<LEAN_FILE>:147:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  Fact False

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.

Response:
by
  ext k
  by_cases h : k = c.index j
  · rw [h]
    let r : Fin (c.blocksFun (c.index j)) → Fin n := c.embedding (c.index j)
    simp only [Function.update_self]
    change p (c.blocksFun (c.index j)) (Function.update v j z ∘ r) = _
    let j' := c.invEmbedding j
    suffices B : Function.update v j z ∘ r = Function.update (v ∘ r) j' z by rw [B]
    suffices C : Function.update v (r j') z ∘ r = Function.update (v ∘ r) j' z by
      convert! C; exact (c.embedding_comp_inv j).symm
    exact Function.update_comp_eq_of_injective _ (c.embedding _).injective _ _
  · simp only [h, Function.update_of_ne, Ne, not_false_iff]
    let r : Fin (c.blocksFun k) → Fin n := c.embedding k
    change p (c.blocksFun k) (Function.update v j z ∘ r) = p (c.blocksFun k) (v ∘ r)
    suffices B : Function.update v j z ∘ r = v ∘ r by rw [B]
    apply Function.update_comp_eq_of_notMem_range
    rwa [c.mem_range_embedding_iff']

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Analysis.Analytic.Basic
public import Mathlib.Analysis.Analytic.CPolynomialDef
public import Mathlib.Combinatorics.Enumerative.Composition

Namespace:
FormalMultilinearSeries

Local safe context:
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
/-!
# Composition of analytic functions

In this file we prove that the composition of analytic functions is analytic.

The argument is the following. Assume `g z = ∑' qₙ (z, ..., z)` and `f y = ∑' pₖ (y, ..., y)`. Then

`g (f y) = ∑' qₙ (∑' pₖ (y, ..., y), ..., ∑' pₖ (y, ..., y))
= ∑' qₙ (p_{i₁} (y, ..., y), ..., p_{iₙ} (y, ..., y))`.

For each `n` and `i₁, ..., iₙ`, define a `i₁ + ... + iₙ` multilinear function mapping
`(y₀, ..., y_{i₁ + ... + iₙ - 1})` to
`qₙ (p_{i₁} (y₀, ..., y_{i₁-1}), p_{i₂} (y_{i₁}, ..., y_{i₁ + i₂ - 1}), ..., p_{iₙ} (....)))`.
Then `g ∘ f` is obtained by summing all these multilinear functions.

To formalize this, we use compositions of an integer `N`, i.e., its decompositions into
a sum `i₁ + ... + iₙ` of positive integers. Given such a composition `c` and two formal
multilinear series `q` and `p`, let `q.compAlongComposition p c` be the above multilinear
function. Then the `N`-th coefficient in the power series expansion of `g ∘ f` is the sum of these
terms over all `c : Composition N`.

To complete the proof, we need to show that this power series has a positive radius of convergence.
This follows from the fact that `Composition N` has cardinality `2^(N-1)` and estimates on
the norm of `qₙ` and `pₖ`, which give summability. We also need to show that it indeed converges to
`g ∘ f`. For this, we note that the composition of partial sums converges to `g ∘ f`, and that it
corresponds to a part of the whole sum, on a subset that increases to the whole space. By
summability of the norms, this implies the overall convergence.

## Main results

* `q.comp p` is the formal composition of the formal multilinear series `q` and `p`.
* `HasFPowerSeriesAt.comp` states that if two functions `g` and `f` admit power series expansions
  `q` and `p`, then `g ∘ f` admits a power series expansion given by `q.comp p`.
* `AnalyticAt.comp` states that the composition of analytic functions is analytic.
* `FormalMultilinearSeries.comp_assoc` states that composition is associative on formal
  multilinear series.

## Implementation details

The main technical difficulty is to write down things. In particular, we need to define precisely
`q.compAlongComposition p c` and to show that it is indeed a continuous multilinear
function. This requires a whole interface built on the class `Composition`. Once this is set,
the main difficulty is to reorder the sums, writing the composition of the partial sums as a sum
over some subset of `Σ n, Composition n`. We need to check that the reordering is a bijection,
running over difficulties due to the dependent nature of the types under consideration, that are
controlled thanks to the interface for `Composition`.

The associativity of composition on formal multilinear series is a nontrivial result: it does not
follow from the associativity of composition of analytic functions, as there is no uniqueness for
the formal multilinear series representing a function (and also, it holds even when the radius of
convergence of the series is `0`). Instead, we give a direct proof, which amounts to reordering
double sums in a careful way. The change of variables is a canonical (combinatorial) bijection
`Composition.sigmaEquivSigmaPi` between `(Σ (a : Composition n), Composition a.length)` and
`(Σ (c : Composition n), Π (i : Fin c.length), Composition (c.blocksFun i))`, and is described
in more details below in the paragraph on associativity.
-/

@[expose] public section

noncomputable section

variable {𝕜 : Type*} {E F G H : Type*}

open Filter List

open scoped Topology NNReal ENNReal

section Topological

variable [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G]
variable [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G]
variable [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G]

/-! ### Composing formal multilinear series -/

namespace FormalMultilinearSeries

variable [IsTopologicalAddGroup E] [ContinuousConstSMul 𝕜 E]
variable [IsTopologicalAddGroup F] [ContinuousConstSMul 𝕜 F]
variable [IsTopologicalAddGroup G] [ContinuousConstSMul 𝕜 G]

/-!
In this paragraph, we define the composition of formal multilinear series, by summing over all
possible compositions of `n`.
-/

/-- Given a formal multilinear series `p`, a composition `c` of `n` and the index `i` of a
block of `c`, we may define a function on `Fin n → E` by picking the variables in the `i`-th block
of `n`, and applying the corresponding coefficient of `p` to these variables. This function is
called `p.applyComposition c v i` for `v : Fin n → E` and `i : Fin c.length`. -/
def applyComposition (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n) :
    (Fin n → E) → Fin c.length → F := fun v i => p (c.blocksFun i) (v ∘ c.embedding i)

theorem applyComposition_ones (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
    p.applyComposition (Composition.ones n) = fun v i =>
      p 1 fun _ => v (Fin.castLE (Composition.length_le _) i) := by
  funext v i
  apply p.congr (Composition.ones_blocksFun _ _)
  intro j hjn hj1
  obtain rfl : j = 0 := by lia
  refine congr_arg v ?_
  rw [Fin.ext_iff, Fin.val_castLE, Composition.ones_embedding, Fin.val_mk]

theorem applyComposition_single (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : 0 < n)
    (v : Fin n → E) : p.applyComposition (Composition.single n hn) v = fun _j => p n v := by
  ext j
  refine p.congr (by simp) fun i hi1 hi2 => ?_
  dsimp
  congr 1
  convert! Composition.single_embedding hn ⟨i, hi2⟩ using 1
  obtain ⟨j_val, j_property⟩ := j
  have : j_val = 0 := le_bot_iff.1 (Nat.lt_succ_iff.1 j_property)
  rw! [this]
  rfl

@[simp]
theorem removeZero_applyComposition (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
    (c : Composition n) : p.removeZero.applyComposition c = p.applyComposition c := by
  ext v i
  simp [applyComposition, zero_lt_one.trans_le (c.one_le_blocksFun i), removeZero_of_pos]

/-- Technical lemma stating how `p.applyComposition` commutes with updating variables. This
will be the key point to show that functions constructed from `applyComposition` retain
multilinearity. -/

Target:
theorem applyComposition_update (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n)
    (j : Fin n) (v : Fin n → E) (z : E) :
    p.applyComposition c (Function.update v j z) =
      Function.update (p.applyComposition c v) (c.index j)
        (p (c.blocksFun (c.index j))
          (Function.update (v ∘ c.embedding (c.index j)) (c.invEmbedding j) z)) :=

Failed candidate:
by
  let __repair_9ad67ad0cb : Fact False := inferInstance
  ext k
  by_cases h : k = c.index j
  · rw [h]
    let r : Fin (c.blocksFun (c.index j)) → Fin n := c.embedding (c.index j)
    simp only [Function.update_self]
    change p (c.blocksFun (c.index j)) (Function.update v j z ∘ r) = _
    let j' := c.invEmbedding j
    suffices B : Function.update v j z ∘ r = Function.update (v ∘ r) j' z by rw [B]
    suffices C : Function.update v (r j') z ∘ r = Function.update (v ∘ r) j' z by
      convert! C; exact (c.embedding_comp_inv j).symm
    exact Function.update_comp_eq_of_injective _ (c.embedding _).injective _ _
  · simp only [h, Function.update_of_ne, Ne, not_false_iff]
    let r : Fin (c.blocksFun k) → Fin n := c.embedding k
    change p (c.blocksFun k) (Function.update v j z ∘ r) = p (c.blocksFun k) (v ∘ r)
    suffices B : Function.update v j z ∘ r = v ∘ r by rw [B]
    apply Function.update_comp_eq_of_notMem_range
    rwa [c.mem_range_embedding_iff']

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:147:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:

[Assistant]
by
  ext k
  by_cases h : k = c.index j
  · rw [h]
    let r : Fin (c.blocksFun (c.index j)) → Fin n := c.embedding (c.index j)
    simp only [Function.update_self]
    change p (c.blocksFun (c.index j)) (Function.update v j z ∘ r) = _
    let j' := c.invEmbedding j
    suffices B : Function.update v j z ∘ r = Function.update (v ∘ r) j' z by rw [B]
    suffices C : Function.update v (r j') z ∘ r = Function.update (v ∘ r) j' z by
      convert! C; exact (c.embedding_comp_inv j).symm
    exact Function.update_comp_eq_of_injective _ (c.embedding _).injective _ _
  · simp only [h, Function.update_of_ne, Ne, not_false_iff]
    let r : Fin (c.blocksFun k) → Fin n := c.embedding k
    change p (c.blocksFun k) (Function.update v j z ∘ r) = p (c.blocksFun k) (v ∘ r)
    suffices B : Function.update v j z ∘ r = v ∘ r by rw [B]
    apply Function.update_comp_eq_of_notMem_range
    rwa [c.mem_range_embedding_iff']

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Analysis.Analytic.Basic
public import Mathlib.Analysis.Analytic.CPolynomialDef
public import Mathlib.Combinatorics.Enumerative.Composition

Namespace:
FormalMultilinearSeries

Local safe context:
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
/-!
# Composition of analytic functions

In this file we prove that the composition of analytic functions is analytic.

The argument is the following. Assume `g z = ∑' qₙ (z, ..., z)` and `f y = ∑' pₖ (y, ..., y)`. Then

`g (f y) = ∑' qₙ (∑' pₖ (y, ..., y), ..., ∑' pₖ (y, ..., y))
= ∑' qₙ (p_{i₁} (y, ..., y), ..., p_{iₙ} (y, ..., y))`.

For each `n` and `i₁, ..., iₙ`, define a `i₁ + ... + iₙ` multilinear function mapping
`(y₀, ..., y_{i₁ + ... + iₙ - 1})` to
`qₙ (p_{i₁} (y₀, ..., y_{i₁-1}), p_{i₂} (y_{i₁}, ..., y_{i₁ + i₂ - 1}), ..., p_{iₙ} (....)))`.
Then `g ∘ f` is obtained by summing all these multilinear functions.

To formalize this, we use compositions of an integer `N`, i.e., its decompositions into
a sum `i₁ + ... + iₙ` of positive integers. Given such a composition `c` and two formal
multilinear series `q` and `p`, let `q.compAlongComposition p c` be the above multilinear
function. Then the `N`-th coefficient in the power series expansion of `g ∘ f` is the sum of these
terms over all `c : Composition N`.

To complete the proof, we need to show that this power series has a positive radius of convergence.
This follows from the fact that `Composition N` has cardinality `2^(N-1)` and estimates on
the norm of `qₙ` and `pₖ`, which give summability. We also need to show that it indeed converges to
`g ∘ f`. For this, we note that the composition of partial sums converges to `g ∘ f`, and that it
corresponds to a part of the whole sum, on a subset that increases to the whole space. By
summability of the norms, this implies the overall convergence.

## Main results

* `q.comp p` is the formal composition of the formal multilinear series `q` and `p`.
* `HasFPowerSeriesAt.comp` states that if two functions `g` and `f` admit power series expansions
  `q` and `p`, then `g ∘ f` admits a power series expansion given by `q.comp p`.
* `AnalyticAt.comp` states that the composition of analytic functions is analytic.
* `FormalMultilinearSeries.comp_assoc` states that composition is associative on formal
  multilinear series.

