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

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

```lean
def findModuleMatches (moduleRef : ModuleDiscrTreeRef α) (ty : Expr) : MetaM (MatchResult α)
:= do ⟨⟩⟨⟩
profileitM Exception  "RefinedDiscrTree local search" (← getOptions) do
    let discrTree ← moduleRef.ref.get
    let (localCandidates, localTree) ← getMatch discrTree ty false false
    moduleRef.ref.set localTree
    MonadExcept.ofExcept localCandidates

/--
`findMatches` combines `findImportMatches` and `findModuleMatches`.

* `ext` should be an environment extension with an `IO.Ref` for caching the `RefinedDiscrTree`.
* `addEntry` is the function for creating `RefinedDiscrTree` entries from constants.
* `ty` is the expression type.
* `constantsPerTask` is the number of constants in imported modules to be used for each
  new task.
* `capacityPerTask` is the initial capacity of the `HashMap` at the root of the
  `RefinedDiscrTree` for each new task.
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=114, messages=[Message(data="Function expected at\n  ModuleDiscrTreeRef\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `ModuleDiscrTreeRef` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=35), pos_end=Position(line=1, column=55), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  MetaM\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  (MatchResult α)\n\nHint: The identifier `MetaM` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=71), pos_end=Position(line=1, column=92), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=114, i_end=884, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=0), pos_end=Position(line=3, column=10), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=19, column=2), pos_end=None, severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

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

```lean
def findModuleMatches (moduleRef : ModuleDiscrTreeRef α) (ty : Expr) : MetaM (MatchResult α)
:= do ⟨⟩⟨⟩
profileitM Exception  "RefinedDiscrTree local search" (← getOptions) do
    let discrTree ← moduleRef.ref.get
    let (localCandidates, localTree) ← getMatch discrTree ty false false
    moduleRef.ref.set localTree
    MonadExcept.ofExcept localCandidates

/--
`findMatches` combines `findImportMatches` and `findModuleMatches`.

* `ext` should be an environment extension with an `IO.Ref` for caching the `RefinedDiscrTree`.
* `addEntry` is the function for creating `RefinedDiscrTree` entries from constants.
* `ty` is the expression type.
* `constantsPerTask` is the number of constants in imported modules to be used for each
  new task.
* `capacityPerTask` is the initial capacity of the `HashMap` at the root of the
  `RefinedDiscrTree` for each new task.
-/
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=114, messages=[Message(data="Function expected at\n  ModuleDiscrTreeRef\nbut this term has type\n  ?m.1\n\nNote: Expected a function because this term is being applied to the argument\n  α\n\nHint: The identifier `ModuleDiscrTreeRef` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=35), pos_end=Position(line=1, column=55), severity=<Severity.ERROR: 3>, kind=None), Message(data="Function expected at\n  MetaM\nbut this term has type\n  ?m.3\n\nNote: Expected a function because this term is being applied to the argument\n  (MatchResult α)\n\nHint: The identifier `MetaM` is unknown, and Lean's `autoImplicit` option causes an unknown identifier to be treated as an implicitly bound variable with an unknown type. However, the unknown type cannot be a function, and a function is what Lean expects here. This is often the result of a typo or a missing `import` or `open` statement.", pos=Position(line=1, column=71), pos_end=Position(line=1, column=92), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=114, i_end=884, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=3, column=0), pos_end=Position(line=3, column=10), severity=<Severity.ERROR: 3>, kind=None), Message(data="unexpected end of input; expected 'lemma'", pos=Position(line=19, column=2), 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.Tactic.FieldSimp.Lemmas).

[User]
Complete the following proof.

```lean
theorem eq_mul_of_eq_eq_eq_mul {M : Type*} [Mul M] {a b c D e f : M}
    (h₁ : a = b) (h₂ : b = c) (h₃ : c = D * e) (h₄ : e = f) :
    a = D * f
```

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

```lean
theorem eq_mul_of_eq_eq_eq_mul {M : Type*} [Mul M] {a b c D e f : M}
    (h₁ : a = b) (h₂ : b = c) (h₃ : c = D * e) (h₄ : e = f) :
    a = D * f
```

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.Logic.Equiv.Option).

[User]
Complete the following proof.

```lean
theorem optionEquivSumPUnit_coe {α} (a : α) : optionEquivSumPUnit α a = Sum.inl a
```

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

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

```lean
theorem optionEquivSumPUnit_coe {α} (a : α) : optionEquivSumPUnit α a = Sum.inl a
```

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


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

[User]
Complete the following proof.

```lean
lemma zero_mul_eq {α} {R : α → α → Prop} [Semiring α] {a b : α} (h : a = 0) (_ : R b 0) :
    a * b = 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.Tactic.Linarith.Lemmas).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
lemma zero_mul_eq {α} {R : α → α → Prop} [Semiring α] {a b : α} (h : a = 0) (_ : R b 0) :
    a * b = 0
```

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


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

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

## Current excerpt (around `Mathlib.Logic.Equiv.Basic.sigmaSigmaSubtypeEq`, L468-L474)
```lean
def sigmaSigmaSubtypeEq {α β : Type*} {γ : α → β → Type*} (a : α) (b : β) :
    {s : (a : α) × (b : β) × γ a b // s.1 = a ∧ s.2.1 = b} ≃ γ a b := have : Unique (@Subtype ((_ : α) × β) (fun ⟨a', b'⟩ ↦ a' = a ∧ b' = b)) := {
    default := ⟨⟨a, b⟩, ⟨rfl, rfl⟩⟩
    uniq := by rintro ⟨⟨a', b'⟩, ⟨rfl, rfl⟩⟩; rfl }
  sigmaSigmaSubtype (fun ⟨a', b'⟩ ↦ a' = a ∧ b' = b) ⟨rfl, rfl⟩
```

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

## Current excerpt (around `Mathlib.Logic.Equiv.Basic.sigmaSigmaSubtypeEq`, L468-L474)
```lean
def sigmaSigmaSubtypeEq {α β : Type*} {γ : α → β → Type*} (a : α) (b : β) :
    {s : (a : α) × (b : β) × γ a b // s.1 = a ∧ s.2.1 = b} ≃ γ a b := have : Unique (@Subtype ((_ : α) × β) (fun ⟨a', b'⟩ ↦ a' = a ∧ b' = b)) := {
    default := ⟨⟨a, b⟩, ⟨rfl, rfl⟩⟩
    uniq := by rintro ⟨⟨a', b'⟩, ⟨rfl, rfl⟩⟩; rfl }
  sigmaSigmaSubtype (fun ⟨a', b'⟩ ↦ a' = a ∧ b' = b) ⟨rfl, rfl⟩
```

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


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

[User]
Complete the following proof.

```lean
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by fun_prop)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one)
```

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

```lean
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by fun_prop)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one)
```

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.Tactic.CategoryTheory.Coherence.Basic).

