)
theorem of_eq_tot {f : ℕ →. ℕ} {g : ℕ → ℕ} (hf : Nat.Partrec f) (H : ∀ n, g n ∈ f n) :
    Nat.Partrec g :=
hf.of_eq fun n => eq_some_iff.2 (H n)
```

-- snippet 16
```lean
-- Mathlib.Computability.Partrec.of_primrec (L181-L203)
theorem of_primrec {f : ℕ → ℕ} (hf : Nat.Primrec f) : Nat.Partrec f := by
induction hf with
  | zero => exact zero
  | succ => exact succ
  | left => exact left
  | right => exact right
  | pair _ _ pf pg =>
    refine (pf.pair pg).of_eq_tot fun n => ?_
    simp [Seq.seq]
  | comp _ _ pf pg =>
    refine (pf.comp pg).of_eq_tot fun n => (by simp)
  | prec _ _ pf pg =>
    refine (pf.prec pg).of_eq_tot fun n => ?_
    simp only [unpaired, PFun.coe_val, bind_eq_bind]
    induction n.unpair.2 with
    | zero => simp
    | succ m IH =>
      simp only [mem_bind_iff, mem_some_iff]
      exact ⟨_, IH, rfl⟩

protected theorem some : Nat.Partrec some :=
  of_primrec Primrec.id
```

-- snippet 17
```lean
-- Mathlib.Computability.Partrec.none (L204-L207)
theorem none : Nat.Partrec fun _ => none :=
(of_primrec (Nat.Primrec.const 1)).rfind.of_eq fun _ =>
    eq_none_iff.2 fun _ ⟨h, _⟩ => by simp at h
```

-- snippet 18
```lean
-- Mathlib.Computability.Partrec.prec' (L208-L213)
theorem prec' {f g h} (hf : Nat.Partrec f) (hg : Nat.Partrec g) (hh : Nat.Partrec h) :
    Nat.Partrec fun a => (f a).bind fun n => n.rec (g a)
      fun y IH => do {let i ← IH; h (Nat.pair a (Nat.pair y i))} :=
((prec hg hh).comp (pair Partrec.some hf)).of_eq fun a =>
    ext fun s => by simp [Seq.seq]
```

-- snippet 19
```lean
-- Mathlib.Computability.Partrec.ppred (L214-L229)
theorem ppred : Nat.Partrec fun n => ppred n :=
have : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1 :=
    (Primrec.ite
      (@PrimrecRel.comp _ _ _ _ _ _ _ _ _
        Primrec.eq Primrec.fst (_root_.Primrec.succ.comp Primrec.snd))
      (_root_.Primrec.const 0) (_root_.Primrec.const 1)).to₂
  (of_primrec (Primrec₂.unpaired'.2 this)).rfind.of_eq fun n => by
    cases n
    · exact eq_none_iff.2 (by simp)
    · exact eq_some_iff.2 (by simp; lia)

end Partrec

end Nat

/-- Partially recursive partial functions `α → σ` between `Primcodable` types -/
```

-- snippet 20
```lean
-- Mathlib.Computability.Partrec.Partrec (L230-L233)
def Partrec {α σ} [Primcodable α] [Primcodable σ] (f : α →. σ) :=
Nat.Partrec fun n => Part.bind (decode (α := α) n) fun a => (f a).map encode

/-- Partially recursive partial functions `α → β → σ` between `Primcodable` types -/
```

-- snippet 21
```lean
-- Mathlib.Computability.Partrec.Partrec₂ (L234-L238)
def Partrec₂ {α β σ} [Primcodable α] [Primcodable β] [Primcodable σ] (f : α → β →. σ) :=
Partrec fun p : α × β => f p.1 p.2

/-- Computable functions `α → σ` between `Primcodable` types:
  a function is computable if and only if it is partially recursive (as a partial function) -/
```

-- snippet 22
```lean
-- Mathlib.Computability.Partrec.Computable (L239-L242)
def Computable {α σ} [Primcodable α] [Primcodable σ] (f : α → σ) :=
Partrec (f : α →. σ)

/-- Computable functions `α → β → σ` between `Primcodable` types -/
```

-- snippet 23
```lean
-- Mathlib.Computability.Partrec.Computable₂ (L243-L245)
def Computable₂ {α β σ} [Primcodable α] [Primcodable β] [Primcodable σ] (f : α → β → σ) :=
Computable fun p : α × β => f p.1 p.2
```

-- snippet 24
```lean
-- Mathlib.Computability.Partrec.Primrec (L246-L267)
theorem Primrec.to_comp {α σ} [Primcodable α] [Primcodable σ] {f : α → σ} (hf : Primrec f) :
    Computable f :=
(Nat.Partrec.ppred.comp (Nat.Partrec.of_primrec hf)).of_eq fun n => by
    simp; cases decode (α := α) n <;> simp

nonrec theorem Primrec₂.to_comp {α β σ} [Primcodable α] [Primcodable β] [Primcodable σ]
    {f : α → β → σ} (hf : Primrec₂ f) : Computable₂ f :=
  hf.to_comp

protected theorem Computable.partrec {α σ} [Primcodable α] [Primcodable σ] {f : α → σ}
    (hf : Computable f) : Partrec (f : α →. σ) :=
  hf

protected theorem Computable₂.partrec₂ {α β σ} [Primcodable α] [Primcodable β] [Primcodable σ]
    {f : α → β → σ} (hf : Computable₂ f) : Partrec₂ fun a => (f a : β →. σ) :=
  hf

namespace Computable

variable {α : Type*} {β : Type*} {γ : Type*} {σ : Type*}
variable [Primcodable α] [Primcodable β] [Primcodable γ] [Primcodable σ]
```

-- snippet 25
```lean
-- Mathlib.Computability.Partrec.of_eq (L268-L270)
theorem of_eq {f g : α → σ} (hf : Computable f) (H : ∀ n, f n = g n) : Computable g :=
(funext H : f = g) ▸ hf
```
[/INST]