## Implementation details

The main technical difficulty is to write down things. In particular, we need to define precisely
`q.compAlongComposition p c` and to show that it is indeed a continuous multilinear
function. This requires a whole interface built on the class `Composition`. Once this is set,
the main difficulty is to reorder the sums, writing the composition of the partial sums as a sum
over some subset of `Σ n, Composition n`. We need to check that the reordering is a bijection,
running over difficulties due to the dependent nature of the types under consideration, that are
controlled thanks to the interface for `Composition`.

The associativity of composition on formal multilinear series is a nontrivial result: it does not
follow from the associativity of composition of analytic functions, as there is no uniqueness for
the formal multilinear series representing a function (and also, it holds even when the radius of
convergence of the series is `0`). Instead, we give a direct proof, which amounts to reordering
double sums in a careful way. The change of variables is a canonical (combinatorial) bijection
`Composition.sigmaEquivSigmaPi` between `(Σ (a : Composition n), Composition a.length)` and
`(Σ (c : Composition n), Π (i : Fin c.length), Composition (c.blocksFun i))`, and is described
in more details below in the paragraph on associativity.
-/

@[expose] public section

noncomputable section

variable {𝕜 : Type*} {E F G H : Type*}

open Filter List

open scoped Topology NNReal ENNReal

section Topological

variable [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G]
variable [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G]
variable [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G]

/-! ### Composing formal multilinear series -/

namespace FormalMultilinearSeries

variable [IsTopologicalAddGroup E] [ContinuousConstSMul 𝕜 E]
variable [IsTopologicalAddGroup F] [ContinuousConstSMul 𝕜 F]
variable [IsTopologicalAddGroup G] [ContinuousConstSMul 𝕜 G]

/-!
In this paragraph, we define the composition of formal multilinear series, by summing over all
possible compositions of `n`.
-/

/-- Given a formal multilinear series `p`, a composition `c` of `n` and the index `i` of a
block of `c`, we may define a function on `Fin n → E` by picking the variables in the `i`-th block
of `n`, and applying the corresponding coefficient of `p` to these variables. This function is
called `p.applyComposition c v i` for `v : Fin n → E` and `i : Fin c.length`. -/
def applyComposition (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n) :
    (Fin n → E) → Fin c.length → F := fun v i => p (c.blocksFun i) (v ∘ c.embedding i)

theorem applyComposition_ones (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
    p.applyComposition (Composition.ones n) = fun v i =>
      p 1 fun _ => v (Fin.castLE (Composition.length_le _) i) := by
  funext v i
  apply p.congr (Composition.ones_blocksFun _ _)
  intro j hjn hj1
  obtain rfl : j = 0 := by lia
  refine congr_arg v ?_
  rw [Fin.ext_iff, Fin.val_castLE, Composition.ones_embedding, Fin.val_mk]

theorem applyComposition_single (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : 0 < n)
    (v : Fin n → E) : p.applyComposition (Composition.single n hn) v = fun _j => p n v := by
  ext j
  refine p.congr (by simp) fun i hi1 hi2 => ?_
  dsimp
  congr 1
  convert! Composition.single_embedding hn ⟨i, hi2⟩ using 1
  obtain ⟨j_val, j_property⟩ := j
  have : j_val = 0 := le_bot_iff.1 (Nat.lt_succ_iff.1 j_property)
  rw! [this]
  rfl

@[simp]
theorem removeZero_applyComposition (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
    (c : Composition n) : p.removeZero.applyComposition c = p.applyComposition c := by
  ext v i
  simp [applyComposition, zero_lt_one.trans_le (c.one_le_blocksFun i), removeZero_of_pos]

/-- Technical lemma stating how `p.applyComposition` commutes with updating variables. This
will be the key point to show that functions constructed from `applyComposition` retain
multilinearity. -/

Target:
theorem applyComposition_update (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n)
    (j : Fin n) (v : Fin n → E) (z : E) :
    p.applyComposition c (Function.update v j z) =
      Function.update (p.applyComposition c v) (c.index j)
        (p (c.blocksFun (c.index j))
          (Function.update (v ∘ c.embedding (c.index j)) (c.invEmbedding j) z)) :=

Failed candidate:
by
  let __repair_9ad67ad0cb : Fact False := inferInstance
  ext k
  by_cases h : k = c.index j
  · rw [h]
    let r : Fin (c.blocksFun (c.index j)) → Fin n := c.embedding (c.index j)
    simp only [Function.update_self]
    change p (c.blocksFun (c.index j)) (Function.update v j z ∘ r) = _
    let j' := c.invEmbedding j
    suffices B : Function.update v j z ∘ r = Function.update (v ∘ r) j' z by rw [B]
    suffices C : Function.update v (r j') z ∘ r = Function.update (v ∘ r) j' z by
      convert! C; exact (c.embedding_comp_inv j).symm
    exact Function.update_comp_eq_of_injective _ (c.embedding _).injective _ _
  · simp only [h, Function.update_of_ne, Ne, not_false_iff]
    let r : Fin (c.blocksFun k) → Fin n := c.embedding k
    change p (c.blocksFun k) (Function.update v j z ∘ r) = p (c.blocksFun k) (v ∘ r)
    suffices B : Function.update v j z ∘ r = v ∘ r by rw [B]
    apply Function.update_comp_eq_of_notMem_range
    rwa [c.mem_range_embedding_iff']

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:147:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:


<LEAN_FILE>:134:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class
  Fact False

Hint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.

Response:
by
  rw [← decompose_symm_of _, Equiv.apply_symm_apply]

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Algebra.DirectSum.Module
public import Mathlib.Algebra.Module.Submodule.Basic

Namespace:
DirectSum

Local safe context:
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
/-!
# Decompositions of additive monoids, groups, and modules into direct sums

## Main definitions

* `DirectSum.Decomposition ℳ`: A typeclass to provide a constructive decomposition from
  an additive monoid `M` into a family of additive submonoids `ℳ`
* `DirectSum.decompose ℳ`: The canonical equivalence provided by the above typeclass

## Main statements

* `DirectSum.Decomposition.isInternal`: The link to `DirectSum.IsInternal`.

## Implementation details

As we want to talk about different types of decomposition (additive monoids, modules, rings, ...),
we choose to avoid heavily bundling `DirectSum.decompose`, instead making copies for the
`AddEquiv`, `LinearEquiv`, etc. This means we have to repeat statements that follow from these
bundled homs, but means we don't have to repeat statements for different types of decomposition.
-/

@[expose] public section

variable {ι R M σ : Type*}

open DirectSum

namespace DirectSum

section AddCommMonoid

variable [DecidableEq ι] [AddCommMonoid M]
variable [SetLike σ M] [AddSubmonoidClass σ M] (ℳ : ι → σ)

/-- A decomposition is an equivalence between an additive monoid `M` and a direct sum of additive
submonoids `ℳ i` of that `M`, such that the "recomposition" is canonical. This definition also
works for additive groups and modules.

This is a version of `DirectSum.IsInternal` which comes with a constructive inverse to the
canonical "recomposition" rather than just a proof that the "recomposition" is bijective.

Often it is easier to construct a term of this type via `Decomposition.ofAddHom` or
`Decomposition.ofLinearMap`. -/
class Decomposition where
  decompose' : M → ⨁ i, ℳ i
  left_inv : Function.LeftInverse (DirectSum.coeAddMonoidHom ℳ) decompose'
  right_inv : Function.RightInverse (DirectSum.coeAddMonoidHom ℳ) decompose'

/-- `DirectSum.Decomposition` instances, while carrying data, are always equal. -/
instance : Subsingleton (Decomposition ℳ) :=
  ⟨fun x y ↦ by
    obtain ⟨_, _, xr⟩ := x
    obtain ⟨_, yl, _⟩ := y
    congr
    exact Function.LeftInverse.eq_rightInverse xr yl⟩

/-- A convenience method to construct a decomposition from an `AddMonoidHom`, such that the proofs
of left and right inverse can be constructed via `ext`. -/
abbrev Decomposition.ofAddHom (decompose : M →+ ⨁ i, ℳ i)
    (h_left_inv : (DirectSum.coeAddMonoidHom ℳ).comp decompose = .id _)
    (h_right_inv : decompose.comp (DirectSum.coeAddMonoidHom ℳ) = .id _) : Decomposition ℳ where
  decompose' := decompose
  left_inv := DFunLike.congr_fun h_left_inv
  right_inv := DFunLike.congr_fun h_right_inv

/-- Noncomputably conjure a decomposition instance from a `DirectSum.IsInternal` proof. -/
@[implicit_reducible]
noncomputable def IsInternal.chooseDecomposition (h : IsInternal ℳ) :
    DirectSum.Decomposition ℳ where
  decompose' := (Equiv.ofBijective _ h).symm
  left_inv := (Equiv.ofBijective _ h).right_inv
  right_inv := (Equiv.ofBijective _ h).left_inv

variable [Decomposition ℳ]

protected theorem Decomposition.isInternal : DirectSum.IsInternal ℳ :=
  ⟨Decomposition.right_inv.injective, Decomposition.left_inv.surjective⟩

/-- If `M` is graded by `ι` with degree `i` component `ℳ i`, then it is isomorphic as
to a direct sum of components. This is the canonical spelling of the `decompose'` field. -/
def decompose : M ≃ ⨁ i, ℳ i where
  toFun := Decomposition.decompose'
  invFun := DirectSum.coeAddMonoidHom ℳ
  left_inv := Decomposition.left_inv
  right_inv := Decomposition.right_inv

omit [AddSubmonoidClass σ M] in
/-- A substructure `p ⊆ M` is homogeneous if for every `m ∈ p`, all homogeneous components
  of `m` are in `p`. -/
def SetLike.IsHomogeneous {P : Type*} [SetLike P M] (p : P) : Prop :=
  ∀ (i : ι) ⦃m : M⦄, m ∈ p → (DirectSum.decompose ℳ m i : M) ∈ p

@[elab_as_elim]
protected theorem Decomposition.inductionOn {motive : M → Prop} (zero : motive 0)
    (homogeneous : ∀ {i} (m : ℳ i), motive (m : M))
    (add : ∀ m m' : M, motive m → motive m' → motive (m + m')) : ∀ m, motive m := by
  let ℳ' : ι → AddSubmonoid M := fun i ↦
    (⟨⟨ℳ i, fun x y ↦ AddMemClass.add_mem x y⟩, (ZeroMemClass.zero_mem _)⟩ : AddSubmonoid M)
  haveI t : DirectSum.Decomposition ℳ' :=
    { decompose' := DirectSum.decompose ℳ
      left_inv := fun _ ↦ (decompose ℳ).left_inv _
      right_inv := fun _ ↦ (decompose ℳ).right_inv _ }
  have mem : ∀ m, m ∈ iSup ℳ' := fun _m ↦
    (DirectSum.IsInternal.addSubmonoid_iSup_eq_top ℳ' (Decomposition.isInternal ℳ')).symm ▸ trivial
  -- Porting note: needs to use @ even though no implicit argument is provided
  exact fun m ↦ @AddSubmonoid.iSup_induction _ _ _ ℳ' _ _ (mem m)
    (fun i m h ↦ homogeneous ⟨m, h⟩) zero add
--  exact fun m ↦
--    AddSubmonoid.iSup_induction ℳ' (mem m) (fun i m h ↦ h_homogeneous ⟨m, h⟩) h_zero h_add

@[simp]
theorem Decomposition.decompose'_eq : Decomposition.decompose' = decompose ℳ := rfl

@[simp]
theorem decompose_symm_of {i : ι} (x : ℳ i) : (decompose ℳ).symm (DirectSum.of _ i x) = x :=
  DirectSum.coeAddMonoidHom_of ℳ _ _

@[simp]

Target:
theorem decompose_coe {i : ι} (x : ℳ i) : decompose ℳ (x : M) = DirectSum.of _ i x :=

Failed candidate:
by
  let __repair_517d504871 : Fact False := inferInstance
  rw [← decompose_symm_of _, Equiv.apply_symm_apply]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:134:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [← decompose_symm_of _, Equiv.apply_symm_apply]

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Algebra.DirectSum.Module
public import Mathlib.Algebra.Module.Submodule.Basic

Namespace:
DirectSum

Local safe context:
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
/-!
# Decompositions of additive monoids, groups, and modules into direct sums

## Main definitions

* `DirectSum.Decomposition ℳ`: A typeclass to provide a constructive decomposition from
  an additive monoid `M` into a family of additive submonoids `ℳ`
* `DirectSum.decompose ℳ`: The canonical equivalence provided by the above typeclass

## Main statements

* `DirectSum.Decomposition.isInternal`: The link to `DirectSum.IsInternal`.