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

```lean
def normalForm (ρ : Type) [Context ρ]
    [MonadMor₁ (CoherenceM ρ)]
    [MonadMor₂Iso (CoherenceM ρ)]
    [MonadNormalExpr (CoherenceM ρ)] [MkEval (CoherenceM ρ)]
    [MkMor₂ (CoherenceM ρ)]
    [MonadMor₂ (CoherenceM ρ)]
    (nm : Name) (mvarId : MVarId) : MetaM (List MVarId)
```

Declaration: `Mathlib.Tactic.CategoryTheory.Coherence.Basic.normalForm` (module Mathlib.Tactic.CategoryTheory.Coherence.Basic).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.CategoryTheory.Coherence.Basic).[/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
def normalForm (ρ : Type) [Context ρ]
    [MonadMor₁ (CoherenceM ρ)]
    [MonadMor₂Iso (CoherenceM ρ)]
    [MonadNormalExpr (CoherenceM ρ)] [MkEval (CoherenceM ρ)]
    [MkMor₂ (CoherenceM ρ)]
    [MonadMor₂ (CoherenceM ρ)]
    (nm : Name) (mvarId : MVarId) : MetaM (List MVarId)
```

Declaration: `Mathlib.Tactic.CategoryTheory.Coherence.Basic.normalForm` (module Mathlib.Tactic.CategoryTheory.Coherence.Basic).[/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.Tactic.NormNum.RealSqrt).

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

```lean
lemma isNat_realSqrt {x : ℝ} {nx ny : ℕ} (h : IsNat x nx) (hy : ny * ny = nx) :
    IsNat √x ny
```

Declaration: `Mathlib.Tactic.NormNum.RealSqrt.isNat_realSqrt` (module Mathlib.Tactic.NormNum.RealSqrt).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Tactic.NormNum.RealSqrt).[/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 isNat_realSqrt {x : ℝ} {nx ny : ℕ} (h : IsNat x nx) (hy : ny * ny = nx) :
    IsNat √x ny
```

Declaration: `Mathlib.Tactic.NormNum.RealSqrt.isNat_realSqrt` (module Mathlib.Tactic.NormNum.RealSqrt).[/INST]


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

[User]
Complete the following proof.

```lean
def ReaderT.mkLabel {α β} (ρ) : Label α m β → Label α (ReaderT ρ m) β
  | ⟨f⟩ => ⟨monadLift ∘ 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.Control.Monad.Cont).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Complete the following proof.

```lean
def ReaderT.mkLabel {α β} (ρ) : Label α m β → Label α (ReaderT ρ m) β
  | ⟨f⟩ => ⟨monadLift ∘ 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.AlgebraicTopology.DoldKan.Projections).

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

```lean
theorem Q_zero : (Q 0 : K[X] ⟶ _) = 0
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_zero (L54-L54)
lemma P_zero : (P 0 : K[X] ⟶ K[X]) = 𝟙 _ := rfl
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_succ (L55-L57)
lemma P_succ (q : ℕ) : (P (q + 1) : K[X] ⟶ K[X]) = P q ≫ (𝟙 _ + Hσ q) := rfl
/-- All the `P q` coincide with `𝟙 _` in degree 0. -/
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_f_0_eq (L59-L66)
theorem P_f_0_eq (q : ℕ) : ((P q).f 0 : X _⦋0⦌ ⟶ X _⦋0⦌) = 𝟙 _ := by
induction q with
  | zero => rfl
  | succ q hq =>
    simp only [P_succ, HomologicalComplex.add_f_apply, HomologicalComplex.comp_f,
      HomologicalComplex.id_f, id_comp, hq, Hσ_eq_zero, add_zero]

/-- `Q q` is the complement projection associated to `P q` -/
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.Q (L67-L69)
def Q (q : ℕ) : K[X] ⟶ K[X] :=
𝟙 _ - P q
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_add_Q (L70-L73)
theorem P_add_Q (q : ℕ) : P q + Q q = 𝟙 K[X] := by
rw [Q]
  abel
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_add_Q_f (L74-L76)
theorem P_add_Q_f (q n : ℕ) : (P q).f n + (Q q).f n = 𝟙 (X _⦋n⦌) :=
HomologicalComplex.congr_hom (P_add_Q q) n
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.Q_succ (L81-L85)
theorem Q_succ (q : ℕ) : (Q (q + 1) : K[X] ⟶ _) = Q q - P q ≫ Hσ q := by
simp only [Q, P_succ, comp_add, comp_id]
  abel

/-- All the `Q q` coincide with `0` in degree 0. -/
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.Q_f_0_eq (L87-L93)
theorem Q_f_0_eq (q : ℕ) : ((Q q).f 0 : X _⦋0⦌ ⟶ X _⦋0⦌) = 0 := by
simp only [HomologicalComplex.sub_f_apply, HomologicalComplex.id_f, Q, P_f_0_eq, sub_self]

namespace HigherFacesVanish

/-- This lemma expresses the vanishing of
`(P q).f (n+1) ≫ X.δ k : X _⦋n+1⦌ ⟶ X _⦋n⦌` when `k≠0` and `k≥n-q+2` -/
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.of_P (L94-L100)
theorem of_P : ∀ q n : ℕ, HigherFacesVanish q ((P q).f (n + 1) : X _⦋n + 1⦌ ⟶ X _⦋n + 1⦌)
  | 0 => fun n j hj₁ => by lia
  | q + 1 => fun n => by
    simp only [P_succ]
    exact (of_P q n).induction

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.comp_P_eq_self (L102-L123)
theorem comp_P_eq_self {Y : C} {n q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} (v : HigherFacesVanish q φ) :
    φ ≫ (P q).f (n + 1) = φ := by
induction q with
  | zero =>
    simp only [P_zero]
    apply comp_id
  | succ q hq =>
    simp only [P_succ, comp_add, HomologicalComplex.comp_f, HomologicalComplex.add_f_apply,
      comp_id, ← assoc, hq v.of_succ, add_eq_left]
    by_cases! hqn : n < q
    · exact v.of_succ.comp_Hσ_eq_zero hqn
    · obtain ⟨a, ha⟩ := Nat.le.dest hqn
      have hnaq : n = a + q := by lia
      simp only [v.of_succ.comp_Hσ_eq hnaq, neg_eq_zero, ← assoc]
      have eq := v ⟨a, by lia⟩ (by
        simp only [hnaq, add_assoc]
        rfl)
      simp only [Fin.succ_mk] at eq
      simp only [eq, zero_comp]

end HigherFacesVanish
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.comp_P_eq_self_iff (L124-L132)
theorem comp_P_eq_self_iff {Y : C} {n q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} :
    φ ≫ (P q).f (n + 1) = φ ↔ HigherFacesVanish q φ := by
constructor
  · intro hφ
    rw [← hφ]
    apply HigherFacesVanish.of_comp
    apply HigherFacesVanish.of_P
  · exact HigherFacesVanish.comp_P_eq_self
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_f_idem (L134-L138)
theorem P_f_idem (q n : ℕ) : ((P q).f n : X _⦋n⦌ ⟶ _) ≫ (P q).f n = (P q).f n := by
rcases n with (_ | n)
  · rw [P_f_0_eq q, comp_id]
  · exact (HigherFacesVanish.of_P q n).comp_P_eq_self
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.Q_f_idem (L140-L142)
theorem Q_f_idem (q n : ℕ) : ((Q q).f n : X _⦋n⦌ ⟶ _) ≫ (Q q).f n = (Q q).f n :=
idem_of_id_sub_idem _ (P_f_idem q n)
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_idem (L144-L147)
theorem P_idem (q : ℕ) : (P q : K[X] ⟶ K[X]) ≫ P q = P q := by
ext n
  exact P_f_idem q n
```

-- snippet 15
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.Q_idem (L149-L154)
theorem Q_idem (q : ℕ) : (Q q : K[X] ⟶ K[X]) ≫ Q q = Q q := by
ext n
  exact Q_f_idem q n

set_option backward.isDefEq.respectTransparency false in
/-- For each `q`, `P q` is a natural transformation. -/
```

-- snippet 16
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.τ₁ (L186-L197)
def τ₁ (η : G ⋙ F ≅ eB.functor) : eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor :=
calc
    eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ (eB.functor ⋙ e'.inverse) ⋙ e'.functor :=
        (associator _ _ _).symm
    _ ≅ (G ⋙ eA.functor) ⋙ e'.functor := isoWhiskerRight hG _
    _ ≅ G ⋙ eA.functor ⋙ e'.functor := associator _ _ _
    _ ≅ G ⋙ F := isoWhiskerLeft _ hF
    _ ≅ eB.functor := η

variable (η : G ⋙ F ≅ eB.functor)

/-- The counit isomorphism of `equivalence`. -/
```