## Implementation details

As we want to talk about different types of decomposition (additive monoids, modules, rings, ...),
we choose to avoid heavily bundling `DirectSum.decompose`, instead making copies for the
`AddEquiv`, `LinearEquiv`, etc. This means we have to repeat statements that follow from these
bundled homs, but means we don't have to repeat statements for different types of decomposition.
-/

@[expose] public section

variable {ι R M σ : Type*}

open DirectSum

namespace DirectSum

section AddCommMonoid

variable [DecidableEq ι] [AddCommMonoid M]
variable [SetLike σ M] [AddSubmonoidClass σ M] (ℳ : ι → σ)

/-- A decomposition is an equivalence between an additive monoid `M` and a direct sum of additive
submonoids `ℳ i` of that `M`, such that the "recomposition" is canonical. This definition also
works for additive groups and modules.

This is a version of `DirectSum.IsInternal` which comes with a constructive inverse to the
canonical "recomposition" rather than just a proof that the "recomposition" is bijective.

Often it is easier to construct a term of this type via `Decomposition.ofAddHom` or
`Decomposition.ofLinearMap`. -/
class Decomposition where
  decompose' : M → ⨁ i, ℳ i
  left_inv : Function.LeftInverse (DirectSum.coeAddMonoidHom ℳ) decompose'
  right_inv : Function.RightInverse (DirectSum.coeAddMonoidHom ℳ) decompose'

/-- `DirectSum.Decomposition` instances, while carrying data, are always equal. -/
instance : Subsingleton (Decomposition ℳ) :=
  ⟨fun x y ↦ by
    obtain ⟨_, _, xr⟩ := x
    obtain ⟨_, yl, _⟩ := y
    congr
    exact Function.LeftInverse.eq_rightInverse xr yl⟩

/-- A convenience method to construct a decomposition from an `AddMonoidHom`, such that the proofs
of left and right inverse can be constructed via `ext`. -/
abbrev Decomposition.ofAddHom (decompose : M →+ ⨁ i, ℳ i)
    (h_left_inv : (DirectSum.coeAddMonoidHom ℳ).comp decompose = .id _)
    (h_right_inv : decompose.comp (DirectSum.coeAddMonoidHom ℳ) = .id _) : Decomposition ℳ where
  decompose' := decompose
  left_inv := DFunLike.congr_fun h_left_inv
  right_inv := DFunLike.congr_fun h_right_inv

/-- Noncomputably conjure a decomposition instance from a `DirectSum.IsInternal` proof. -/
@[implicit_reducible]
noncomputable def IsInternal.chooseDecomposition (h : IsInternal ℳ) :
    DirectSum.Decomposition ℳ where
  decompose' := (Equiv.ofBijective _ h).symm
  left_inv := (Equiv.ofBijective _ h).right_inv
  right_inv := (Equiv.ofBijective _ h).left_inv

variable [Decomposition ℳ]

protected theorem Decomposition.isInternal : DirectSum.IsInternal ℳ :=
  ⟨Decomposition.right_inv.injective, Decomposition.left_inv.surjective⟩

/-- If `M` is graded by `ι` with degree `i` component `ℳ i`, then it is isomorphic as
to a direct sum of components. This is the canonical spelling of the `decompose'` field. -/
def decompose : M ≃ ⨁ i, ℳ i where
  toFun := Decomposition.decompose'
  invFun := DirectSum.coeAddMonoidHom ℳ
  left_inv := Decomposition.left_inv
  right_inv := Decomposition.right_inv

omit [AddSubmonoidClass σ M] in
/-- A substructure `p ⊆ M` is homogeneous if for every `m ∈ p`, all homogeneous components
  of `m` are in `p`. -/
def SetLike.IsHomogeneous {P : Type*} [SetLike P M] (p : P) : Prop :=
  ∀ (i : ι) ⦃m : M⦄, m ∈ p → (DirectSum.decompose ℳ m i : M) ∈ p

@[elab_as_elim]
protected theorem Decomposition.inductionOn {motive : M → Prop} (zero : motive 0)
    (homogeneous : ∀ {i} (m : ℳ i), motive (m : M))
    (add : ∀ m m' : M, motive m → motive m' → motive (m + m')) : ∀ m, motive m := by
  let ℳ' : ι → AddSubmonoid M := fun i ↦
    (⟨⟨ℳ i, fun x y ↦ AddMemClass.add_mem x y⟩, (ZeroMemClass.zero_mem _)⟩ : AddSubmonoid M)
  haveI t : DirectSum.Decomposition ℳ' :=
    { decompose' := DirectSum.decompose ℳ
      left_inv := fun _ ↦ (decompose ℳ).left_inv _
      right_inv := fun _ ↦ (decompose ℳ).right_inv _ }
  have mem : ∀ m, m ∈ iSup ℳ' := fun _m ↦
    (DirectSum.IsInternal.addSubmonoid_iSup_eq_top ℳ' (Decomposition.isInternal ℳ')).symm ▸ trivial
  -- Porting note: needs to use @ even though no implicit argument is provided
  exact fun m ↦ @AddSubmonoid.iSup_induction _ _ _ ℳ' _ _ (mem m)
    (fun i m h ↦ homogeneous ⟨m, h⟩) zero add
--  exact fun m ↦
--    AddSubmonoid.iSup_induction ℳ' (mem m) (fun i m h ↦ h_homogeneous ⟨m, h⟩) h_zero h_add

@[simp]
theorem Decomposition.decompose'_eq : Decomposition.decompose' = decompose ℳ := rfl

@[simp]
theorem decompose_symm_of {i : ι} (x : ℳ i) : (decompose ℳ).symm (DirectSum.of _ i x) = x :=
  DirectSum.coeAddMonoidHom_of ℳ _ _

@[simp]

Target:
theorem decompose_coe {i : ι} (x : ℳ i) : decompose ℳ (x : M) = DirectSum.of _ i x :=

Failed candidate:
by
  let __repair_517d504871 : Fact False := inferInstance
  rw [← decompose_symm_of _, Equiv.apply_symm_apply]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:134:42: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n  Fact False\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.", "returncode": 1}

Corrected proof:


<LEAN_FILE>:109:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 6455266) 0
in the target expression
  IntermediateField.fixedField ⊤ = ⊥

k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
⊢ IntermediateField.fixedField ⊤ = ⊥

Response:
by
  rw [← IntermediateField.fixingSubgroup_bot, fixedField_fixingSubgroup]

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.FieldTheory.KrullTopology
public import Mathlib.FieldTheory.Galois.GaloisClosure
public import Mathlib.Topology.Algebra.Group.ClosedSubgroup

Namespace:
InfiniteGalois

Local safe context:
/-
Copyright (c) 2024 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
/-!

# The Fundamental Theorem of Infinite Galois Theory

In this file, we prove the fundamental theorem of infinite Galois theory and the special case for
open subgroups and normal subgroups. We first verify that `IntermediateField.fixingSubgroup` and
`IntermediateField.fixedField` are inverses of each other between intermediate fields and
closed subgroups of the Galois group.

## Main definitions and results

In `K/k`, for any intermediate field `L` :

* `fixingSubgroup_isClosed` : the subgroup fixing `L` (`Gal(K/L)`) is closed.

* `fixedField_fixingSubgroup` : the field fixed by the
  subgroup fixing `L` is equal to `L` itself.

For any subgroup `H` of `Gal(K/k)` :

* `restrict_fixedField` : For a Galois intermediate field `M`, the fixed field of the image of `H`
  restricted to `M` is equal to the fixed field of `H` intersected with `M`.
* `fixingSubgroup_fixedField` : If `H` is closed, the fixing subgroup of the fixed field of `H`
  is equal to `H` itself.

The fundamental theorem of infinite Galois theory :

* `IntermediateFieldEquivClosedSubgroup` : The order equivalence is given by mapping any
  intermediate field `L` to the subgroup fixing `L`, and the inverse maps any
  closed subgroup of `Gal(K/k)` `H` to the fixed field of `H`. The composition is equal to
  the identity as described in the lemmas above, and compatibility with the order follows easily.

Special cases :

* `isOpen_iff_finite` : The fixing subgroup of an intermediate field `L` is open if and only if
  `L` is finite-dimensional.

* `normal_iff_isGalois` : The fixing subgroup of an intermediate field `L` is normal if and only if
  `L` is Galois.

-/

@[expose] public section

variable {k K : Type*} [Field k] [Field K] [Algebra k K]

namespace InfiniteGalois

open scoped Pointwise
open FiniteGaloisIntermediateField AlgEquiv
--Note: The `adjoin`s below are `FiniteGaloisIntermediateField.adjoin`

lemma fixingSubgroup_isClosed (L : IntermediateField k K) [IsGalois k K] :
    IsClosed (L.fixingSubgroup : Set Gal(K/k)) where
  isOpen_compl := isOpen_iff_mem_nhds.mpr fun σ h => by
    apply mem_nhds_iff.mpr
    rcases Set.not_subset.mp ((mem_fixingSubgroup_iff Gal(K/k)).not.mp h) with ⟨y, yL, ne⟩
    use σ • ((adjoin k {y}).1.fixingSubgroup : Set Gal(K/k))
    constructor
    · intro f hf
      rcases (Set.mem_smul_set.mp hf) with ⟨g, hg, eq⟩
      simp only [Set.mem_compl_iff, SetLike.mem_coe, ← eq]
      apply (mem_fixingSubgroup_iff Gal(K/k)).not.mpr
      push Not
      use y
      simp only [yL, smul_eq_mul, AlgEquiv.smul_def, AlgEquiv.mul_apply, ne_eq, true_and]
      have : g y = y := (mem_fixingSubgroup_iff Gal(K/k)).mp hg y <|
        adjoin_simple_le_iff.mp le_rfl
      simpa only [this, ne_eq, AlgEquiv.smul_def] using! ne
    · simp only [(IntermediateField.fixingSubgroup_isOpen (adjoin k {y}).1).smul σ, true_and]
      use 1
      simp only [SetLike.mem_coe, smul_eq_mul, mul_one, and_true, Subgroup.one_mem]

lemma fixedField_fixingSubgroup (L : IntermediateField k K) [IsGalois k K] :
    IntermediateField.fixedField L.fixingSubgroup = L := by
  apply le_antisymm
  · intro x hx
    rw [IntermediateField.mem_fixedField_iff] at hx
    have mem : x ∈ (adjoin L {x}).1 := subset_adjoin _ _ rfl
    have : IntermediateField.fixedField (⊤ : Subgroup ((adjoin L {x}) ≃ₐ[L] (adjoin L {x}))) = ⊥ :=
      (IsGalois.tfae.out 0 1).mp (by infer_instance)
    have : ⟨x, mem⟩ ∈ (⊥ : IntermediateField L (adjoin L {x})) := by
      rw [← this, IntermediateField.mem_fixedField_iff]
      intro f _
      rcases restrictNormalHom_surjective K f with ⟨σ, hσ⟩
      apply Subtype.val_injective
      rw [← hσ, restrictNormalHom_apply (adjoin L {x}).1 σ ⟨x, mem⟩]
      have := hx ((IntermediateField.fixingSubgroupEquiv L).symm σ)
      simpa only [SetLike.coe_mem, true_implies]
    rcases IntermediateField.mem_bot.mp this with ⟨y, hy⟩
    obtain ⟨rfl⟩ : y = x := congrArg Subtype.val hy
    exact y.2
  · exact (IntermediateField.le_iff_le L.fixingSubgroup L).mpr le_rfl

Target:
lemma fixedField_bot [IsGalois k K] :
    IntermediateField.fixedField (⊤ : Subgroup Gal(K/k)) = ⊥ :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 6455266) 0) = ((fun x : Nat => x + 6455266) 0) by rfl]
  rw [← IntermediateField.fixingSubgroup_bot, fixedField_fixingSubgroup]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:109:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 6455266) 0\nin the target expression\n  IntermediateField.fixedField ⊤ = ⊥\n\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\n⊢ IntermediateField.fixedField ⊤ = ⊥", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [← IntermediateField.fixingSubgroup_bot, fixedField_fixingSubgroup]

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.FieldTheory.KrullTopology
public import Mathlib.FieldTheory.Galois.GaloisClosure
public import Mathlib.Topology.Algebra.Group.ClosedSubgroup

Namespace:
InfiniteGalois

Local safe context:
/-
Copyright (c) 2024 Nailin Guan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nailin Guan
-/
/-!