-- snippet 17
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso_eq (L131-L136)
theorem equivalence₂CounitIso_eq :
    (equivalence₂ eB hF).counitIso = equivalence₂CounitIso eB hF := by
ext Y'
  simp [equivalence₂, equivalence₁CounitIso_eq]

/-- The unit isomorphism of the equivalence `equivalence₂` between `A` and `B`. -/
```

-- snippet 18
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₀ (L60-L66)
def equivalence₀ : A ≌ B' :=
eA.trans e'

variable {eA} {e'}

/-- An intermediate equivalence `A ≌ B'` whose functor is `F` and whose inverse is
`e'.inverse ⋙ eA.inverse`. -/
```

-- snippet 19
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso (L92-L102)
def equivalence₁UnitIso : 𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse :=
calc
    𝟭 A ≅ eA.functor ⋙ eA.inverse := eA.unitIso
    _ ≅ eA.functor ⋙ 𝟭 A' ⋙ eA.inverse := isoWhiskerLeft _ (leftUnitor _).symm
    _ ≅ eA.functor ⋙ (e'.functor ⋙ e'.inverse) ⋙ eA.inverse :=
      isoWhiskerLeft _ (isoWhiskerRight e'.unitIso _)
    _ ≅ eA.functor ⋙ (e'.functor ⋙ e'.inverse ⋙ eA.inverse) :=
      isoWhiskerLeft _ (associator _ _ _)
    _ ≅ (eA.functor ⋙ e'.functor) ⋙ e'.inverse ⋙ eA.inverse := (associator _ _ _).symm
    _ ≅ F ⋙ e'.inverse ⋙ eA.inverse := isoWhiskerRight hF _
```

-- snippet 20
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.υ (L229-L238)
def υ : eA.functor ≅ F ⋙ e'.inverse :=
calc
    eA.functor ≅ eA.functor ⋙ 𝟭 A' := (rightUnitor _).symm
    _ ≅ eA.functor ⋙ e'.functor ⋙ e'.inverse := isoWhiskerLeft _ e'.unitIso
    _ ≅ (eA.functor ⋙ e'.functor) ⋙ e'.inverse := (associator _ _ _).symm
    _ ≅ F ⋙ e'.inverse := isoWhiskerRight hF _

variable (ε : eA.functor ≅ F ⋙ e'.inverse) (hG)

/-- The unit isomorphism of `equivalence`. -/
```

-- snippet 21
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence (L161-L169)
def equivalence : A ≌ B :=
((equivalence₂ eB hF).changeInverse
    (calc eB.functor ⋙ e'.inverse ⋙ eA.inverse ≅
        (eB.functor ⋙ e'.inverse) ⋙ eA.inverse := (associator _ _ _).symm
    _ ≅ (G ⋙ eA.functor) ⋙ eA.inverse := isoWhiskerRight hG _
    _ ≅ G ⋙ eA.functor ⋙ eA.inverse := associator _ _ _
    _ ≅ G ⋙ 𝟭 A := isoWhiskerLeft _ eA.unitIso.symm
    _ ≅ G := G.rightUnitor))
```

-- snippet 22
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso (L75-L85)
def equivalence₁CounitIso : (e'.inverse ⋙ eA.inverse) ⋙ F ≅ 𝟭 B' :=
calc
    (e'.inverse ⋙ eA.inverse) ⋙ F ≅ (e'.inverse ⋙ eA.inverse) ⋙ eA.functor ⋙ e'.functor :=
      isoWhiskerLeft _ hF.symm
    _ ≅ e'.inverse ⋙ (eA.inverse ⋙ eA.functor ⋙ e'.functor) := associator _ _ _
    _ ≅ e'.inverse ⋙ (eA.inverse ⋙ eA.functor) ⋙ e'.functor :=
      isoWhiskerLeft _ (associator _ _ _).symm
    _ ≅ e'.inverse ⋙ 𝟭 _ ⋙ e'.functor := isoWhiskerLeft _ (isoWhiskerRight eA.counitIso _)
    _ ≅ e'.inverse ⋙ e'.functor := isoWhiskerLeft _ (leftUnitor _)
    _ ≅ 𝟭 B' := e'.counitIso
```

-- snippet 23
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso_eq (L103-L108)
theorem equivalence₁UnitIso_eq : (equivalence₁ hF).unitIso = equivalence₁UnitIso hF := by
ext X
  simp [equivalence₁]

/-- An intermediate equivalence `A ≌ B` obtained as the composition of `equivalence₁` and
the inverse of `eB : B ≌ B'`. -/
```

-- snippet 24
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁ (L68-L69)
def equivalence₁ : A ≌ B' := (equivalence₀ eA e').changeFunctor hF
```

-- snippet 25
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso_eq (L86-L90)
theorem equivalence₁CounitIso_eq : (equivalence₁ hF).counitIso = equivalence₁CounitIso hF := by
ext Y
  simp [equivalence₁, equivalence₀]

/-- The unit isomorphism of the equivalence `equivalence₁` between `A` and `B'`. -/
```

-- snippet 26
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalenceCounitIso (L199-L207)
def equivalenceCounitIso : G ⋙ F ⋙ eB.inverse ≅ 𝟭 B :=
calc
    G ⋙ F ⋙ eB.inverse ≅ (G ⋙ F) ⋙ eB.inverse := (associator _ _ _).symm
    _ ≅ eB.functor ⋙ eB.inverse := isoWhiskerRight η _
    _ ≅ 𝟭 B := eB.unitIso.symm

variable {η hF hG}

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 27
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Decomposition.id (L109-L113)
def id : MorphComponents X n (X _⦋n + 1⦌) where
  a := PInfty.f (n + 1)
b i := X.σ i

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 28
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Decomposition.φ (L100-L107)
def φ {Z : C} (f : MorphComponents X n Z) : X _⦋n + 1⦌ ⟶ Z :=
PInfty.f (n + 1) ≫ f.a + ∑ i : Fin (n + 1), (P i).f (n + 1) ≫ X.δ i.rev.succ ≫
    f.b (Fin.rev i)

variable (X n)

/-- the canonical `MorphComponents` whose associated morphism is the identity
(see `F_id`) thanks to `decomposition_Q n (n+1)` -/
```

-- snippet 29
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂_inverse (L113-L117)
theorem equivalence₂_inverse :
    (equivalence₂ eB hF).inverse = eB.functor ⋙ e'.inverse ⋙ eA.inverse :=
rfl

/-- The counit isomorphism of the equivalence `equivalence₂` between `A` and `B`. -/
```

-- snippet 30
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Decomposition.decomposition_Q (L55-L89)
theorem decomposition_Q (n q : ℕ) :
    ((Q q).f (n + 1) : X _⦋n + 1⦌ ⟶ X _⦋n + 1⦌) =
      ∑ i : Fin (n + 1) with i.val < q, (P i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ (Fin.rev i) := by
induction q with
  | zero =>
    simp only [Q_zero, HomologicalComplex.zero_f_apply, Nat.not_lt_zero,
      Finset.filter_false, Finset.sum_empty]
  | succ q hq =>
    by_cases! hqn : n < q
    · rw [Q_is_eventually_constant (show n + 1 ≤ q by lia), hq]
      congr 1
      ext ⟨x, hx⟩
      simp_rw [Finset.mem_filter_univ]
      lia
    · obtain ⟨a, ha⟩ := Nat.le.dest hqn
      rw [Q_succ, HomologicalComplex.sub_f_apply, HomologicalComplex.comp_f, hq]
      symm
      conv_rhs => rw [sub_eq_add_neg, add_comm]
      let q' : Fin (n + 1) := ⟨q, Nat.lt_succ_of_le hqn⟩
      rw [← @Finset.add_sum_erase _ _ _ _ _ _ q' (by simp [q'])]
      congr
      · have hnaq' : n = a + q := by lia
        simp only [(HigherFacesVanish.of_P q n).comp_Hσ_eq hnaq', q'.rev_eq hnaq', neg_neg]
        rfl
      · ext ⟨i, hi⟩
        simp_rw [Finset.mem_erase, Finset.mem_filter_univ, q', ne_eq, Fin.mk.injEq]
        lia

variable (X)

/-- The structure `MorphComponents` is an ad hoc structure that is used in
the proof that `N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ))`
reflects isomorphisms. The fields are the data that are needed in order to
construct a morphism `X _⦋n+1⦌ ⟶ Z` (see `φ`) using the decomposition of the
identity given by `decomposition_Q n (n+1)`. -/
```