# The Fundamental Theorem of Infinite Galois Theory

In this file, we prove the fundamental theorem of infinite Galois theory and the special case for
open subgroups and normal subgroups. We first verify that `IntermediateField.fixingSubgroup` and
`IntermediateField.fixedField` are inverses of each other between intermediate fields and
closed subgroups of the Galois group.

## Main definitions and results

In `K/k`, for any intermediate field `L` :

* `fixingSubgroup_isClosed` : the subgroup fixing `L` (`Gal(K/L)`) is closed.

* `fixedField_fixingSubgroup` : the field fixed by the
  subgroup fixing `L` is equal to `L` itself.

For any subgroup `H` of `Gal(K/k)` :

* `restrict_fixedField` : For a Galois intermediate field `M`, the fixed field of the image of `H`
  restricted to `M` is equal to the fixed field of `H` intersected with `M`.
* `fixingSubgroup_fixedField` : If `H` is closed, the fixing subgroup of the fixed field of `H`
  is equal to `H` itself.

The fundamental theorem of infinite Galois theory :

* `IntermediateFieldEquivClosedSubgroup` : The order equivalence is given by mapping any
  intermediate field `L` to the subgroup fixing `L`, and the inverse maps any
  closed subgroup of `Gal(K/k)` `H` to the fixed field of `H`. The composition is equal to
  the identity as described in the lemmas above, and compatibility with the order follows easily.

Special cases :

* `isOpen_iff_finite` : The fixing subgroup of an intermediate field `L` is open if and only if
  `L` is finite-dimensional.

* `normal_iff_isGalois` : The fixing subgroup of an intermediate field `L` is normal if and only if
  `L` is Galois.

-/

@[expose] public section

variable {k K : Type*} [Field k] [Field K] [Algebra k K]

namespace InfiniteGalois

open scoped Pointwise
open FiniteGaloisIntermediateField AlgEquiv
--Note: The `adjoin`s below are `FiniteGaloisIntermediateField.adjoin`

lemma fixingSubgroup_isClosed (L : IntermediateField k K) [IsGalois k K] :
    IsClosed (L.fixingSubgroup : Set Gal(K/k)) where
  isOpen_compl := isOpen_iff_mem_nhds.mpr fun σ h => by
    apply mem_nhds_iff.mpr
    rcases Set.not_subset.mp ((mem_fixingSubgroup_iff Gal(K/k)).not.mp h) with ⟨y, yL, ne⟩
    use σ • ((adjoin k {y}).1.fixingSubgroup : Set Gal(K/k))
    constructor
    · intro f hf
      rcases (Set.mem_smul_set.mp hf) with ⟨g, hg, eq⟩
      simp only [Set.mem_compl_iff, SetLike.mem_coe, ← eq]
      apply (mem_fixingSubgroup_iff Gal(K/k)).not.mpr
      push Not
      use y
      simp only [yL, smul_eq_mul, AlgEquiv.smul_def, AlgEquiv.mul_apply, ne_eq, true_and]
      have : g y = y := (mem_fixingSubgroup_iff Gal(K/k)).mp hg y <|
        adjoin_simple_le_iff.mp le_rfl
      simpa only [this, ne_eq, AlgEquiv.smul_def] using! ne
    · simp only [(IntermediateField.fixingSubgroup_isOpen (adjoin k {y}).1).smul σ, true_and]
      use 1
      simp only [SetLike.mem_coe, smul_eq_mul, mul_one, and_true, Subgroup.one_mem]

lemma fixedField_fixingSubgroup (L : IntermediateField k K) [IsGalois k K] :
    IntermediateField.fixedField L.fixingSubgroup = L := by
  apply le_antisymm
  · intro x hx
    rw [IntermediateField.mem_fixedField_iff] at hx
    have mem : x ∈ (adjoin L {x}).1 := subset_adjoin _ _ rfl
    have : IntermediateField.fixedField (⊤ : Subgroup ((adjoin L {x}) ≃ₐ[L] (adjoin L {x}))) = ⊥ :=
      (IsGalois.tfae.out 0 1).mp (by infer_instance)
    have : ⟨x, mem⟩ ∈ (⊥ : IntermediateField L (adjoin L {x})) := by
      rw [← this, IntermediateField.mem_fixedField_iff]
      intro f _
      rcases restrictNormalHom_surjective K f with ⟨σ, hσ⟩
      apply Subtype.val_injective
      rw [← hσ, restrictNormalHom_apply (adjoin L {x}).1 σ ⟨x, mem⟩]
      have := hx ((IntermediateField.fixingSubgroupEquiv L).symm σ)
      simpa only [SetLike.coe_mem, true_implies]
    rcases IntermediateField.mem_bot.mp this with ⟨y, hy⟩
    obtain ⟨rfl⟩ : y = x := congrArg Subtype.val hy
    exact y.2
  · exact (IntermediateField.le_iff_le L.fixingSubgroup L).mpr le_rfl

Target:
lemma fixedField_bot [IsGalois k K] :
    IntermediateField.fixedField (⊤ : Subgroup Gal(K/k)) = ⊥ :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 6455266) 0) = ((fun x : Nat => x + 6455266) 0) by rfl]
  rw [← IntermediateField.fixingSubgroup_bot, fixedField_fixingSubgroup]

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:109:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 6455266) 0\nin the target expression\n  IntermediateField.fixedField ⊤ = ⊥\n\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\n⊢ IntermediateField.fixedField ⊤ = ⊥", "returncode": 1}

Corrected proof:


<LEAN_FILE>:104:2: warning: declaration uses `sorry`
<LEAN_FILE>:108:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingdb6a4e6103.value`
<LEAN_FILE>:107:0: error: unsolved goals
α : Type v
M : Type u
inst✝¹ : Monoid M
inst✝ : MulAction M α
m : M
a : α
⊢ period m a ∣ Monoid.exponent M

Response:
by
  rw [← pow_smul_eq_iff_period_dvd, Monoid.pow_exponent_eq_one, one_smul]

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Dynamics.PeriodicPts.Lemmas
public import Mathlib.GroupTheory.Exponent
public import Mathlib.GroupTheory.GroupAction.Basic

Namespace:
MulAction

Local safe context:
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
/-!
# Period of a group action

This module defines some helpful lemmas around [`MulAction.period`] and [`AddAction.period`].
The period of a point `a` by a group element `g` is the smallest `m` such that `g ^ m • a = a`
(resp. `(m • g) +ᵥ a = a`) for a given `g : G` and `a : α`.

If such an `m` does not exist,
then by convention `MulAction.period` and `AddAction.period` return 0.
-/

public section

namespace MulAction

universe u v
variable {α : Type v}
variable {G : Type u} [Group G] [MulAction G α]
variable {M : Type u} [Monoid M] [MulAction M α]

/-- If the action is periodic, then a lower bound for its period can be computed. -/
@[to_additive /-- If the action is periodic, then a lower bound for its period can be computed. -/]
theorem le_period {m : M} {a : α} {n : ℕ} (period_pos : 0 < period m a)
    (moved : ∀ k, 0 < k → k < n → m ^ k • a ≠ a) : n ≤ period m a :=
  le_of_not_gt fun period_lt_n =>
    moved _ period_pos period_lt_n <| pow_period_smul m a

/-- If for some `n`, `m ^ n • a = a`, then `period m a ≤ n`. -/
@[to_additive /-- If for some `n`, `(n • m) +ᵥ a = a`, then `period m a ≤ n`. -/]
theorem period_le_of_fixed {m : M} {a : α} {n : ℕ} (n_pos : 0 < n) (fixed : m ^ n • a = a) :
    period m a ≤ n :=
  (isPeriodicPt_smul_iff.mpr fixed).minimalPeriod_le n_pos

/-- If for some `n`, `m ^ n • a = a`, then `0 < period m a`. -/
@[to_additive /-- If for some `n`, `(n • m) +ᵥ a = a`, then `0 < period m a`. -/]
theorem period_pos_of_fixed {m : M} {a : α} {n : ℕ} (n_pos : 0 < n) (fixed : m ^ n • a = a) :
    0 < period m a :=
  (isPeriodicPt_smul_iff.mpr fixed).minimalPeriod_pos n_pos

@[to_additive]
theorem period_eq_one_iff {m : M} {a : α} : period m a = 1 ↔ m • a = a :=
  ⟨fun eq_one => pow_one m ▸ eq_one ▸ pow_period_smul m a,
   fun fixed => le_antisymm
    (period_le_of_fixed one_pos (by simpa))
    (period_pos_of_fixed one_pos (by simpa))⟩

/-- For any non-zero `n` less than the period of `m` on `a`, `a` is moved by `m ^ n`. -/
@[to_additive
/-- For any non-zero `n` less than the period of `m` on `a`, `a` is moved by `n • m`. -/]
theorem pow_smul_ne_of_lt_period {m : M} {a : α} {n : ℕ} (n_pos : 0 < n)
    (n_lt_period : n < period m a) : m ^ n • a ≠ a := fun a_fixed =>
  not_le_of_gt n_lt_period <| period_le_of_fixed n_pos a_fixed

section Identities

/-! ### `MulAction.period` for common group elements
-/

variable (M) in
@[to_additive (attr := simp)]
theorem period_one (a : α) : period (1 : M) a = 1 := period_eq_one_iff.mpr (one_smul M a)

@[to_additive (attr := simp)]
theorem period_inv (g : G) (a : α) : period g⁻¹ a = period g a := by
  simp only [period_eq_minimalPeriod, Function.minimalPeriod_eq_minimalPeriod_iff,
    isPeriodicPt_smul_iff]
  intro n
  rw [smul_eq_iff_eq_inv_smul, eq_comm, ← zpow_natCast, inv_zpow, inv_inv, zpow_natCast]

end Identities

section MonoidExponent

/-! ### `MulAction.period` and group exponents

The period of a given element `m : M` can be bounded by the `Monoid.exponent M` or `orderOf m`.
-/

@[to_additive]
theorem period_dvd_orderOf (m : M) (a : α) : period m a ∣ orderOf m := by
  rw [← pow_smul_eq_iff_period_dvd, pow_orderOf_eq_one, one_smul]

@[to_additive]
theorem period_pos_of_orderOf_pos {m : M} (order_pos : 0 < orderOf m) (a : α) :
    0 < period m a :=
  Nat.pos_of_dvd_of_pos (period_dvd_orderOf m a) order_pos

@[to_additive]
theorem period_le_orderOf {m : M} (order_pos : 0 < orderOf m) (a : α) :
    period m a ≤ orderOf m :=
  Nat.le_of_dvd order_pos (period_dvd_orderOf m a)

@[to_additive]

Target:
theorem period_dvd_exponent (m : M) (a : α) : period m a ∣ Monoid.exponent M :=

Failed candidate:
by
  have __repair_db6a4e6103 := OProverMissingdb6a4e6103.value
  rw [← pow_smul_eq_iff_period_dvd, Monoid.pow_exponent_eq_one, one_smul]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:104:2: warning: declaration uses `sorry`\n<LEAN_FILE>:108:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingdb6a4e6103.value`\n<LEAN_FILE>:107:0: error: unsolved goals\nα : Type v\nM : Type u\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\nm : M\na : α\n⊢ period m a ∣ Monoid.exponent M", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [← pow_smul_eq_iff_period_dvd, Monoid.pow_exponent_eq_one, one_smul]

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Dynamics.PeriodicPts.Lemmas
public import Mathlib.GroupTheory.Exponent
public import Mathlib.GroupTheory.GroupAction.Basic

Namespace:
MulAction

Local safe context:
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
/-!
# Period of a group action

This module defines some helpful lemmas around [`MulAction.period`] and [`AddAction.period`].
The period of a point `a` by a group element `g` is the smallest `m` such that `g ^ m • a = a`
(resp. `(m • g) +ᵥ a = a`) for a given `g : G` and `a : α`.