-- snippet 31
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalenceUnitIso (L240-L264)
def equivalenceUnitIso : 𝟭 A ≅ (F ⋙ eB.inverse) ⋙ G :=
calc
    𝟭 A ≅ eA.functor ⋙ eA.inverse := eA.unitIso
    _ ≅ (F ⋙ e'.inverse) ⋙ eA.inverse := isoWhiskerRight ε _
    _ ≅ F ⋙ e'.inverse ⋙ eA.inverse := associator _ _ _
    _ ≅ F ⋙ 𝟭 B' ⋙ e'.inverse ⋙ eA.inverse := isoWhiskerLeft _ (leftUnitor _).symm
    _ ≅ F ⋙ (eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
      isoWhiskerLeft _ (isoWhiskerRight eB.counitIso.symm _)
    _ ≅ (F ⋙ eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse := (associator _ _ _).symm
    _ ≅ ((F ⋙ eB.inverse) ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
      isoWhiskerRight (associator _ _ _).symm _
    _ ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse := associator _ _ _
    _ ≅ (F ⋙ eB.inverse) ⋙ (eB.functor ⋙ e'.inverse) ⋙ eA.inverse :=
      isoWhiskerLeft _ (associator _ _ _).symm
    _ ≅ (F ⋙ eB.inverse) ⋙ (G ⋙ eA.functor) ⋙ eA.inverse :=
      isoWhiskerLeft _ (isoWhiskerRight hG _)
    _ ≅ ((F ⋙ eB.inverse) ⋙ G ⋙ eA.functor) ⋙ eA.inverse := (associator _ _ _).symm
    _ ≅ (((F ⋙ eB.inverse) ⋙ G) ⋙ eA.functor) ⋙ eA.inverse :=
      isoWhiskerRight (associator _ _ _).symm _
    _ ≅ ((F ⋙ eB.inverse) ⋙ G) ⋙ eA.functor ⋙ eA.inverse := associator _ _ _
    _ ≅ ((F ⋙ eB.inverse) ⋙ G) ⋙ 𝟭 A := isoWhiskerLeft _ eA.unitIso.symm
    _ ≅ (F ⋙ eB.inverse) ⋙ G := rightUnitor _

variable {ε hF hG}
```

-- snippet 32
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂UnitIso (L138-L151)
def equivalence₂UnitIso : 𝟭 A ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse :=
calc
    𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse := equivalence₁UnitIso hF
    _ ≅ F ⋙ 𝟭 B' ⋙ e'.inverse ⋙ eA.inverse :=
      isoWhiskerLeft _ (leftUnitor _).symm
    _ ≅ F ⋙ (eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
      isoWhiskerLeft _ (isoWhiskerRight eB.counitIso.symm _)
    _ ≅ (F ⋙ eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
      (associator _ _ _).symm
    _ ≅ ((F ⋙ eB.inverse) ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
      isoWhiskerRight (associator _ _ _).symm _
    _ ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse :=
      associator _ _ _
```

-- snippet 33
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁_inverse (L70-L73)
theorem equivalence₁_inverse : (equivalence₁ hF).inverse = e'.inverse ⋙ eA.inverse :=
rfl

/-- The counit isomorphism of the equivalence `equivalence₁` between `A` and `B'`. -/
```

-- snippet 34
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂UnitIso_eq (L152-L159)
theorem equivalence₂UnitIso_eq : (equivalence₂ eB hF).unitIso = equivalence₂UnitIso eB hF := by
ext X
  simp [equivalence₂, equivalence₁]

variable {eB}

/-- The equivalence `A ≌ B` whose functor is `F ⋙ eB.inverse` and
whose inverse functor is `G : B ⥤ A`. -/
```

-- snippet 35
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.τ₀ (L176-L184)
def τ₀ : eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor :=
calc
    eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor ⋙ 𝟭 _ := isoWhiskerLeft _ e'.counitIso
    _ ≅ eB.functor := Functor.rightUnitor _

/-- The isomorphism `eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor` deduced
from the isomorphisms `hF : eA.functor ⋙ e'.functor ≅ F`,
`hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor` and the datum of
an isomorphism `η : G ⋙ F ≅ eB.functor`. -/
```

-- snippet 36
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso (L119-L130)
def equivalence₂CounitIso : (eB.functor ⋙ e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse ≅ 𝟭 B :=
calc
    (eB.functor ⋙ e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse
      ≅ eB.functor ⋙ (e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse := associator _ _ _
    _ ≅ eB.functor ⋙ ((e'.inverse ⋙ eA.inverse) ⋙ F) ⋙ eB.inverse :=
      isoWhiskerLeft _ (associator _ _ _).symm
    _ ≅ eB.functor ⋙ 𝟭 _ ⋙ eB.inverse :=
      isoWhiskerLeft _ (isoWhiskerRight (equivalence₁CounitIso hF) _)
    _ ≅ eB.functor ⋙ eB.inverse := isoWhiskerLeft _ (leftUnitor _)
    _ ≅ 𝟭 B := eB.unitIso.symm

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 37
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalenceUnitIso_eq (L265-L277)
theorem equivalenceUnitIso_eq (hε : υ hF = ε) :
    (equivalence hF hG).unitIso = equivalenceUnitIso hG ε := by
ext1; apply NatTrans.ext; ext X
  dsimp [equivalence]
  simp only [assoc, comp_id, equivalenceUnitIso_hom_app, equivalence₂_inverse, Functor.comp_obj,
    id_comp, equivalence₂UnitIso_eq eB hF, equivalence₂UnitIso_hom_app,
    ← eA.inverse.map_comp_assoc, assoc, ← hε, υ_hom_app]

end Compatibility

end DoldKan

end AlgebraicTopology
```

-- snippet 38
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence_functor (L170-L174)
theorem equivalence_functor : (equivalence hF hG).functor = F ⋙ eB.inverse :=
rfl

/-- The isomorphism `eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor` deduced
from the counit isomorphism of `e'`. -/
```

-- snippet 39
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂ (L110-L112)
def equivalence₂ : A ≌ B :=
(equivalence₁ hF).trans eB.symm
```

-- snippet 40
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalenceCounitIso_eq (L208-L227)
theorem equivalenceCounitIso_eq (hη : τ₀ = τ₁ hF hG η) :
    (equivalence hF hG).counitIso = equivalenceCounitIso η := by
ext1; apply NatTrans.ext; ext Y
  dsimp [equivalence]
  simp only [comp_id, id_comp, Functor.map_comp, equivalence₂CounitIso_eq,
    equivalence₂CounitIso_hom_app, assoc, equivalenceCounitIso_hom_app]
  simp only [equivalence₂_inverse, comp_obj, ← τ₀_hom_app, hη, τ₁_hom_app, ←
    eB.inverse.map_comp_assoc]
  rw [hF.inv.naturality_assoc, hF.inv.naturality_assoc]
  congr 2
  simp only [← e'.functor.map_comp_assoc]
  simp only [Functor.comp_map, Equivalence.fun_inv_map, comp_obj, id_obj, map_comp, assoc]
  simp only [← e'.functor.map_comp_assoc]
  simp only [Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app, comp_obj, comp_id,
    Equivalence.functor_unit_comp, map_id, id_comp]

variable (hF)

/-- The isomorphism `eA.functor ≅ F ⋙ e'.inverse` deduced from the
unit isomorphism of `e'` and the isomorphism `hF : eA.functor ⋙ e'.functor ≅ F`. -/
```

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

```lean
theorem Q_zero : (Q 0 : K[X] ⟶ _) = 0
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_zero (L54-L54)
lemma P_zero : (P 0 : K[X] ⟶ K[X]) = 𝟙 _ := rfl
```

-- snippet 2
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_succ (L55-L57)
lemma P_succ (q : ℕ) : (P (q + 1) : K[X] ⟶ K[X]) = P q ≫ (𝟙 _ + Hσ q) := rfl
/-- All the `P q` coincide with `𝟙 _` in degree 0. -/
```

-- snippet 3
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_f_0_eq (L59-L66)
theorem P_f_0_eq (q : ℕ) : ((P q).f 0 : X _⦋0⦌ ⟶ X _⦋0⦌) = 𝟙 _ := by
induction q with
  | zero => rfl
  | succ q hq =>
    simp only [P_succ, HomologicalComplex.add_f_apply, HomologicalComplex.comp_f,
      HomologicalComplex.id_f, id_comp, hq, Hσ_eq_zero, add_zero]

/-- `Q q` is the complement projection associated to `P q` -/
```

-- snippet 4
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.Q (L67-L69)
def Q (q : ℕ) : K[X] ⟶ K[X] :=
𝟙 _ - P q
```

-- snippet 5
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_add_Q (L70-L73)
theorem P_add_Q (q : ℕ) : P q + Q q = 𝟙 K[X] := by
rw [Q]
  abel
```

-- snippet 6
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_add_Q_f (L74-L76)
theorem P_add_Q_f (q n : ℕ) : (P q).f n + (Q q).f n = 𝟙 (X _⦋n⦌) :=
HomologicalComplex.congr_hom (P_add_Q q) n
```

-- snippet 7
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.Q_succ (L81-L85)
theorem Q_succ (q : ℕ) : (Q (q + 1) : K[X] ⟶ _) = Q q - P q ≫ Hσ q := by
simp only [Q, P_succ, comp_add, comp_id]
  abel

/-- All the `Q q` coincide with `0` in degree 0. -/
```

-- snippet 8
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.Q_f_0_eq (L87-L93)
theorem Q_f_0_eq (q : ℕ) : ((Q q).f 0 : X _⦋0⦌ ⟶ X _⦋0⦌) = 0 := by
simp only [HomologicalComplex.sub_f_apply, HomologicalComplex.id_f, Q, P_f_0_eq, sub_self]

namespace HigherFacesVanish

/-- This lemma expresses the vanishing of
`(P q).f (n+1) ≫ X.δ k : X _⦋n+1⦌ ⟶ X _⦋n⦌` when `k≠0` and `k≥n-q+2` -/
```

-- snippet 9
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.of_P (L94-L100)
theorem of_P : ∀ q n : ℕ, HigherFacesVanish q ((P q).f (n + 1) : X _⦋n + 1⦌ ⟶ X _⦋n + 1⦌)
  | 0 => fun n j hj₁ => by lia
  | q + 1 => fun n => by
    simp only [P_succ]
    exact (of_P q n).induction

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 10
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.comp_P_eq_self (L102-L123)
theorem comp_P_eq_self {Y : C} {n q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} (v : HigherFacesVanish q φ) :
    φ ≫ (P q).f (n + 1) = φ := by
induction q with
  | zero =>
    simp only [P_zero]
    apply comp_id
  | succ q hq =>
    simp only [P_succ, comp_add, HomologicalComplex.comp_f, HomologicalComplex.add_f_apply,
      comp_id, ← assoc, hq v.of_succ, add_eq_left]
    by_cases! hqn : n < q
    · exact v.of_succ.comp_Hσ_eq_zero hqn
    · obtain ⟨a, ha⟩ := Nat.le.dest hqn
      have hnaq : n = a + q := by lia
      simp only [v.of_succ.comp_Hσ_eq hnaq, neg_eq_zero, ← assoc]
      have eq := v ⟨a, by lia⟩ (by
        simp only [hnaq, add_assoc]
        rfl)
      simp only [Fin.succ_mk] at eq
      simp only [eq, zero_comp]

end HigherFacesVanish
```

-- snippet 11
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.comp_P_eq_self_iff (L124-L132)
theorem comp_P_eq_self_iff {Y : C} {n q : ℕ} {φ : Y ⟶ X _⦋n + 1⦌} :
    φ ≫ (P q).f (n + 1) = φ ↔ HigherFacesVanish q φ := by
constructor
  · intro hφ
    rw [← hφ]
    apply HigherFacesVanish.of_comp
    apply HigherFacesVanish.of_P
  · exact HigherFacesVanish.comp_P_eq_self
```

-- snippet 12
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_f_idem (L134-L138)
theorem P_f_idem (q n : ℕ) : ((P q).f n : X _⦋n⦌ ⟶ _) ≫ (P q).f n = (P q).f n := by
rcases n with (_ | n)
  · rw [P_f_0_eq q, comp_id]
  · exact (HigherFacesVanish.of_P q n).comp_P_eq_self
```

-- snippet 13
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.Q_f_idem (L140-L142)
theorem Q_f_idem (q n : ℕ) : ((Q q).f n : X _⦋n⦌ ⟶ _) ≫ (Q q).f n = (Q q).f n :=
idem_of_id_sub_idem _ (P_f_idem q n)
```

-- snippet 14
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.P_idem (L144-L147)
theorem P_idem (q : ℕ) : (P q : K[X] ⟶ K[X]) ≫ P q = P q := by
ext n
  exact P_f_idem q n
```

-- snippet 15
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Projections.Q_idem (L149-L154)
theorem Q_idem (q : ℕ) : (Q q : K[X] ⟶ K[X]) ≫ Q q = Q q := by
ext n
  exact Q_f_idem q n

set_option backward.isDefEq.respectTransparency false in
/-- For each `q`, `P q` is a natural transformation. -/
```

-- snippet 16
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.τ₁ (L186-L197)
def τ₁ (η : G ⋙ F ≅ eB.functor) : eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor :=
calc
    eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ (eB.functor ⋙ e'.inverse) ⋙ e'.functor :=
        (associator _ _ _).symm
    _ ≅ (G ⋙ eA.functor) ⋙ e'.functor := isoWhiskerRight hG _
    _ ≅ G ⋙ eA.functor ⋙ e'.functor := associator _ _ _
    _ ≅ G ⋙ F := isoWhiskerLeft _ hF
    _ ≅ eB.functor := η

variable (η : G ⋙ F ≅ eB.functor)

/-- The counit isomorphism of `equivalence`. -/
```

-- snippet 17
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso_eq (L131-L136)
theorem equivalence₂CounitIso_eq :
    (equivalence₂ eB hF).counitIso = equivalence₂CounitIso eB hF := by
ext Y'
  simp [equivalence₂, equivalence₁CounitIso_eq]

/-- The unit isomorphism of the equivalence `equivalence₂` between `A` and `B`. -/
```