If such an `m` does not exist,
then by convention `MulAction.period` and `AddAction.period` return 0.
-/

public section

namespace MulAction

universe u v
variable {α : Type v}
variable {G : Type u} [Group G] [MulAction G α]
variable {M : Type u} [Monoid M] [MulAction M α]

/-- If the action is periodic, then a lower bound for its period can be computed. -/
@[to_additive /-- If the action is periodic, then a lower bound for its period can be computed. -/]
theorem le_period {m : M} {a : α} {n : ℕ} (period_pos : 0 < period m a)
    (moved : ∀ k, 0 < k → k < n → m ^ k • a ≠ a) : n ≤ period m a :=
  le_of_not_gt fun period_lt_n =>
    moved _ period_pos period_lt_n <| pow_period_smul m a

/-- If for some `n`, `m ^ n • a = a`, then `period m a ≤ n`. -/
@[to_additive /-- If for some `n`, `(n • m) +ᵥ a = a`, then `period m a ≤ n`. -/]
theorem period_le_of_fixed {m : M} {a : α} {n : ℕ} (n_pos : 0 < n) (fixed : m ^ n • a = a) :
    period m a ≤ n :=
  (isPeriodicPt_smul_iff.mpr fixed).minimalPeriod_le n_pos

/-- If for some `n`, `m ^ n • a = a`, then `0 < period m a`. -/
@[to_additive /-- If for some `n`, `(n • m) +ᵥ a = a`, then `0 < period m a`. -/]
theorem period_pos_of_fixed {m : M} {a : α} {n : ℕ} (n_pos : 0 < n) (fixed : m ^ n • a = a) :
    0 < period m a :=
  (isPeriodicPt_smul_iff.mpr fixed).minimalPeriod_pos n_pos

@[to_additive]
theorem period_eq_one_iff {m : M} {a : α} : period m a = 1 ↔ m • a = a :=
  ⟨fun eq_one => pow_one m ▸ eq_one ▸ pow_period_smul m a,
   fun fixed => le_antisymm
    (period_le_of_fixed one_pos (by simpa))
    (period_pos_of_fixed one_pos (by simpa))⟩

/-- For any non-zero `n` less than the period of `m` on `a`, `a` is moved by `m ^ n`. -/
@[to_additive
/-- For any non-zero `n` less than the period of `m` on `a`, `a` is moved by `n • m`. -/]
theorem pow_smul_ne_of_lt_period {m : M} {a : α} {n : ℕ} (n_pos : 0 < n)
    (n_lt_period : n < period m a) : m ^ n • a ≠ a := fun a_fixed =>
  not_le_of_gt n_lt_period <| period_le_of_fixed n_pos a_fixed

section Identities

/-! ### `MulAction.period` for common group elements
-/

variable (M) in
@[to_additive (attr := simp)]
theorem period_one (a : α) : period (1 : M) a = 1 := period_eq_one_iff.mpr (one_smul M a)

@[to_additive (attr := simp)]
theorem period_inv (g : G) (a : α) : period g⁻¹ a = period g a := by
  simp only [period_eq_minimalPeriod, Function.minimalPeriod_eq_minimalPeriod_iff,
    isPeriodicPt_smul_iff]
  intro n
  rw [smul_eq_iff_eq_inv_smul, eq_comm, ← zpow_natCast, inv_zpow, inv_inv, zpow_natCast]

end Identities

section MonoidExponent

/-! ### `MulAction.period` and group exponents

The period of a given element `m : M` can be bounded by the `Monoid.exponent M` or `orderOf m`.
-/

@[to_additive]
theorem period_dvd_orderOf (m : M) (a : α) : period m a ∣ orderOf m := by
  rw [← pow_smul_eq_iff_period_dvd, pow_orderOf_eq_one, one_smul]

@[to_additive]
theorem period_pos_of_orderOf_pos {m : M} (order_pos : 0 < orderOf m) (a : α) :
    0 < period m a :=
  Nat.pos_of_dvd_of_pos (period_dvd_orderOf m a) order_pos

@[to_additive]
theorem period_le_orderOf {m : M} (order_pos : 0 < orderOf m) (a : α) :
    period m a ≤ orderOf m :=
  Nat.le_of_dvd order_pos (period_dvd_orderOf m a)

@[to_additive]

Target:
theorem period_dvd_exponent (m : M) (a : α) : period m a ∣ Monoid.exponent M :=

Failed candidate:
by
  have __repair_db6a4e6103 := OProverMissingdb6a4e6103.value
  rw [← pow_smul_eq_iff_period_dvd, Monoid.pow_exponent_eq_one, one_smul]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:104:2: warning: declaration uses `sorry`\n<LEAN_FILE>:108:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingdb6a4e6103.value`\n<LEAN_FILE>:107:0: error: unsolved goals\nα : Type v\nM : Type u\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\nm : M\na : α\n⊢ period m a ∣ Monoid.exponent M", "returncode": 1}

Corrected proof:


<LEAN_FILE>:238:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingd10574c978.value`
<LEAN_FILE>:237:0: error: unsolved goals
Γ : Type u_1
inst✝ : Inhabited Γ
l : ListBlank Γ
n : ℕ
⊢ l.nth (n + 1) = l.tail.nth n

Response:
by
  rw [← ListBlank.cons_head_tail l]
  induction l.tail using Quotient.inductionOn'
  rfl

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Logic.Function.Iterate
public import Mathlib.Tactic.ApplyFun
public import Mathlib.Data.List.GetD
public import Mathlib.Algebra.Group.Int.Defs
public import Mathlib.Algebra.Group.Nat.Defs
public import Mathlib.Data.List.Basic

Namespace:
Turing

Local safe context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
/-!
# Turing machine tapes

This file defines the notion of a Turing machine tape, and the operations on it. A tape is a
bidirectional infinite sequence of cells, each of which stores an element of a given alphabet `Γ`.
All but finitely many of the cells are required to hold the blank symbol `default : Γ`.

## Main definitions

* `ListBlank Γ` is the type of one-directional tapes with alphabet `Γ`. Implemented as a quotient
  of `List Γ` by extension by blanks at the end.
* `Tape Γ` is the type of Turing machine tapes with alphabet `Γ`. Implemented as two
  `ListBlank Γ` instances, one for each direction, as well as a head symbol.

-/

@[expose] public section

assert_not_exists MonoidWithZero

open Function (iterate_succ iterate_succ_apply iterate_zero_apply)

namespace Turing

section ListBlank

/-- The `BlankExtends` partial order holds of `l₁` and `l₂` if `l₂` is obtained by adding
blanks (`default : Γ`) to the end of `l₁`. -/
def BlankExtends {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) : Prop :=
  ∃ n, l₂ = l₁ ++ List.replicate n default

@[refl]
theorem BlankExtends.refl {Γ} [Inhabited Γ] (l : List Γ) : BlankExtends l l :=
  ⟨0, by simp⟩

@[trans]
theorem BlankExtends.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
    BlankExtends l₁ l₂ → BlankExtends l₂ l₃ → BlankExtends l₁ l₃ := by
  rintro ⟨i, rfl⟩ ⟨j, rfl⟩
  exact ⟨i + j, by simp⟩

theorem BlankExtends.below_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
    BlankExtends l l₁ → BlankExtends l l₂ → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by
  rintro ⟨i, rfl⟩ ⟨j, rfl⟩ h; use j - i
  simp only [List.length_append, Nat.add_le_add_iff_left, List.length_replicate] at h
  simp only [← List.replicate_add, Nat.add_sub_cancel' h, List.append_assoc]

/-- Any two extensions by blank `l₁,l₂` of `l` have a common join (which can be taken to be the
longer of `l₁` and `l₂`). -/
def BlankExtends.above {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} (h₁ : BlankExtends l l₁)
    (h₂ : BlankExtends l l₂) : { l' // BlankExtends l₁ l' ∧ BlankExtends l₂ l' } :=
  if h : l₁.length ≤ l₂.length then ⟨l₂, h₁.below_of_le h₂ h, BlankExtends.refl _⟩
  else ⟨l₁, BlankExtends.refl _, h₂.below_of_le h₁ (le_of_not_ge h)⟩

theorem BlankExtends.above_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
    BlankExtends l₁ l → BlankExtends l₂ l → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by
  rintro ⟨i, rfl⟩ ⟨j, e⟩ h; use i - j
  refine List.append_cancel_right (e.symm.trans ?_)
  rw [List.append_assoc, ← List.replicate_add, Nat.sub_add_cancel]
  apply_fun List.length at e
  simp only [List.length_append, List.length_replicate] at e
  rwa [← Nat.add_le_add_iff_left, e, Nat.add_le_add_iff_right]

/-- `BlankRel` is the symmetric closure of `BlankExtends`, turning it into an equivalence
relation. Two lists are related by `BlankRel` if one extends the other by blanks. -/
def BlankRel {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) : Prop :=
  BlankExtends l₁ l₂ ∨ BlankExtends l₂ l₁

@[refl]
theorem BlankRel.refl {Γ} [Inhabited Γ] (l : List Γ) : BlankRel l l :=
  Or.inl (BlankExtends.refl _)

@[symm]
theorem BlankRel.symm {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} : BlankRel l₁ l₂ → BlankRel l₂ l₁ :=
  Or.symm

@[trans]
theorem BlankRel.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
    BlankRel l₁ l₂ → BlankRel l₂ l₃ → BlankRel l₁ l₃ := by
  grind [eq_def, BlankExtends.below_of_le, BlankExtends.above_of_le, BlankExtends.trans]

/-- Given two `BlankRel` lists, there exists (constructively) a common join. -/
def BlankRel.above {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} (h : BlankRel l₁ l₂) :
    { l // BlankExtends l₁ l ∧ BlankExtends l₂ l } := by
  refine
    if hl : l₁.length ≤ l₂.length then ⟨l₂, Or.elim h id fun h' ↦ ?_, BlankExtends.refl _⟩
    else ⟨l₁, BlankExtends.refl _, Or.elim h (fun h' ↦ ?_) id⟩
  · exact (BlankExtends.refl _).above_of_le h' hl
  · exact (BlankExtends.refl _).above_of_le h' (le_of_not_ge hl)

/-- Given two `BlankRel` lists, there exists (constructively) a common meet. -/
def BlankRel.below {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} (h : BlankRel l₁ l₂) :
    { l // BlankExtends l l₁ ∧ BlankExtends l l₂ } := by
  refine
    if hl : l₁.length ≤ l₂.length then ⟨l₁, BlankExtends.refl _, Or.elim h id fun h' ↦ ?_⟩
    else ⟨l₂, Or.elim h (fun h' ↦ ?_) id, BlankExtends.refl _⟩
  · exact (BlankExtends.refl _).above_of_le h' hl
  · exact (BlankExtends.refl _).above_of_le h' (le_of_not_ge hl)

theorem BlankRel.equivalence (Γ) [Inhabited Γ] : Equivalence (@BlankRel Γ _) :=
  ⟨BlankRel.refl, @BlankRel.symm _ _, @BlankRel.trans _ _⟩

/-- Construct a setoid instance for `BlankRel`. -/
@[implicit_reducible]
def BlankRel.setoid (Γ) [Inhabited Γ] : Setoid (List Γ) :=
  ⟨_, BlankRel.equivalence _⟩

/-- A `ListBlank Γ` is a quotient of `List Γ` by extension by blanks at the end. This is used to
represent half-tapes of a Turing machine, so that we can pretend that the list continues
infinitely with blanks. -/
def ListBlank (Γ) [Inhabited Γ] :=
  Quotient (BlankRel.setoid Γ)

instance ListBlank.inhabited {Γ} [Inhabited Γ] : Inhabited (ListBlank Γ) :=
  ⟨Quotient.mk'' []⟩

instance ListBlank.hasEmptyc {Γ} [Inhabited Γ] : EmptyCollection (ListBlank Γ) :=
  ⟨Quotient.mk'' []⟩

/-- A modified version of `Quotient.liftOn'` specialized for `ListBlank`, with the stronger
precondition `BlankExtends` instead of `BlankRel`. -/
protected abbrev ListBlank.liftOn {Γ} [Inhabited Γ] {α} (l : ListBlank Γ) (f : List Γ → α)
    (H : ∀ a b, BlankExtends a b → f a = f b) : α :=
  l.liftOn' f <| by rintro a b (h | h) <;> [exact H _ _ h; exact (H _ _ h).symm]

/-- The quotient map turning a `List` into a `ListBlank`. -/
def ListBlank.mk {Γ} [Inhabited Γ] : List Γ → ListBlank Γ :=
  Quotient.mk''

@[elab_as_elim]
protected theorem ListBlank.induction_on {Γ} [Inhabited Γ] {p : ListBlank Γ → Prop}
    (q : ListBlank Γ) (h : ∀ a, p (ListBlank.mk a)) : p q :=
  Quotient.inductionOn' q h

/-- The head of a `ListBlank` is well defined. -/
def ListBlank.head {Γ} [Inhabited Γ] (l : ListBlank Γ) : Γ := by
  apply l.liftOn List.headI
  rintro a _ ⟨i, rfl⟩
  cases a
  · cases i <;> rfl
  rfl