-- snippet 18
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₀ (L60-L66)
def equivalence₀ : A ≌ B' :=
eA.trans e'

variable {eA} {e'}

/-- An intermediate equivalence `A ≌ B'` whose functor is `F` and whose inverse is
`e'.inverse ⋙ eA.inverse`. -/
```

-- snippet 19
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso (L92-L102)
def equivalence₁UnitIso : 𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse :=
calc
    𝟭 A ≅ eA.functor ⋙ eA.inverse := eA.unitIso
    _ ≅ eA.functor ⋙ 𝟭 A' ⋙ eA.inverse := isoWhiskerLeft _ (leftUnitor _).symm
    _ ≅ eA.functor ⋙ (e'.functor ⋙ e'.inverse) ⋙ eA.inverse :=
      isoWhiskerLeft _ (isoWhiskerRight e'.unitIso _)
    _ ≅ eA.functor ⋙ (e'.functor ⋙ e'.inverse ⋙ eA.inverse) :=
      isoWhiskerLeft _ (associator _ _ _)
    _ ≅ (eA.functor ⋙ e'.functor) ⋙ e'.inverse ⋙ eA.inverse := (associator _ _ _).symm
    _ ≅ F ⋙ e'.inverse ⋙ eA.inverse := isoWhiskerRight hF _
```

-- snippet 20
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.υ (L229-L238)
def υ : eA.functor ≅ F ⋙ e'.inverse :=
calc
    eA.functor ≅ eA.functor ⋙ 𝟭 A' := (rightUnitor _).symm
    _ ≅ eA.functor ⋙ e'.functor ⋙ e'.inverse := isoWhiskerLeft _ e'.unitIso
    _ ≅ (eA.functor ⋙ e'.functor) ⋙ e'.inverse := (associator _ _ _).symm
    _ ≅ F ⋙ e'.inverse := isoWhiskerRight hF _

variable (ε : eA.functor ≅ F ⋙ e'.inverse) (hG)

/-- The unit isomorphism of `equivalence`. -/
```

-- snippet 21
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence (L161-L169)
def equivalence : A ≌ B :=
((equivalence₂ eB hF).changeInverse
    (calc eB.functor ⋙ e'.inverse ⋙ eA.inverse ≅
        (eB.functor ⋙ e'.inverse) ⋙ eA.inverse := (associator _ _ _).symm
    _ ≅ (G ⋙ eA.functor) ⋙ eA.inverse := isoWhiskerRight hG _
    _ ≅ G ⋙ eA.functor ⋙ eA.inverse := associator _ _ _
    _ ≅ G ⋙ 𝟭 A := isoWhiskerLeft _ eA.unitIso.symm
    _ ≅ G := G.rightUnitor))
```

-- snippet 22
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso (L75-L85)
def equivalence₁CounitIso : (e'.inverse ⋙ eA.inverse) ⋙ F ≅ 𝟭 B' :=
calc
    (e'.inverse ⋙ eA.inverse) ⋙ F ≅ (e'.inverse ⋙ eA.inverse) ⋙ eA.functor ⋙ e'.functor :=
      isoWhiskerLeft _ hF.symm
    _ ≅ e'.inverse ⋙ (eA.inverse ⋙ eA.functor ⋙ e'.functor) := associator _ _ _
    _ ≅ e'.inverse ⋙ (eA.inverse ⋙ eA.functor) ⋙ e'.functor :=
      isoWhiskerLeft _ (associator _ _ _).symm
    _ ≅ e'.inverse ⋙ 𝟭 _ ⋙ e'.functor := isoWhiskerLeft _ (isoWhiskerRight eA.counitIso _)
    _ ≅ e'.inverse ⋙ e'.functor := isoWhiskerLeft _ (leftUnitor _)
    _ ≅ 𝟭 B' := e'.counitIso
```

-- snippet 23
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁UnitIso_eq (L103-L108)
theorem equivalence₁UnitIso_eq : (equivalence₁ hF).unitIso = equivalence₁UnitIso hF := by
ext X
  simp [equivalence₁]

/-- An intermediate equivalence `A ≌ B` obtained as the composition of `equivalence₁` and
the inverse of `eB : B ≌ B'`. -/
```

-- snippet 24
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁ (L68-L69)
def equivalence₁ : A ≌ B' := (equivalence₀ eA e').changeFunctor hF
```

-- snippet 25
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁CounitIso_eq (L86-L90)
theorem equivalence₁CounitIso_eq : (equivalence₁ hF).counitIso = equivalence₁CounitIso hF := by
ext Y
  simp [equivalence₁, equivalence₀]

/-- The unit isomorphism of the equivalence `equivalence₁` between `A` and `B'`. -/
```

-- snippet 26
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalenceCounitIso (L199-L207)
def equivalenceCounitIso : G ⋙ F ⋙ eB.inverse ≅ 𝟭 B :=
calc
    G ⋙ F ⋙ eB.inverse ≅ (G ⋙ F) ⋙ eB.inverse := (associator _ _ _).symm
    _ ≅ eB.functor ⋙ eB.inverse := isoWhiskerRight η _
    _ ≅ 𝟭 B := eB.unitIso.symm

variable {η hF hG}

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 27
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Decomposition.id (L109-L113)
def id : MorphComponents X n (X _⦋n + 1⦌) where
  a := PInfty.f (n + 1)
b i := X.σ i

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 28
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Decomposition.φ (L100-L107)
def φ {Z : C} (f : MorphComponents X n Z) : X _⦋n + 1⦌ ⟶ Z :=
PInfty.f (n + 1) ≫ f.a + ∑ i : Fin (n + 1), (P i).f (n + 1) ≫ X.δ i.rev.succ ≫
    f.b (Fin.rev i)

variable (X n)

/-- the canonical `MorphComponents` whose associated morphism is the identity
(see `F_id`) thanks to `decomposition_Q n (n+1)` -/
```

-- snippet 29
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂_inverse (L113-L117)
theorem equivalence₂_inverse :
    (equivalence₂ eB hF).inverse = eB.functor ⋙ e'.inverse ⋙ eA.inverse :=
rfl

/-- The counit isomorphism of the equivalence `equivalence₂` between `A` and `B`. -/
```

-- snippet 30
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Decomposition.decomposition_Q (L55-L89)
theorem decomposition_Q (n q : ℕ) :
    ((Q q).f (n + 1) : X _⦋n + 1⦌ ⟶ X _⦋n + 1⦌) =
      ∑ i : Fin (n + 1) with i.val < q, (P i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ (Fin.rev i) := by
induction q with
  | zero =>
    simp only [Q_zero, HomologicalComplex.zero_f_apply, Nat.not_lt_zero,
      Finset.filter_false, Finset.sum_empty]
  | succ q hq =>
    by_cases! hqn : n < q
    · rw [Q_is_eventually_constant (show n + 1 ≤ q by lia), hq]
      congr 1
      ext ⟨x, hx⟩
      simp_rw [Finset.mem_filter_univ]
      lia
    · obtain ⟨a, ha⟩ := Nat.le.dest hqn
      rw [Q_succ, HomologicalComplex.sub_f_apply, HomologicalComplex.comp_f, hq]
      symm
      conv_rhs => rw [sub_eq_add_neg, add_comm]
      let q' : Fin (n + 1) := ⟨q, Nat.lt_succ_of_le hqn⟩
      rw [← @Finset.add_sum_erase _ _ _ _ _ _ q' (by simp [q'])]
      congr
      · have hnaq' : n = a + q := by lia
        simp only [(HigherFacesVanish.of_P q n).comp_Hσ_eq hnaq', q'.rev_eq hnaq', neg_neg]
        rfl
      · ext ⟨i, hi⟩
        simp_rw [Finset.mem_erase, Finset.mem_filter_univ, q', ne_eq, Fin.mk.injEq]
        lia

variable (X)

/-- The structure `MorphComponents` is an ad hoc structure that is used in
the proof that `N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ))`
reflects isomorphisms. The fields are the data that are needed in order to
construct a morphism `X _⦋n+1⦌ ⟶ Z` (see `φ`) using the decomposition of the
identity given by `decomposition_Q n (n+1)`. -/
```

-- snippet 31
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalenceUnitIso (L240-L264)
def equivalenceUnitIso : 𝟭 A ≅ (F ⋙ eB.inverse) ⋙ G :=
calc
    𝟭 A ≅ eA.functor ⋙ eA.inverse := eA.unitIso
    _ ≅ (F ⋙ e'.inverse) ⋙ eA.inverse := isoWhiskerRight ε _
    _ ≅ F ⋙ e'.inverse ⋙ eA.inverse := associator _ _ _
    _ ≅ F ⋙ 𝟭 B' ⋙ e'.inverse ⋙ eA.inverse := isoWhiskerLeft _ (leftUnitor _).symm
    _ ≅ F ⋙ (eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
      isoWhiskerLeft _ (isoWhiskerRight eB.counitIso.symm _)
    _ ≅ (F ⋙ eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse := (associator _ _ _).symm
    _ ≅ ((F ⋙ eB.inverse) ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
      isoWhiskerRight (associator _ _ _).symm _
    _ ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse := associator _ _ _
    _ ≅ (F ⋙ eB.inverse) ⋙ (eB.functor ⋙ e'.inverse) ⋙ eA.inverse :=
      isoWhiskerLeft _ (associator _ _ _).symm
    _ ≅ (F ⋙ eB.inverse) ⋙ (G ⋙ eA.functor) ⋙ eA.inverse :=
      isoWhiskerLeft _ (isoWhiskerRight hG _)
    _ ≅ ((F ⋙ eB.inverse) ⋙ G ⋙ eA.functor) ⋙ eA.inverse := (associator _ _ _).symm
    _ ≅ (((F ⋙ eB.inverse) ⋙ G) ⋙ eA.functor) ⋙ eA.inverse :=
      isoWhiskerRight (associator _ _ _).symm _
    _ ≅ ((F ⋙ eB.inverse) ⋙ G) ⋙ eA.functor ⋙ eA.inverse := associator _ _ _
    _ ≅ ((F ⋙ eB.inverse) ⋙ G) ⋙ 𝟭 A := isoWhiskerLeft _ eA.unitIso.symm
    _ ≅ (F ⋙ eB.inverse) ⋙ G := rightUnitor _

variable {ε hF hG}
```

-- snippet 32
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂UnitIso (L138-L151)
def equivalence₂UnitIso : 𝟭 A ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse :=
calc
    𝟭 A ≅ F ⋙ e'.inverse ⋙ eA.inverse := equivalence₁UnitIso hF
    _ ≅ F ⋙ 𝟭 B' ⋙ e'.inverse ⋙ eA.inverse :=
      isoWhiskerLeft _ (leftUnitor _).symm
    _ ≅ F ⋙ (eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
      isoWhiskerLeft _ (isoWhiskerRight eB.counitIso.symm _)
    _ ≅ (F ⋙ eB.inverse ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
      (associator _ _ _).symm
    _ ≅ ((F ⋙ eB.inverse) ⋙ eB.functor) ⋙ e'.inverse ⋙ eA.inverse :=
      isoWhiskerRight (associator _ _ _).symm _
    _ ≅ (F ⋙ eB.inverse) ⋙ eB.functor ⋙ e'.inverse ⋙ eA.inverse :=
      associator _ _ _
```

-- snippet 33
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₁_inverse (L70-L73)
theorem equivalence₁_inverse : (equivalence₁ hF).inverse = e'.inverse ⋙ eA.inverse :=
rfl

/-- The counit isomorphism of the equivalence `equivalence₁` between `A` and `B'`. -/
```

-- snippet 34
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂UnitIso_eq (L152-L159)
theorem equivalence₂UnitIso_eq : (equivalence₂ eB hF).unitIso = equivalence₂UnitIso eB hF := by
ext X
  simp [equivalence₂, equivalence₁]

variable {eB}

/-- The equivalence `A ≌ B` whose functor is `F ⋙ eB.inverse` and
whose inverse functor is `G : B ⥤ A`. -/
```

-- snippet 35
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.τ₀ (L176-L184)
def τ₀ : eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor :=
calc
    eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor ⋙ 𝟭 _ := isoWhiskerLeft _ e'.counitIso
    _ ≅ eB.functor := Functor.rightUnitor _

/-- The isomorphism `eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor` deduced
from the isomorphisms `hF : eA.functor ⋙ e'.functor ≅ F`,
`hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor` and the datum of
an isomorphism `η : G ⋙ F ≅ eB.functor`. -/
```

-- snippet 36
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂CounitIso (L119-L130)
def equivalence₂CounitIso : (eB.functor ⋙ e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse ≅ 𝟭 B :=
calc
    (eB.functor ⋙ e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse
      ≅ eB.functor ⋙ (e'.inverse ⋙ eA.inverse) ⋙ F ⋙ eB.inverse := associator _ _ _
    _ ≅ eB.functor ⋙ ((e'.inverse ⋙ eA.inverse) ⋙ F) ⋙ eB.inverse :=
      isoWhiskerLeft _ (associator _ _ _).symm
    _ ≅ eB.functor ⋙ 𝟭 _ ⋙ eB.inverse :=
      isoWhiskerLeft _ (isoWhiskerRight (equivalence₁CounitIso hF) _)
    _ ≅ eB.functor ⋙ eB.inverse := isoWhiskerLeft _ (leftUnitor _)
    _ ≅ 𝟭 B := eB.unitIso.symm

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 37
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalenceUnitIso_eq (L265-L277)
theorem equivalenceUnitIso_eq (hε : υ hF = ε) :
    (equivalence hF hG).unitIso = equivalenceUnitIso hG ε := by
ext1; apply NatTrans.ext; ext X
  dsimp [equivalence]
  simp only [assoc, comp_id, equivalenceUnitIso_hom_app, equivalence₂_inverse, Functor.comp_obj,
    id_comp, equivalence₂UnitIso_eq eB hF, equivalence₂UnitIso_hom_app,
    ← eA.inverse.map_comp_assoc, assoc, ← hε, υ_hom_app]

end Compatibility

end DoldKan

end AlgebraicTopology
```

-- snippet 38
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence_functor (L170-L174)
theorem equivalence_functor : (equivalence hF hG).functor = F ⋙ eB.inverse :=
rfl

/-- The isomorphism `eB.functor ⋙ e'.inverse ⋙ e'.functor ≅ eB.functor` deduced
from the counit isomorphism of `e'`. -/
```

-- snippet 39
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalence₂ (L110-L112)
def equivalence₂ : A ≌ B :=
(equivalence₁ hF).trans eB.symm
```

-- snippet 40
```lean
-- Mathlib.AlgebraicTopology.DoldKan.Compatibility.equivalenceCounitIso_eq (L208-L227)
theorem equivalenceCounitIso_eq (hη : τ₀ = τ₁ hF hG η) :
    (equivalence hF hG).counitIso = equivalenceCounitIso η := by
ext1; apply NatTrans.ext; ext Y
  dsimp [equivalence]
  simp only [comp_id, id_comp, Functor.map_comp, equivalence₂CounitIso_eq,
    equivalence₂CounitIso_hom_app, assoc, equivalenceCounitIso_hom_app]
  simp only [equivalence₂_inverse, comp_obj, ← τ₀_hom_app, hη, τ₁_hom_app, ←
    eB.inverse.map_comp_assoc]
  rw [hF.inv.naturality_assoc, hF.inv.naturality_assoc]
  congr 2
  simp only [← e'.functor.map_comp_assoc]
  simp only [Functor.comp_map, Equivalence.fun_inv_map, comp_obj, id_obj, map_comp, assoc]
  simp only [← e'.functor.map_comp_assoc]
  simp only [Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app, comp_obj, comp_id,
    Equivalence.functor_unit_comp, map_id, id_comp]

variable (hF)