@[simp]
theorem ListBlank.head_mk {Γ} [Inhabited Γ] (l : List Γ) :
    ListBlank.head (ListBlank.mk l) = l.headI :=
  rfl

/-- The tail of a `ListBlank` is well defined (up to the tail of blanks). -/
def ListBlank.tail {Γ} [Inhabited Γ] (l : ListBlank Γ) : ListBlank Γ := by
  apply l.liftOn (fun l ↦ ListBlank.mk l.tail)
  rintro a _ ⟨i, rfl⟩
  refine Quotient.sound' (Or.inl ?_)
  cases a
  · rcases i with - | i <;> [exact ⟨0, rfl⟩; exact ⟨i, rfl⟩]
  exact ⟨i, rfl⟩

@[simp]
theorem ListBlank.tail_mk {Γ} [Inhabited Γ] (l : List Γ) :
    ListBlank.tail (ListBlank.mk l) = ListBlank.mk l.tail :=
  rfl

/-- We can cons an element onto a `ListBlank`. -/
def ListBlank.cons {Γ} [Inhabited Γ] (a : Γ) (l : ListBlank Γ) : ListBlank Γ := by
  apply l.liftOn (fun l ↦ ListBlank.mk (List.cons a l))
  rintro _ _ ⟨i, rfl⟩
  exact Quotient.sound' (Or.inl ⟨i, rfl⟩)

@[simp]
theorem ListBlank.cons_mk {Γ} [Inhabited Γ] (a : Γ) (l : List Γ) :
    ListBlank.cons a (ListBlank.mk l) = ListBlank.mk (a :: l) :=
  rfl

@[simp]
theorem ListBlank.head_cons {Γ} [Inhabited Γ] (a : Γ) : ∀ l : ListBlank Γ, (l.cons a).head = a :=
  Quotient.ind' fun _ ↦ rfl

@[simp]
theorem ListBlank.tail_cons {Γ} [Inhabited Γ] (a : Γ) : ∀ l : ListBlank Γ, (l.cons a).tail = l :=
  Quotient.ind' fun _ ↦ rfl

/-- The `cons` and `head`/`tail` functions are mutually inverse, unlike in the case of `List` where
this only holds for nonempty lists. -/
@[simp]
theorem ListBlank.cons_head_tail {Γ} [Inhabited Γ] : ∀ l : ListBlank Γ, l.tail.cons l.head = l := by
  apply Quotient.ind'
  refine fun l ↦ Quotient.sound' (Or.inr ?_)
  cases l
  · exact ⟨1, rfl⟩
  · rfl

/-- The `cons` and `head`/`tail` functions are mutually inverse, unlike in the case of `List` where
this only holds for nonempty lists. -/
theorem ListBlank.exists_cons {Γ} [Inhabited Γ] (l : ListBlank Γ) :
    ∃ a l', l = ListBlank.cons a l' :=
  ⟨_, _, (ListBlank.cons_head_tail _).symm⟩

/-- The n-th element of a `ListBlank` is well defined for all `n : ℕ`, unlike in a `List`. -/
def ListBlank.nth {Γ} [Inhabited Γ] (l : ListBlank Γ) (n : ℕ) : Γ := by
  apply l.liftOn (fun l ↦ List.getI l n)
  rintro l _ ⟨i, rfl⟩
  rcases lt_or_ge n _ with h | h
  · rw [List.getI_append _ _ _ h]
  rw [List.getI_eq_default _ h]
  rcases le_or_gt _ n with h₂ | h₂
  · rw [List.getI_eq_default _ h₂]
  rw [List.getI_eq_getElem _ h₂, List.getElem_append_right h, List.getElem_replicate]

@[simp]
theorem ListBlank.nth_mk {Γ} [Inhabited Γ] (l : List Γ) (n : ℕ) :
    (ListBlank.mk l).nth n = l.getI n :=
  rfl

@[simp]
theorem ListBlank.nth_zero {Γ} [Inhabited Γ] (l : ListBlank Γ) : l.nth 0 = l.head := by
  rw [← ListBlank.cons_head_tail l]
  induction l.tail using Quotient.inductionOn'
  rfl

@[simp]

Target:
theorem ListBlank.nth_succ {Γ} [Inhabited Γ] (l : ListBlank Γ) (n : ℕ) :
    l.nth (n + 1) = l.tail.nth n :=

Failed candidate:
by
  have __repair_d10574c978 := OProverMissingd10574c978.value
  rw [← ListBlank.cons_head_tail l]
  induction l.tail using Quotient.inductionOn'
  rfl

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:238:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingd10574c978.value`\n<LEAN_FILE>:237:0: error: unsolved goals\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl : ListBlank Γ\nn : ℕ\n⊢ l.nth (n + 1) = l.tail.nth n", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [← ListBlank.cons_head_tail l]
  induction l.tail using Quotient.inductionOn'
  rfl

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Logic.Function.Iterate
public import Mathlib.Tactic.ApplyFun
public import Mathlib.Data.List.GetD
public import Mathlib.Algebra.Group.Int.Defs
public import Mathlib.Algebra.Group.Nat.Defs
public import Mathlib.Data.List.Basic

Namespace:
Turing

Local safe context:
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
/-!
# Turing machine tapes

This file defines the notion of a Turing machine tape, and the operations on it. A tape is a
bidirectional infinite sequence of cells, each of which stores an element of a given alphabet `Γ`.
All but finitely many of the cells are required to hold the blank symbol `default : Γ`.

## Main definitions

* `ListBlank Γ` is the type of one-directional tapes with alphabet `Γ`. Implemented as a quotient
  of `List Γ` by extension by blanks at the end.
* `Tape Γ` is the type of Turing machine tapes with alphabet `Γ`. Implemented as two
  `ListBlank Γ` instances, one for each direction, as well as a head symbol.

-/

@[expose] public section

assert_not_exists MonoidWithZero

open Function (iterate_succ iterate_succ_apply iterate_zero_apply)

namespace Turing

section ListBlank

/-- The `BlankExtends` partial order holds of `l₁` and `l₂` if `l₂` is obtained by adding
blanks (`default : Γ`) to the end of `l₁`. -/
def BlankExtends {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) : Prop :=
  ∃ n, l₂ = l₁ ++ List.replicate n default

@[refl]
theorem BlankExtends.refl {Γ} [Inhabited Γ] (l : List Γ) : BlankExtends l l :=
  ⟨0, by simp⟩

@[trans]
theorem BlankExtends.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
    BlankExtends l₁ l₂ → BlankExtends l₂ l₃ → BlankExtends l₁ l₃ := by
  rintro ⟨i, rfl⟩ ⟨j, rfl⟩
  exact ⟨i + j, by simp⟩

theorem BlankExtends.below_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
    BlankExtends l l₁ → BlankExtends l l₂ → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by
  rintro ⟨i, rfl⟩ ⟨j, rfl⟩ h; use j - i
  simp only [List.length_append, Nat.add_le_add_iff_left, List.length_replicate] at h
  simp only [← List.replicate_add, Nat.add_sub_cancel' h, List.append_assoc]

/-- Any two extensions by blank `l₁,l₂` of `l` have a common join (which can be taken to be the
longer of `l₁` and `l₂`). -/
def BlankExtends.above {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} (h₁ : BlankExtends l l₁)
    (h₂ : BlankExtends l l₂) : { l' // BlankExtends l₁ l' ∧ BlankExtends l₂ l' } :=
  if h : l₁.length ≤ l₂.length then ⟨l₂, h₁.below_of_le h₂ h, BlankExtends.refl _⟩
  else ⟨l₁, BlankExtends.refl _, h₂.below_of_le h₁ (le_of_not_ge h)⟩

theorem BlankExtends.above_of_le {Γ} [Inhabited Γ] {l l₁ l₂ : List Γ} :
    BlankExtends l₁ l → BlankExtends l₂ l → l₁.length ≤ l₂.length → BlankExtends l₁ l₂ := by
  rintro ⟨i, rfl⟩ ⟨j, e⟩ h; use i - j
  refine List.append_cancel_right (e.symm.trans ?_)
  rw [List.append_assoc, ← List.replicate_add, Nat.sub_add_cancel]
  apply_fun List.length at e
  simp only [List.length_append, List.length_replicate] at e
  rwa [← Nat.add_le_add_iff_left, e, Nat.add_le_add_iff_right]

/-- `BlankRel` is the symmetric closure of `BlankExtends`, turning it into an equivalence
relation. Two lists are related by `BlankRel` if one extends the other by blanks. -/
def BlankRel {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) : Prop :=
  BlankExtends l₁ l₂ ∨ BlankExtends l₂ l₁

@[refl]
theorem BlankRel.refl {Γ} [Inhabited Γ] (l : List Γ) : BlankRel l l :=
  Or.inl (BlankExtends.refl _)

@[symm]
theorem BlankRel.symm {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} : BlankRel l₁ l₂ → BlankRel l₂ l₁ :=
  Or.symm

@[trans]
theorem BlankRel.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
    BlankRel l₁ l₂ → BlankRel l₂ l₃ → BlankRel l₁ l₃ := by
  grind [eq_def, BlankExtends.below_of_le, BlankExtends.above_of_le, BlankExtends.trans]

/-- Given two `BlankRel` lists, there exists (constructively) a common join. -/
def BlankRel.above {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} (h : BlankRel l₁ l₂) :
    { l // BlankExtends l₁ l ∧ BlankExtends l₂ l } := by
  refine
    if hl : l₁.length ≤ l₂.length then ⟨l₂, Or.elim h id fun h' ↦ ?_, BlankExtends.refl _⟩
    else ⟨l₁, BlankExtends.refl _, Or.elim h (fun h' ↦ ?_) id⟩
  · exact (BlankExtends.refl _).above_of_le h' hl
  · exact (BlankExtends.refl _).above_of_le h' (le_of_not_ge hl)

/-- Given two `BlankRel` lists, there exists (constructively) a common meet. -/
def BlankRel.below {Γ} [Inhabited Γ] {l₁ l₂ : List Γ} (h : BlankRel l₁ l₂) :
    { l // BlankExtends l l₁ ∧ BlankExtends l l₂ } := by
  refine
    if hl : l₁.length ≤ l₂.length then ⟨l₁, BlankExtends.refl _, Or.elim h id fun h' ↦ ?_⟩
    else ⟨l₂, Or.elim h (fun h' ↦ ?_) id, BlankExtends.refl _⟩
  · exact (BlankExtends.refl _).above_of_le h' hl
  · exact (BlankExtends.refl _).above_of_le h' (le_of_not_ge hl)

theorem BlankRel.equivalence (Γ) [Inhabited Γ] : Equivalence (@BlankRel Γ _) :=
  ⟨BlankRel.refl, @BlankRel.symm _ _, @BlankRel.trans _ _⟩

/-- Construct a setoid instance for `BlankRel`. -/
@[implicit_reducible]
def BlankRel.setoid (Γ) [Inhabited Γ] : Setoid (List Γ) :=
  ⟨_, BlankRel.equivalence _⟩

/-- A `ListBlank Γ` is a quotient of `List Γ` by extension by blanks at the end. This is used to
represent half-tapes of a Turing machine, so that we can pretend that the list continues
infinitely with blanks. -/
def ListBlank (Γ) [Inhabited Γ] :=
  Quotient (BlankRel.setoid Γ)

instance ListBlank.inhabited {Γ} [Inhabited Γ] : Inhabited (ListBlank Γ) :=
  ⟨Quotient.mk'' []⟩

instance ListBlank.hasEmptyc {Γ} [Inhabited Γ] : EmptyCollection (ListBlank Γ) :=
  ⟨Quotient.mk'' []⟩

/-- A modified version of `Quotient.liftOn'` specialized for `ListBlank`, with the stronger
precondition `BlankExtends` instead of `BlankRel`. -/
protected abbrev ListBlank.liftOn {Γ} [Inhabited Γ] {α} (l : ListBlank Γ) (f : List Γ → α)
    (H : ∀ a b, BlankExtends a b → f a = f b) : α :=
  l.liftOn' f <| by rintro a b (h | h) <;> [exact H _ _ h; exact (H _ _ h).symm]

/-- The quotient map turning a `List` into a `ListBlank`. -/
def ListBlank.mk {Γ} [Inhabited Γ] : List Γ → ListBlank Γ :=
  Quotient.mk''