/-- The isomorphism `eA.functor ≅ F ⋙ e'.inverse` deduced from the
unit isomorphism of `e'` and the isomorphism `hF : eA.functor ⋙ e'.functor ≅ F`. -/
```
[/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]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.AlgebraicTopology.SimplexCategory.Basic.eq_σ_comp_of_not_injective`:
State 0: ⊢ theorem eq_σ_comp_of_not_injective {n : ℕ} {Δ' : SimplexCategory} (θ : ⦋n + 1⦌ ⟶ Δ')
    (hθ : ¬Function.Injective θ.toOrderHom) :
    ∃ (i : Fin (n + 1)) (θ' : ⦋n⦌ ⟶ Δ'), θ = σ i ≫ θ'
Tactic: simp only [Function.Injective, exists_prop, not_forall] at hθ  -- as θ is not injective, there exists `x<y` such that `θ x = θ y`  -- and then, `θ x = θ (x+1)`
State 1: ⊢ <goal after simp only [Function.Injective, exists_prop, not_forall] at hθ>
Tactic: have hθ₂ : ∃ x y : Fin (n + 2), (Hom.toOrderHom θ) x = (Hom.toOrderHom θ) y ∧ x < y := by  rcases hθ with ⟨x, y, ⟨h₁, h₂⟩⟩  by_cases h : x < y  · exact ⟨x, y, ⟨h₁, h⟩⟩  · refine ⟨y, x, ⟨h₁.symm, ?_⟩⟩  lia  rcases hθ₂ with ⟨x, y, ⟨h₁, h₂⟩⟩
State 2: ⊢ <goal after have hθ₂ : ∃ x y : Fin (n + 2), (Hom.toOrderHom θ) x = (Hom.toOrderHom θ) y ∧ x < y := by>
Tactic: use x.castPred ((Fin.le_last _).trans_lt' h₂).ne
State 3: ⊢ <goal after use x.castPred ((Fin.le_last _).trans_lt' h₂).ne>
Tactic: apply eq_σ_comp_of_not_injective'
State 4: ⊢ <goal after apply eq_σ_comp_of_not_injective'>
Tactic: apply le_antisymm  · exact θ.toOrderHom.monotone (le_of_lt Fin.castSucc_lt_succ)  · rw [Fin.castSucc_castPred, h₁]
State 5: ⊢ <goal after apply le_antisymm>
Tactic: exact θ.toOrderHom.monotone ((Fin.succ_castPred_le_iff _).mpr h₂)
```

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

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

```text
Proof-state transcript for `Mathlib.AlgebraicTopology.SimplexCategory.Basic.eq_σ_comp_of_not_injective`:
State 0: ⊢ theorem eq_σ_comp_of_not_injective {n : ℕ} {Δ' : SimplexCategory} (θ : ⦋n + 1⦌ ⟶ Δ')
    (hθ : ¬Function.Injective θ.toOrderHom) :
    ∃ (i : Fin (n + 1)) (θ' : ⦋n⦌ ⟶ Δ'), θ = σ i ≫ θ'
Tactic: simp only [Function.Injective, exists_prop, not_forall] at hθ  -- as θ is not injective, there exists `x<y` such that `θ x = θ y`  -- and then, `θ x = θ (x+1)`
State 1: ⊢ <goal after simp only [Function.Injective, exists_prop, not_forall] at hθ>
Tactic: have hθ₂ : ∃ x y : Fin (n + 2), (Hom.toOrderHom θ) x = (Hom.toOrderHom θ) y ∧ x < y := by  rcases hθ with ⟨x, y, ⟨h₁, h₂⟩⟩  by_cases h : x < y  · exact ⟨x, y, ⟨h₁, h⟩⟩  · refine ⟨y, x, ⟨h₁.symm, ?_⟩⟩  lia  rcases hθ₂ with ⟨x, y, ⟨h₁, h₂⟩⟩
State 2: ⊢ <goal after have hθ₂ : ∃ x y : Fin (n + 2), (Hom.toOrderHom θ) x = (Hom.toOrderHom θ) y ∧ x < y := by>
Tactic: use x.castPred ((Fin.le_last _).trans_lt' h₂).ne
State 3: ⊢ <goal after use x.castPred ((Fin.le_last _).trans_lt' h₂).ne>
Tactic: apply eq_σ_comp_of_not_injective'
State 4: ⊢ <goal after apply eq_σ_comp_of_not_injective'>
Tactic: apply le_antisymm  · exact θ.toOrderHom.monotone (le_of_lt Fin.castSucc_lt_succ)  · rw [Fin.castSucc_castPred, h₁]
State 5: ⊢ <goal after apply le_antisymm>
Tactic: exact θ.toOrderHom.monotone ((Fin.succ_castPred_le_iff _).mpr h₂)
```

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.Basic).

[User]
Complete the following proof.

```lean
def intEquivNatSumNat : ℤ ≃ ℕ ⊕ ℕ where
  toFun z
```

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

```lean
def intEquivNatSumNat : ℤ ≃ ℕ ⊕ ℕ where
  toFun z
```

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.Primrec.List).

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

```lean
theorem list_getD (d : α) : Primrec₂ fun l n => List.getD l n d
  frobnicateTacticThatDoesNotExist := by
simp only [List.getD_eq_getElem?_getD]
  exact option_getD.comp₂ list_getElem? (const _)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=149, messages=[Message(data='Function expected at\n  l.getD n d\nbut this term has type\n  α\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist', pos=Position(line=1, column=48), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Primcodable (List α)\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=28), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='`simp` made no progress', pos=Position(line=3, column=0), pos_end=Position(line=3, column=38), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=149, i_end=198, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=2), pos_end=Position(line=4, column=7), 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.Computability.Primrec.List).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
theorem list_getD (d : α) : Primrec₂ fun l n => List.getD l n d
  frobnicateTacticThatDoesNotExist := by
simp only [List.getD_eq_getElem?_getD]
  exact option_getD.comp₂ list_getElem? (const _)
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=149, messages=[Message(data='Function expected at\n  l.getD n d\nbut this term has type\n  α\n\nNote: Expected a function because this term is being applied to the argument\n  frobnicateTacticThatDoesNotExist', pos=Position(line=1, column=48), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind=None), Message(data='failed to synthesize instance of type class\n  Primcodable (List α)\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.', pos=Position(line=1, column=28), pos_end=Position(line=2, column=34), severity=<Severity.ERROR: 3>, kind='lean.synthInstanceFailed._namedError'), Message(data='`simp` made no progress', pos=Position(line=3, column=0), pos_end=Position(line=3, column=38), severity=<Severity.ERROR: 3>, kind=None)], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None), CompilationUnit(i_begin=149, i_end=198, messages=[Message(data='unexpected identifier; expected command', pos=Position(line=4, column=2), pos_end=Position(line=4, column=7), 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.Tactic.FunProp.Core).

[User]
Complete the following proof.

```lean
def tryTheoremWithHint? (e : Expr) (thmOrigin : Origin)
    (hint : Array (Nat × Expr))
    (funProp : Expr → FunPropM (Option Result)) (newMCtxDepth : Bool
```

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

```lean
def tryTheoremWithHint? (e : Expr) (thmOrigin : Origin)
    (hint : Array (Nat × Expr))
    (funProp : Expr → FunPropM (Option Result)) (newMCtxDepth : Bool
```

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