@[elab_as_elim]
protected theorem ListBlank.induction_on {Γ} [Inhabited Γ] {p : ListBlank Γ → Prop}
    (q : ListBlank Γ) (h : ∀ a, p (ListBlank.mk a)) : p q :=
  Quotient.inductionOn' q h

/-- The head of a `ListBlank` is well defined. -/
def ListBlank.head {Γ} [Inhabited Γ] (l : ListBlank Γ) : Γ := by
  apply l.liftOn List.headI
  rintro a _ ⟨i, rfl⟩
  cases a
  · cases i <;> rfl
  rfl

@[simp]
theorem ListBlank.head_mk {Γ} [Inhabited Γ] (l : List Γ) :
    ListBlank.head (ListBlank.mk l) = l.headI :=
  rfl

/-- The tail of a `ListBlank` is well defined (up to the tail of blanks). -/
def ListBlank.tail {Γ} [Inhabited Γ] (l : ListBlank Γ) : ListBlank Γ := by
  apply l.liftOn (fun l ↦ ListBlank.mk l.tail)
  rintro a _ ⟨i, rfl⟩
  refine Quotient.sound' (Or.inl ?_)
  cases a
  · rcases i with - | i <;> [exact ⟨0, rfl⟩; exact ⟨i, rfl⟩]
  exact ⟨i, rfl⟩

@[simp]
theorem ListBlank.tail_mk {Γ} [Inhabited Γ] (l : List Γ) :
    ListBlank.tail (ListBlank.mk l) = ListBlank.mk l.tail :=
  rfl

/-- We can cons an element onto a `ListBlank`. -/
def ListBlank.cons {Γ} [Inhabited Γ] (a : Γ) (l : ListBlank Γ) : ListBlank Γ := by
  apply l.liftOn (fun l ↦ ListBlank.mk (List.cons a l))
  rintro _ _ ⟨i, rfl⟩
  exact Quotient.sound' (Or.inl ⟨i, rfl⟩)

@[simp]
theorem ListBlank.cons_mk {Γ} [Inhabited Γ] (a : Γ) (l : List Γ) :
    ListBlank.cons a (ListBlank.mk l) = ListBlank.mk (a :: l) :=
  rfl

@[simp]
theorem ListBlank.head_cons {Γ} [Inhabited Γ] (a : Γ) : ∀ l : ListBlank Γ, (l.cons a).head = a :=
  Quotient.ind' fun _ ↦ rfl

@[simp]
theorem ListBlank.tail_cons {Γ} [Inhabited Γ] (a : Γ) : ∀ l : ListBlank Γ, (l.cons a).tail = l :=
  Quotient.ind' fun _ ↦ rfl

/-- The `cons` and `head`/`tail` functions are mutually inverse, unlike in the case of `List` where
this only holds for nonempty lists. -/
@[simp]
theorem ListBlank.cons_head_tail {Γ} [Inhabited Γ] : ∀ l : ListBlank Γ, l.tail.cons l.head = l := by
  apply Quotient.ind'
  refine fun l ↦ Quotient.sound' (Or.inr ?_)
  cases l
  · exact ⟨1, rfl⟩
  · rfl

/-- The `cons` and `head`/`tail` functions are mutually inverse, unlike in the case of `List` where
this only holds for nonempty lists. -/
theorem ListBlank.exists_cons {Γ} [Inhabited Γ] (l : ListBlank Γ) :
    ∃ a l', l = ListBlank.cons a l' :=
  ⟨_, _, (ListBlank.cons_head_tail _).symm⟩

/-- The n-th element of a `ListBlank` is well defined for all `n : ℕ`, unlike in a `List`. -/
def ListBlank.nth {Γ} [Inhabited Γ] (l : ListBlank Γ) (n : ℕ) : Γ := by
  apply l.liftOn (fun l ↦ List.getI l n)
  rintro l _ ⟨i, rfl⟩
  rcases lt_or_ge n _ with h | h
  · rw [List.getI_append _ _ _ h]
  rw [List.getI_eq_default _ h]
  rcases le_or_gt _ n with h₂ | h₂
  · rw [List.getI_eq_default _ h₂]
  rw [List.getI_eq_getElem _ h₂, List.getElem_append_right h, List.getElem_replicate]

@[simp]
theorem ListBlank.nth_mk {Γ} [Inhabited Γ] (l : List Γ) (n : ℕ) :
    (ListBlank.mk l).nth n = l.getI n :=
  rfl

@[simp]
theorem ListBlank.nth_zero {Γ} [Inhabited Γ] (l : ListBlank Γ) : l.nth 0 = l.head := by
  rw [← ListBlank.cons_head_tail l]
  induction l.tail using Quotient.inductionOn'
  rfl

@[simp]

Target:
theorem ListBlank.nth_succ {Γ} [Inhabited Γ] (l : ListBlank Γ) (n : ℕ) :
    l.nth (n + 1) = l.tail.nth n :=

Failed candidate:
by
  have __repair_d10574c978 := OProverMissingd10574c978.value
  rw [← ListBlank.cons_head_tail l]
  induction l.tail using Quotient.inductionOn'
  rfl

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:238:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissingd10574c978.value`\n<LEAN_FILE>:237:0: error: unsolved goals\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl : ListBlank Γ\nn : ℕ\n⊢ l.nth (n + 1) = l.tail.nth n", "returncode": 1}

Corrected proof:


<LEAN_FILE>:189:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 16065343) 0
in the target expression
  edist f g = (if edist f.fst g.fst = 0 then 0 else 1) + if edist f.snd g.snd = 0 then 0 else 1

α : Type u_2
β : Type u_3
inst✝¹ : EDist α
inst✝ : EDist β
f g : WithLp 0 (α × β)
⊢ edist f g = (if edist f.fst g.fst = 0 then 0 else 1) + if edist f.snd g.snd = 0 then 0 else 1

Response:
by
  convert! if_pos rfl

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Analysis.MeanInequalities
public import Mathlib.Analysis.Normed.Lp.WithLp

Namespace:
WithLp

Local safe context:
/-
Copyright (c) 2023 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Sébastien Gouëzel, Jireh Loreaux
-/
/-!
# `L^p` distance on products of two metric spaces
Given two metric spaces, one can put the max distance on their product, but there is also
a whole family of natural distances, indexed by a parameter `p : ℝ≥0∞`, that also induce
the product topology. We define them in this file. For `0 < p < ∞`, the distance on `α × β`
is given by
$$
d(x, y) = \left(d(x_1, y_1)^p + d(x_2, y_2)^p\right)^{1/p}.
$$
For `p = ∞` the distance is the supremum of the distances and `p = 0` the distance is the
cardinality of the elements that are not equal.

We give instances of this construction for emetric spaces, metric spaces, normed groups and normed
spaces.

To avoid conflicting instances, all these are defined on a copy of the original Prod-type, named
`WithLp p (α × β)`. The assumption `[Fact (1 ≤ p)]` is required for the metric and normed space
instances.

We ensure that the topology, bornology and uniform structure on `WithLp p (α × β)` are (defeq to)
the product topology, product bornology and product uniformity, to be able to use freely continuity
statements for the coordinate functions, for instance.

If you wish to endow a type synonym of `α × β` with the `L^p` distance, you can use
`pseudoMetricSpaceToProd` and the declarations below that one.

## Implementation notes

This file is a straight-forward adaptation of `Mathlib/Analysis/Normed/Lp/PiLp.lean`.

## TODO

TODO: the results about uniformity and bornology in the `Aux` section should be using the tools in
`Mathlib.Topology.MetricSpace.Bilipschitz`, so that they can be inlined in the next section and
the only remaining results are about `Lipschitz` and `Antilipschitz`.

-/

@[expose] public section

open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal

noncomputable section

variable (p : ℝ≥0∞) (𝕜 α β : Type*)

namespace WithLp

section algebra

/- Register simplification lemmas for the applications of `WithLp p (α × β)` elements, as the usual
lemmas for `Prod` will not trigger. -/

variable {p 𝕜 α β}
variable [Semiring 𝕜] [AddCommGroup α] [AddCommGroup β]
variable (x y : WithLp p (α × β)) (c : 𝕜)

/-- The projection on the first coordinate in `WithLp`. -/
protected def fst (x : WithLp p (α × β)) : α := (ofLp x).fst

/-- The projection on the second coordinate in `WithLp`. -/
protected def snd (x : WithLp p (α × β)) : β := (ofLp x).snd

@[simp]
theorem zero_fst : (0 : WithLp p (α × β)).fst = 0 :=
  rfl

@[simp]
theorem zero_snd : (0 : WithLp p (α × β)).snd = 0 :=
  rfl

@[simp]
theorem add_fst : (x + y).fst = x.fst + y.fst :=
  rfl

@[simp]
theorem add_snd : (x + y).snd = x.snd + y.snd :=
  rfl

@[simp]
theorem sub_fst : (x - y).fst = x.fst - y.fst :=
  rfl

@[simp]
theorem sub_snd : (x - y).snd = x.snd - y.snd :=
  rfl

@[simp]
theorem neg_fst : (-x).fst = -x.fst :=
  rfl

@[simp]
theorem neg_snd : (-x).snd = -x.snd :=
  rfl

variable [Module 𝕜 α] [Module 𝕜 β]

@[simp]
theorem smul_fst : (c • x).fst = c • x.fst :=
  rfl

@[simp]
theorem smul_snd : (c • x).snd = c • x.snd :=
  rfl

variable (p 𝕜 α β)

/-- `WithLp.fst` as a linear map. -/
@[simps]
def fstₗ : WithLp p (α × β) →ₗ[𝕜] α where
  toFun := WithLp.fst
  map_add' _ _ := rfl
  map_smul' _ _ := rfl

/-- `WithLp.snd` as a linear map. -/
@[simps]
def sndₗ : WithLp p (α × β) →ₗ[𝕜] β where
  toFun := WithLp.snd
  map_add' _ _ := rfl
  map_smul' _ _ := rfl

end algebra

/-! Note that the unapplied versions of these lemmas are deliberately omitted, as they break
the use of the type synonym. -/

section equiv

variable {p α β}

@[simp] lemma toLp_fst (x : α × β) : (toLp p x).fst = x.fst := rfl
@[simp] lemma toLp_snd (x : α × β) : (toLp p x).snd = x.snd := rfl
@[simp] lemma ofLp_fst (x : WithLp p (α × β)) : (ofLp x).fst = x.fst := rfl
@[simp] lemma ofLp_snd (x : WithLp p (α × β)) : (ofLp x).snd = x.snd := rfl

end equiv

section DistNorm

/-!
### Definition of `edist`, `dist` and `norm` on `WithLp p (α × β)`

In this section we define the `edist`, `dist` and `norm` functions on `WithLp p (α × β)` without
assuming `[Fact (1 ≤ p)]` or metric properties of the spaces `α` and `β`. This allows us to provide
the rewrite lemmas for each of three cases `p = 0`, `p = ∞` and `0 < p.toReal`.
-/

section EDist

variable [EDist α] [EDist β]

open scoped Classical in
/-- Endowing the space `WithLp p (α × β)` with the `L^p` edistance. We register this instance
separate from `WithLp.instProdPseudoEMetric` since the latter requires the type class hypothesis
`[Fact (1 ≤ p)]` in order to prove the triangle inequality.

Registering this separately allows for a future emetric-like structure on `WithLp p (α × β)` for
`p < 1` satisfying a relaxed triangle inequality. The terminology for this varies throughout the
literature, but it is sometimes called a *quasi-metric* or *semi-metric*. -/
instance instProdEDist : EDist (WithLp p (α × β)) where
  edist f g :=
    if _hp : p = 0 then
      (if edist f.fst g.fst = 0 then 0 else 1) + (if edist f.snd g.snd = 0 then 0 else 1)
    else if p = ∞ then
      edist f.fst g.fst ⊔ edist f.snd g.snd
    else
      (edist f.fst g.fst ^ p.toReal + edist f.snd g.snd ^ p.toReal) ^ (1 / p.toReal)

variable {p α β}

@[simp]

Target:
theorem prod_edist_eq_card (f g : WithLp 0 (α × β)) :
    edist f g =
      (if edist f.fst g.fst = 0 then 0 else 1) + (if edist f.snd g.snd = 0 then 0 else 1) :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 16065343) 0) = ((fun x : Nat => x + 16065343) 0) by rfl]
  convert! if_pos rfl

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:189:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 16065343) 0\nin the target expression\n  edist f g = (if edist f.fst g.fst = 0 then 0 else 1) + if edist f.snd g.snd = 0 then 0 else 1\n\nα : Type u_2\nβ : Type u_3\ninst✝¹ : EDist α\ninst✝ : EDist β\nf g : WithLp 0 (α × β)\n⊢ edist f g = (if edist f.fst g.fst = 0 then 0 else 1) + if edist f.snd g.snd = 0 then 0 else 1", "returncode": 1}

Corrected proof:

[Assistant]
by
  convert! if_pos rfl

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.Analysis.MeanInequalities
public import Mathlib.Analysis.Normed.Lp.WithLp

Namespace:
WithLp

Local safe context:
/-
Copyright (c) 2023 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Sébastien Gouëzel, Jireh Loreaux
-/
/-!
# `L^p` distance on products of two metric spaces
Given two metric spaces, one can put the max distance on their product, but there is also
a whole family of natural distances, indexed by a parameter `p : ℝ≥0∞`, that also induce
the product topology. We define them in this file. For `0 < p < ∞`, the distance on `α × β`
is given by
$$
d(x, y) = \left(d(x_1, y_1)^p + d(x_2, y_2)^p\right)^{1/p}.
$$
For `p = ∞` the distance is the supremum of the distances and `p = 0` the distance is the
cardinality of the elements that are not equal.

We give instances of this construction for emetric spaces, metric spaces, normed groups and normed
spaces.

To avoid conflicting instances, all these are defined on a copy of the original Prod-type, named
`WithLp p (α × β)`. The assumption `[Fact (1 ≤ p)]` is required for the metric and normed space
instances.

We ensure that the topology, bornology and uniform structure on `WithLp p (α × β)` are (defeq to)
the product topology, product bornology and product uniformity, to be able to use freely continuity
statements for the coordinate functions, for instance.

If you wish to endow a type synonym of `α × β` with the `L^p` distance, you can use
`pseudoMetricSpaceToProd` and the declarations below that one.

## Implementation notes

This file is a straight-forward adaptation of `Mathlib/Analysis/Normed/Lp/PiLp.lean`.

## TODO

TODO: the results about uniformity and bornology in the `Aux` section should be using the tools in
`Mathlib.Topology.MetricSpace.Bilipschitz`, so that they can be inlined in the next section and
the only remaining results are about `Lipschitz` and `Antilipschitz`.

-/

@[expose] public section

open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal

noncomputable section

variable (p : ℝ≥0∞) (𝕜 α β : Type*)

namespace WithLp

section algebra

/- Register simplification lemmas for the applications of `WithLp p (α × β)` elements, as the usual
lemmas for `Prod` will not trigger. -/

variable {p 𝕜 α β}
variable [Semiring 𝕜] [AddCommGroup α] [AddCommGroup β]
variable (x y : WithLp p (α × β)) (c : 𝕜)

/-- The projection on the first coordinate in `WithLp`. -/
protected def fst (x : WithLp p (α × β)) : α := (ofLp x).fst

/-- The projection on the second coordinate in `WithLp`. -/
protected def snd (x : WithLp p (α × β)) : β := (ofLp x).snd

@[simp]
theorem zero_fst : (0 : WithLp p (α × β)).fst = 0 :=
  rfl

@[simp]
theorem zero_snd : (0 : WithLp p (α × β)).snd = 0 :=
  rfl

@[simp]
theorem add_fst : (x + y).fst = x.fst + y.fst :=
  rfl

@[simp]
theorem add_snd : (x + y).snd = x.snd + y.snd :=
  rfl

@[simp]
theorem sub_fst : (x - y).fst = x.fst - y.fst :=
  rfl

@[simp]
theorem sub_snd : (x - y).snd = x.snd - y.snd :=
  rfl

@[simp]
theorem neg_fst : (-x).fst = -x.fst :=
  rfl

@[simp]
theorem neg_snd : (-x).snd = -x.snd :=
  rfl

variable [Module 𝕜 α] [Module 𝕜 β]

@[simp]
theorem smul_fst : (c • x).fst = c • x.fst :=
  rfl

@[simp]
theorem smul_snd : (c • x).snd = c • x.snd :=
  rfl

variable (p 𝕜 α β)

/-- `WithLp.fst` as a linear map. -/
@[simps]
def fstₗ : WithLp p (α × β) →ₗ[𝕜] α where
  toFun := WithLp.fst
  map_add' _ _ := rfl
  map_smul' _ _ := rfl

/-- `WithLp.snd` as a linear map. -/
@[simps]
def sndₗ : WithLp p (α × β) →ₗ[𝕜] β where
  toFun := WithLp.snd
  map_add' _ _ := rfl
  map_smul' _ _ := rfl

end algebra

/-! Note that the unapplied versions of these lemmas are deliberately omitted, as they break
the use of the type synonym. -/

section equiv

variable {p α β}

@[simp] lemma toLp_fst (x : α × β) : (toLp p x).fst = x.fst := rfl
@[simp] lemma toLp_snd (x : α × β) : (toLp p x).snd = x.snd := rfl
@[simp] lemma ofLp_fst (x : WithLp p (α × β)) : (ofLp x).fst = x.fst := rfl
@[simp] lemma ofLp_snd (x : WithLp p (α × β)) : (ofLp x).snd = x.snd := rfl

end equiv

section DistNorm

/-!
### Definition of `edist`, `dist` and `norm` on `WithLp p (α × β)`

In this section we define the `edist`, `dist` and `norm` functions on `WithLp p (α × β)` without
assuming `[Fact (1 ≤ p)]` or metric properties of the spaces `α` and `β`. This allows us to provide
the rewrite lemmas for each of three cases `p = 0`, `p = ∞` and `0 < p.toReal`.
-/

section EDist

variable [EDist α] [EDist β]

open scoped Classical in
/-- Endowing the space `WithLp p (α × β)` with the `L^p` edistance. We register this instance
separate from `WithLp.instProdPseudoEMetric` since the latter requires the type class hypothesis
`[Fact (1 ≤ p)]` in order to prove the triangle inequality.

Registering this separately allows for a future emetric-like structure on `WithLp p (α × β)` for
`p < 1` satisfying a relaxed triangle inequality. The terminology for this varies throughout the
literature, but it is sometimes called a *quasi-metric* or *semi-metric*. -/
instance instProdEDist : EDist (WithLp p (α × β)) where
  edist f g :=
    if _hp : p = 0 then
      (if edist f.fst g.fst = 0 then 0 else 1) + (if edist f.snd g.snd = 0 then 0 else 1)
    else if p = ∞ then
      edist f.fst g.fst ⊔ edist f.snd g.snd
    else
      (edist f.fst g.fst ^ p.toReal + edist f.snd g.snd ^ p.toReal) ^ (1 / p.toReal)

variable {p α β}

@[simp]

Target:
theorem prod_edist_eq_card (f g : WithLp 0 (α × β)) :
    edist f g =
      (if edist f.fst g.fst = 0 then 0 else 1) + (if edist f.snd g.snd = 0 then 0 else 1) :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 16065343) 0) = ((fun x : Nat => x + 16065343) 0) by rfl]
  convert! if_pos rfl

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:189:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 16065343) 0\nin the target expression\n  edist f g = (if edist f.fst g.fst = 0 then 0 else 1) + if edist f.snd g.snd = 0 then 0 else 1\n\nα : Type u_2\nβ : Type u_3\ninst✝¹ : EDist α\ninst✝ : EDist β\nf g : WithLp 0 (α × β)\n⊢ edist f g = (if edist f.fst g.fst = 0 then 0 else 1) + if edist f.snd g.snd = 0 then 0 else 1", "returncode": 1}

Corrected proof:


<LEAN_FILE>:35:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing75c449811d.value`
<LEAN_FILE>:34:0: error: unsolved goals
R : Type u_1
I : Type u_2
inst✝² : Semiring R
inst✝¹ : DecidableEq I
V : I → Ideal R
inst✝ : Decomposition V
x : R
i : I
⊢ ↑(((decompose V) x) i) = x * idempotent V i

Response:
by
  rw [← smul_eq_mul (a := x), idempotent, ← Submodule.coe_smul, ← smul_apply, ← decompose_smul,
    smul_eq_mul, mul_one]

[User]
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.RingTheory.Idempotents
public import Mathlib.Algebra.DirectSum.Decomposition

Namespace:
DirectSum

Local safe context:
/-
Copyright (c) 2025 Yunzhou Xie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yunzhou Xie, Jujian Zhang, Andrew Yang
-/
/-!
# Decomposition of the identity of a semiring into orthogonal idempotents

In this file we show that if a semiring `R` can be decomposed into a direct sum
of (left) ideals `R = V₁ ⊕ V₂ ⊕ ⋯ ⊕ Vₙ` then in the corresponding decomposition
`1 = e₁ + e₂ + ⋯ + eₙ` with `eᵢ ∈ Vᵢ`, each `eᵢ` is an idempotent and the
`eᵢ`'s form a family of complete orthogonal idempotents.
-/

@[expose] public section

namespace DirectSum

section OrthogonalIdempotents

variable {R I : Type*} [Semiring R] [DecidableEq I] (V : I → Ideal R) [Decomposition V]

/-- The decomposition of `(1 : R)` where `1 = e₁ + e₂ + ⋯ + eₙ` which is induced by
  the decomposition of the semiring `R = V1 ⊕ V2 ⊕ ⋯ ⊕ Vn`. -/
def idempotent (i : I) : R :=
  decompose V 1 i

Target:
lemma decompose_eq_mul_idempotent (x : R) (i : I) : decompose V x i = x * idempotent V i :=

Failed candidate:
by
  have __repair_75c449811d := OProverMissing75c449811d.value
  rw [← smul_eq_mul (a := x), idempotent, ← Submodule.coe_smul, ← smul_apply, ← decompose_smul,
    smul_eq_mul, mul_one]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:35:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing75c449811d.value`\n<LEAN_FILE>:34:0: error: unsolved goals\nR : Type u_1\nI : Type u_2\ninst✝² : Semiring R\ninst✝¹ : DecidableEq I\nV : I → Ideal R\ninst✝ : Decomposition V\nx : R\ni : I\n⊢ ↑(((decompose V) x) i) = x * idempotent V i", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [← smul_eq_mul (a := x), idempotent, ← Submodule.coe_smul, ← smul_apply, ← decompose_smul,
    smul_eq_mul, mul_one]

Prompt:
Repair the failed Lean proof using the exact public context below. Return only one corrected proof body beginning with `by`; no prose or fences.

Imports:
public import Mathlib.RingTheory.Idempotents
public import Mathlib.Algebra.DirectSum.Decomposition

Namespace:
DirectSum

Local safe context:
/-
Copyright (c) 2025 Yunzhou Xie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yunzhou Xie, Jujian Zhang, Andrew Yang
-/
/-!
# Decomposition of the identity of a semiring into orthogonal idempotents

In this file we show that if a semiring `R` can be decomposed into a direct sum
of (left) ideals `R = V₁ ⊕ V₂ ⊕ ⋯ ⊕ Vₙ` then in the corresponding decomposition
`1 = e₁ + e₂ + ⋯ + eₙ` with `eᵢ ∈ Vᵢ`, each `eᵢ` is an idempotent and the
`eᵢ`'s form a family of complete orthogonal idempotents.
-/

@[expose] public section

namespace DirectSum

section OrthogonalIdempotents

variable {R I : Type*} [Semiring R] [DecidableEq I] (V : I → Ideal R) [Decomposition V]

/-- The decomposition of `(1 : R)` where `1 = e₁ + e₂ + ⋯ + eₙ` which is induced by
  the decomposition of the semiring `R = V1 ⊕ V2 ⊕ ⋯ ⊕ Vn`. -/
def idempotent (i : I) : R :=
  decompose V 1 i

Target:
lemma decompose_eq_mul_idempotent (x : R) (i : I) : decompose V x i = x * idempotent V i :=

Failed candidate:
by
  have __repair_75c449811d := OProverMissing75c449811d.value
  rw [← smul_eq_mul (a := x), idempotent, ← Submodule.coe_smul, ← smul_apply, ← decompose_smul,
    smul_eq_mul, mul_one]

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:35:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing75c449811d.value`\n<LEAN_FILE>:34:0: error: unsolved goals\nR : Type u_1\nI : Type u_2\ninst✝² : Semiring R\ninst✝¹ : DecidableEq I\nV : I → Ideal R\ninst✝ : Decomposition V\nx : R\ni : I\n⊢ ↑(((decompose V) x) i) = x * idempotent V i", "returncode": 1}

Corrected proof: