<LEAN_FILE>:109:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_46b502979a`
<LEAN_FILE>:84:0: error: unsolved goals
N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
⊢ LSeriesSummable (⇑χ.zetaMul) s

Response:
by
  refine ArithmeticFunction.LSeriesSummable_mul (LSeriesSummable_zeta_iff.mpr hs) <|
    LSeriesSummable_of_bounded_of_one_lt_re (m := 1) (fun n hn ↦ ?_) hs
  simpa only [toArithmeticFunction, coe_mk, hn, ↓reduceIte]
  using norm_le_one χ _

[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.SpecialFunctions.Complex.LogBounds
public import Mathlib.NumberTheory.Harmonic.ZetaAsymp
public import Mathlib.NumberTheory.LSeries.Dirichlet
public import Mathlib.NumberTheory.LSeries.DirichletContinuation
public import Mathlib.NumberTheory.LSeries.Positivity

Namespace:
DirichletCharacter

Local safe context:
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, David Loeffler
-/
/-!
# The L-function of a Dirichlet character does not vanish on Re(s) ≥ 1

The main result in this file is `DirichletCharacter.LFunction_ne_zero_of_one_le_re`:
if `χ` is a Dirichlet character, `s ∈ ℂ` with `1 ≤ s.re`, and either `χ` is nontrivial or `s ≠ 1`,
then the L-function of `χ` does not vanish at `s`.

As a consequence, we have the corresponding statement for the Riemann ζ function:
`riemannZeta_ne_zero_of_one_le_re` (which does not require `s ≠ 1`, since the junk value at `s = 1`
happens to be non-zero).

These results are prerequisites for the **Prime Number Theorem** and
**Dirichlet's Theorem** on primes in arithmetic progressions.

## Outline of proofs

We split into two cases: first, the special case of (non-trivial) quadratic characters at `s = 1`;
then the remaining case when either `s ≠ 1` or `χ ^ 2 ≠ 1`.

The first case is handled using a positivity argument applied to the series `L χ s * ζ s`: we show
that this function has non-negative Dirichlet coefficients, is strictly positive for `s ≫ 0`, but
vanishes at `s = -2`, so it must have a pole somewhere in between.

The second case is dealt with using the product
`L(χ^0, 1 + x)^3 L(χ, 1 + x + I * y)^4 L(χ^2, 1 + x + 2 * I * y)`, which
we show has absolute value `≥ 1` for all positive `x` and real `y`; if `L(χ, 1 + I * y) = 0` then
this product would have to tend to 0 as `x → 0`, which is a contradiction.
-/

@[expose] public section

/- NB: Many lemmas (and some defs) in this file are private, since they concern properties of
hypothetical objects which we eventually deduce cannot exist. We have only made public the lemmas
whose hypotheses do not turn out to be contradictory.
-/

open Complex Asymptotics Topology Filter
open ArithmeticFunction hiding log

-- We use the ordering on `ℂ` given by comparing real parts for fixed imaginary part
open scoped ComplexOrder

variable {N : ℕ}

namespace DirichletCharacter

section quadratic

/-!
### Convolution of a Dirichlet character with ζ

We define `DirichletCharacter.zetaMul χ` to be the arithmetic function obtained by
taking the product (as arithmetic functions = Dirichlet convolution) of the
arithmetic function `ζ` with `χ`.

We then show that for a quadratic character `χ`, this arithmetic function is multiplicative
and takes nonnegative real values.
-/

/-- The complex-valued arithmetic function that is the convolution of the constant
function `1` with `χ`. -/
noncomputable def zetaMul (χ : DirichletCharacter ℂ N) : ArithmeticFunction ℂ :=
  .zeta * toArithmeticFunction (χ ·)

/-- The arithmetic function `zetaMul χ` is multiplicative. -/
lemma isMultiplicative_zetaMul (χ : DirichletCharacter ℂ N) : χ.zetaMul.IsMultiplicative :=
  isMultiplicative_zeta.natCast.mul <| isMultiplicative_toArithmeticFunction χ

Target:
lemma LSeriesSummable_zetaMul (χ : DirichletCharacter ℂ N) {s : ℂ} (hs : 1 < s.re) :
    LSeriesSummable χ.zetaMul s :=

Failed candidate:
by
  have __repair_pad_0_46b502979a : True := by trivial
  have __repair_pad_1_46b502979a : True := by trivial
  have __repair_pad_2_46b502979a : True := by trivial
  have __repair_pad_3_46b502979a : True := by trivial
  have __repair_pad_4_46b502979a : True := by trivial
  have __repair_pad_5_46b502979a : True := by trivial
  have __repair_pad_6_46b502979a : True := by trivial
  have __repair_pad_7_46b502979a : True := by trivial
  have __repair_pad_8_46b502979a : True := by trivial
  have __repair_pad_9_46b502979a : True := by trivial
  have __repair_pad_10_46b502979a : True := by trivial
  have __repair_pad_11_46b502979a : True := by trivial
  have __repair_pad_12_46b502979a : True := by trivial
  have __repair_pad_13_46b502979a : True := by trivial
  have __repair_pad_14_46b502979a : True := by trivial
  have __repair_pad_15_46b502979a : True := by trivial
  have __repair_pad_16_46b502979a : True := by trivial
  have __repair_pad_17_46b502979a : True := by trivial
  have __repair_pad_18_46b502979a : True := by trivial
  have __repair_pad_19_46b502979a : True := by trivial
  have __repair_pad_20_46b502979a : True := by trivial
  have __repair_pad_21_46b502979a : True := by trivial
  have __repair_pad_22_46b502979a : True := by trivial
  have __repair_pad_23_46b502979a : True := by trivial
  have __repair_fail_46b502979a := __oprover_overlong_46b502979a
  refine ArithmeticFunction.LSeriesSummable_mul (LSeriesSummable_zeta_iff.mpr hs) <|
    LSeriesSummable_of_bounded_of_one_lt_re (m := 1) (fun n hn ↦ ?_) hs
  simpa only [toArithmeticFunction, coe_mk, hn, ↓reduceIte]
  using norm_le_one χ _

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:109:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_46b502979a`\n<LEAN_FILE>:84:0: error: unsolved goals\nN : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\n⊢ LSeriesSummable (⇑χ.zetaMul) s", "returncode": 1}

Corrected proof:

[Assistant]
by
  refine ArithmeticFunction.LSeriesSummable_mul (LSeriesSummable_zeta_iff.mpr hs) <|
    LSeriesSummable_of_bounded_of_one_lt_re (m := 1) (fun n hn ↦ ?_) hs
  simpa only [toArithmeticFunction, coe_mk, hn, ↓reduceIte]
  using norm_le_one χ _

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.SpecialFunctions.Complex.LogBounds
public import Mathlib.NumberTheory.Harmonic.ZetaAsymp
public import Mathlib.NumberTheory.LSeries.Dirichlet
public import Mathlib.NumberTheory.LSeries.DirichletContinuation
public import Mathlib.NumberTheory.LSeries.Positivity

Namespace:
DirichletCharacter

Local safe context:
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, David Loeffler
-/
/-!
# The L-function of a Dirichlet character does not vanish on Re(s) ≥ 1

The main result in this file is `DirichletCharacter.LFunction_ne_zero_of_one_le_re`:
if `χ` is a Dirichlet character, `s ∈ ℂ` with `1 ≤ s.re`, and either `χ` is nontrivial or `s ≠ 1`,
then the L-function of `χ` does not vanish at `s`.

As a consequence, we have the corresponding statement for the Riemann ζ function:
`riemannZeta_ne_zero_of_one_le_re` (which does not require `s ≠ 1`, since the junk value at `s = 1`
happens to be non-zero).

These results are prerequisites for the **Prime Number Theorem** and
**Dirichlet's Theorem** on primes in arithmetic progressions.

## Outline of proofs

We split into two cases: first, the special case of (non-trivial) quadratic characters at `s = 1`;
then the remaining case when either `s ≠ 1` or `χ ^ 2 ≠ 1`.

The first case is handled using a positivity argument applied to the series `L χ s * ζ s`: we show
that this function has non-negative Dirichlet coefficients, is strictly positive for `s ≫ 0`, but
vanishes at `s = -2`, so it must have a pole somewhere in between.

The second case is dealt with using the product
`L(χ^0, 1 + x)^3 L(χ, 1 + x + I * y)^4 L(χ^2, 1 + x + 2 * I * y)`, which
we show has absolute value `≥ 1` for all positive `x` and real `y`; if `L(χ, 1 + I * y) = 0` then
this product would have to tend to 0 as `x → 0`, which is a contradiction.
-/

@[expose] public section

/- NB: Many lemmas (and some defs) in this file are private, since they concern properties of
hypothetical objects which we eventually deduce cannot exist. We have only made public the lemmas
whose hypotheses do not turn out to be contradictory.
-/

open Complex Asymptotics Topology Filter
open ArithmeticFunction hiding log

-- We use the ordering on `ℂ` given by comparing real parts for fixed imaginary part
open scoped ComplexOrder

variable {N : ℕ}

namespace DirichletCharacter

section quadratic

/-!
### Convolution of a Dirichlet character with ζ

We define `DirichletCharacter.zetaMul χ` to be the arithmetic function obtained by
taking the product (as arithmetic functions = Dirichlet convolution) of the
arithmetic function `ζ` with `χ`.

We then show that for a quadratic character `χ`, this arithmetic function is multiplicative
and takes nonnegative real values.
-/

/-- The complex-valued arithmetic function that is the convolution of the constant
function `1` with `χ`. -/
noncomputable def zetaMul (χ : DirichletCharacter ℂ N) : ArithmeticFunction ℂ :=
  .zeta * toArithmeticFunction (χ ·)

/-- The arithmetic function `zetaMul χ` is multiplicative. -/
lemma isMultiplicative_zetaMul (χ : DirichletCharacter ℂ N) : χ.zetaMul.IsMultiplicative :=
  isMultiplicative_zeta.natCast.mul <| isMultiplicative_toArithmeticFunction χ

Target:
lemma LSeriesSummable_zetaMul (χ : DirichletCharacter ℂ N) {s : ℂ} (hs : 1 < s.re) :
    LSeriesSummable χ.zetaMul s :=

Failed candidate:
by
  have __repair_pad_0_46b502979a : True := by trivial
  have __repair_pad_1_46b502979a : True := by trivial
  have __repair_pad_2_46b502979a : True := by trivial
  have __repair_pad_3_46b502979a : True := by trivial
  have __repair_pad_4_46b502979a : True := by trivial
  have __repair_pad_5_46b502979a : True := by trivial
  have __repair_pad_6_46b502979a : True := by trivial
  have __repair_pad_7_46b502979a : True := by trivial
  have __repair_pad_8_46b502979a : True := by trivial
  have __repair_pad_9_46b502979a : True := by trivial
  have __repair_pad_10_46b502979a : True := by trivial
  have __repair_pad_11_46b502979a : True := by trivial
  have __repair_pad_12_46b502979a : True := by trivial
  have __repair_pad_13_46b502979a : True := by trivial
  have __repair_pad_14_46b502979a : True := by trivial
  have __repair_pad_15_46b502979a : True := by trivial
  have __repair_pad_16_46b502979a : True := by trivial
  have __repair_pad_17_46b502979a : True := by trivial
  have __repair_pad_18_46b502979a : True := by trivial
  have __repair_pad_19_46b502979a : True := by trivial
  have __repair_pad_20_46b502979a : True := by trivial
  have __repair_pad_21_46b502979a : True := by trivial
  have __repair_pad_22_46b502979a : True := by trivial
  have __repair_pad_23_46b502979a : True := by trivial
  have __repair_fail_46b502979a := __oprover_overlong_46b502979a
  refine ArithmeticFunction.LSeriesSummable_mul (LSeriesSummable_zeta_iff.mpr hs) <|
    LSeriesSummable_of_bounded_of_one_lt_re (m := 1) (fun n hn ↦ ?_) hs
  simpa only [toArithmeticFunction, coe_mk, hn, ↓reduceIte]
  using norm_le_one χ _

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:109:35: error(lean.unknownIdentifier): Unknown identifier `__oprover_overlong_46b502979a`\n<LEAN_FILE>:84:0: error: unsolved goals\nN : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\n⊢ LSeriesSummable (⇑χ.zetaMul) s", "returncode": 1}

Corrected proof:


<LEAN_FILE>:83:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 11395771) 0
in the target expression
  ((B₁.tmul B₂) x) y = ((B₁.tmul B₂) y) x

R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
N₁ : Type uN₁
N₂ : Type uN₂
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : CommSemiring A
inst✝¹⁴ : AddCommMonoid M₁
inst✝¹³ : AddCommMonoid M₂
inst✝¹² : AddCommMonoid N₁
inst✝¹¹ : AddCommMonoid N₂
inst✝¹⁰ : Algebra R A
inst✝⁹ : Module R M₁
inst✝⁸ : Module A M₁
inst✝⁷ : Module R N₁
inst✝⁶ : Module A N₁
inst✝⁵ : SMulCommClass R A M₁
inst✝⁴ : IsScalarTower R A M₁
inst✝³ : SMulCommClass R A N₁
inst✝² : IsScalarTower R A N₁
inst✝¹ : Module R M₂
inst✝ : Module R N₂
B₁ : BilinMap A M₁ N₁
B₂ : BilinMap R M₂ N₂
hB₁ : ∀ (x y : M₁), (B₁ x) y = (B₁ y) x
hB₂ : ∀ (x y : M₂), (B₂ x) y = (B₂ y) x
x y : M₁ ⊗[R] M₂
⊢ ((B₁.tmul B₂) x) y = ((B₁.tmul B₂) y) x

Response:
by
  revert x y
  rw [isSymm_iff_eq_flip]
  aesop

variable (A) in
/-- The base change of a bilinear map (also known as "extension of scalars"). -/
protected def baseChange (B : BilinMap R M₂ N₂) : BilinMap A (A ⊗[R] M₂) (A ⊗[R] N₂) :=
  BilinMap.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (LinearMap.mul A A) B

[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.LinearAlgebra.BilinearForm.Hom
public import Mathlib.LinearAlgebra.Dual.Lemmas
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.RingTheory.TensorProduct.Finite

Namespace:
LinearMap.BilinMap

Local safe context:
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
/-!
# The bilinear form on a tensor product

## Main definitions

* `LinearMap.BilinMap.tensorDistrib (B₁ ⊗ₜ B₂)`: the bilinear form on `M₁ ⊗ M₂` constructed by
  applying `B₁` on `M₁` and `B₂` on `M₂`.
* `LinearMap.BilinMap.tensorDistribEquiv`: `BilinForm.tensorDistrib` as an equivalence on finite
  free modules.

-/

@[expose] public section

universe u v w uR uA uM₁ uM₂ uN₁ uN₂

variable {R : Type uR} {A : Type uA} {M₁ : Type uM₁} {M₂ : Type uM₂} {N₁ : Type uN₁} {N₂ : Type uN₂}

open TensorProduct

namespace LinearMap

open LinearMap (BilinMap BilinForm)

section CommSemiring

variable [CommSemiring R] [CommSemiring A]
variable [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid N₁] [AddCommMonoid N₂]
variable [Algebra R A] [Module R M₁] [Module A M₁] [Module R N₁] [Module A N₁]
variable [SMulCommClass R A M₁] [IsScalarTower R A M₁]
variable [SMulCommClass R A N₁] [IsScalarTower R A N₁]
variable [Module R M₂] [Module R N₂]

namespace BilinMap

variable (R A) in
/-- The tensor product of two bilinear maps injects into bilinear maps on tensor products.

Note this is heterobasic; the bilinear map on the left can take values in a module over a
(commutative) algebra over the ring of the module in which the right bilinear map is valued. -/
def tensorDistrib :
    (BilinMap A M₁ N₁ ⊗[R] BilinMap R M₂ N₂) →ₗ[A] BilinMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=
  (TensorProduct.lift.equiv (.id A) (M₁ ⊗[R] M₂) (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂)).symm.toLinearMap ∘ₗ
  ((LinearMap.llcomp A _ _ _).flip
    (TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R R A A M₁ M₂ M₁ M₂).toLinearMap)
  ∘ₗ TensorProduct.AlgebraTensorModule.homTensorHomMap R _ _ _ _ _ _
  ∘ₗ (TensorProduct.AlgebraTensorModule.congr
    (TensorProduct.lift.equiv (.id A) M₁ M₁ N₁)
    (TensorProduct.lift.equiv (.id R) _ _ _)).toLinearMap

@[simp]
theorem tensorDistrib_tmul (B₁ : BilinMap A M₁ N₁) (B₂ : BilinMap R M₂ N₂) (m₁ : M₁) (m₂ : M₂)
    (m₁' : M₁) (m₂' : M₂) :
    tensorDistrib R A (B₁ ⊗ₜ B₂) (m₁ ⊗ₜ m₂) (m₁' ⊗ₜ m₂')
      = B₁ m₁ m₁' ⊗ₜ B₂ m₂ m₂' :=
  rfl

/-- The tensor product of two bilinear forms, a shorthand for dot notation. -/
protected abbrev tmul (B₁ : BilinMap A M₁ N₁) (B₂ : BilinMap R M₂ N₂) :
    BilinMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=
  tensorDistrib R A (B₁ ⊗ₜ[R] B₂)

attribute [local ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear maps is symmetric. -/

Target:
lemma tmul_isSymm {B₁ : BilinMap A M₁ N₁} {B₂ : BilinMap R M₂ N₂}
    (hB₁ : ∀ x y, B₁ x y = B₁ y x) (hB₂ : ∀ x y, B₂ x y = B₂ y x)
    (x y : M₁ ⊗[R] M₂) :
    B₁.tmul B₂ x y = B₁.tmul B₂ y x :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 11395771) 0) = ((fun x : Nat => x + 11395771) 0) by rfl]
  revert x y
  rw [isSymm_iff_eq_flip]
  aesop

variable (A) in
/-- The base change of a bilinear map (also known as "extension of scalars"). -/
protected def baseChange (B : BilinMap R M₂ N₂) : BilinMap A (A ⊗[R] M₂) (A ⊗[R] N₂) :=
  BilinMap.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (LinearMap.mul A A) B

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:83:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 11395771) 0\nin the target expression\n  ((B₁.tmul B₂) x) y = ((B₁.tmul B₂) y) x\n\nR : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\nN₁ : Type uN₁\nN₂ : Type uN₂\ninst✝¹⁶ : CommSemiring R\ninst✝¹⁵ : CommSemiring A\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module A M₁\ninst✝⁷ : Module R N₁\ninst✝⁶ : Module A N₁\ninst✝⁵ : SMulCommClass R A M₁\ninst✝⁴ : IsScalarTower R A M₁\ninst✝³ : SMulCommClass R A N₁\ninst✝² : IsScalarTower R A N₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N₂\nB₁ : BilinMap A M₁ N₁\nB₂ : BilinMap R M₂ N₂\nhB₁ : ∀ (x y : M₁), (B₁ x) y = (B₁ y) x\nhB₂ : ∀ (x y : M₂), (B₂ x) y = (B₂ y) x\nx y : M₁ ⊗[R] M₂\n⊢ ((B₁.tmul B₂) x) y = ((B₁.tmul B₂) y) x", "returncode": 1}

Corrected proof:

[Assistant]
by
  revert x y
  rw [isSymm_iff_eq_flip]
  aesop

variable (A) in
/-- The base change of a bilinear map (also known as "extension of scalars"). -/
protected def baseChange (B : BilinMap R M₂ N₂) : BilinMap A (A ⊗[R] M₂) (A ⊗[R] N₂) :=
  BilinMap.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (LinearMap.mul A A) B

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.LinearAlgebra.BilinearForm.Hom
public import Mathlib.LinearAlgebra.Dual.Lemmas
public import Mathlib.LinearAlgebra.TensorProduct.Tower
public import Mathlib.RingTheory.TensorProduct.Finite

Namespace:
LinearMap.BilinMap

Local safe context:
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
/-!
# The bilinear form on a tensor product

## Main definitions

* `LinearMap.BilinMap.tensorDistrib (B₁ ⊗ₜ B₂)`: the bilinear form on `M₁ ⊗ M₂` constructed by
  applying `B₁` on `M₁` and `B₂` on `M₂`.
* `LinearMap.BilinMap.tensorDistribEquiv`: `BilinForm.tensorDistrib` as an equivalence on finite
  free modules.

-/

@[expose] public section

universe u v w uR uA uM₁ uM₂ uN₁ uN₂

variable {R : Type uR} {A : Type uA} {M₁ : Type uM₁} {M₂ : Type uM₂} {N₁ : Type uN₁} {N₂ : Type uN₂}

open TensorProduct

namespace LinearMap

open LinearMap (BilinMap BilinForm)

section CommSemiring

variable [CommSemiring R] [CommSemiring A]
variable [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid N₁] [AddCommMonoid N₂]
variable [Algebra R A] [Module R M₁] [Module A M₁] [Module R N₁] [Module A N₁]
variable [SMulCommClass R A M₁] [IsScalarTower R A M₁]
variable [SMulCommClass R A N₁] [IsScalarTower R A N₁]
variable [Module R M₂] [Module R N₂]

namespace BilinMap

variable (R A) in
/-- The tensor product of two bilinear maps injects into bilinear maps on tensor products.

Note this is heterobasic; the bilinear map on the left can take values in a module over a
(commutative) algebra over the ring of the module in which the right bilinear map is valued. -/
def tensorDistrib :
    (BilinMap A M₁ N₁ ⊗[R] BilinMap R M₂ N₂) →ₗ[A] BilinMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=
  (TensorProduct.lift.equiv (.id A) (M₁ ⊗[R] M₂) (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂)).symm.toLinearMap ∘ₗ
  ((LinearMap.llcomp A _ _ _).flip
    (TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R R A A M₁ M₂ M₁ M₂).toLinearMap)
  ∘ₗ TensorProduct.AlgebraTensorModule.homTensorHomMap R _ _ _ _ _ _
  ∘ₗ (TensorProduct.AlgebraTensorModule.congr
    (TensorProduct.lift.equiv (.id A) M₁ M₁ N₁)
    (TensorProduct.lift.equiv (.id R) _ _ _)).toLinearMap

@[simp]
theorem tensorDistrib_tmul (B₁ : BilinMap A M₁ N₁) (B₂ : BilinMap R M₂ N₂) (m₁ : M₁) (m₂ : M₂)
    (m₁' : M₁) (m₂' : M₂) :
    tensorDistrib R A (B₁ ⊗ₜ B₂) (m₁ ⊗ₜ m₂) (m₁' ⊗ₜ m₂')
      = B₁ m₁ m₁' ⊗ₜ B₂ m₂ m₂' :=
  rfl

/-- The tensor product of two bilinear forms, a shorthand for dot notation. -/
protected abbrev tmul (B₁ : BilinMap A M₁ N₁) (B₂ : BilinMap R M₂ N₂) :
    BilinMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=
  tensorDistrib R A (B₁ ⊗ₜ[R] B₂)

attribute [local ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear maps is symmetric. -/

Target:
lemma tmul_isSymm {B₁ : BilinMap A M₁ N₁} {B₂ : BilinMap R M₂ N₂}
    (hB₁ : ∀ x y, B₁ x y = B₁ y x) (hB₂ : ∀ x y, B₂ x y = B₂ y x)
    (x y : M₁ ⊗[R] M₂) :
    B₁.tmul B₂ x y = B₁.tmul B₂ y x :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 11395771) 0) = ((fun x : Nat => x + 11395771) 0) by rfl]
  revert x y
  rw [isSymm_iff_eq_flip]
  aesop

variable (A) in
/-- The base change of a bilinear map (also known as "extension of scalars"). -/
protected def baseChange (B : BilinMap R M₂ N₂) : BilinMap A (A ⊗[R] M₂) (A ⊗[R] N₂) :=
  BilinMap.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (LinearMap.mul A A) B

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:83:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 11395771) 0\nin the target expression\n  ((B₁.tmul B₂) x) y = ((B₁.tmul B₂) y) x\n\nR : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\nN₁ : Type uN₁\nN₂ : Type uN₂\ninst✝¹⁶ : CommSemiring R\ninst✝¹⁵ : CommSemiring A\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module A M₁\ninst✝⁷ : Module R N₁\ninst✝⁶ : Module A N₁\ninst✝⁵ : SMulCommClass R A M₁\ninst✝⁴ : IsScalarTower R A M₁\ninst✝³ : SMulCommClass R A N₁\ninst✝² : IsScalarTower R A N₁\ninst✝¹ : Module R M₂\ninst✝ : Module R N₂\nB₁ : BilinMap A M₁ N₁\nB₂ : BilinMap R M₂ N₂\nhB₁ : ∀ (x y : M₁), (B₁ x) y = (B₁ y) x\nhB₂ : ∀ (x y : M₂), (B₂ x) y = (B₂ y) x\nx y : M₁ ⊗[R] M₂\n⊢ ((B₁.tmul B₂) x) y = ((B₁.tmul B₂) y) x", "returncode": 1}

Corrected proof:


<LEAN_FILE>:189:42: 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
  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
  let __repair_f567df944e : Fact False := inferInstance
  convert! if_pos rfl

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:189:42: 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
  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
  let __repair_f567df944e : Fact False := inferInstance
  convert! if_pos rfl

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:189:42: 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>:102:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 1719088) 0
in the target expression
  Continuous[_, inst✝] g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous[inst✝¹ i, inst✝] (g ∘ ⇑equiv.symm ∘ ⇑f)

ι : Type t
X : ι → Type u
inst✝¹ : (i : ι) → TopologicalSpace (X i)
Y : Type v
tY : TopologicalSpace Y
Z : Type v'
inst✝ : TopologicalSpace Z
g : WithGeneratedByTopology X Y → Z
⊢ Continuous[_, inst✝] g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous[inst✝¹ i, inst✝] (g ∘ ⇑equiv.symm ∘ ⇑f)

Response:
by
  simp only [continuous_iSup_dom]
  exact forall_congr' (fun i ↦ forall_congr'
    (fun f ↦ (by rw [continuous_coinduced_dom]; simp [equiv, Equiv.refl])))

[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.Topology.ContinuousMap.Basic

Namespace:
Topology.WithGeneratedByTopology

Local safe context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Dagur Asgeirsson, Ben Eltschig
-/
/-!
# The topology that is generated by a family of topological spaces

Let `X : ι → Type u` be a family of topological spaces.
Let `Y` be a topological space. We introduce a type synonym
`WithGeneratedByTopology X Y` for `Y`. This type is endowed
with the `X`-generated topology, which is coinduced by
all continuous maps `X i → Y`. When the bijection
`WithGeneratedByTopology X Y ≃ Y` is a homeomorphism,
we say that `Y` is `X`-generated (typeclass `IsGeneratedBy X Y`).

## TODO (@joelriou)
* Redefine compactly generated spaces and delta generated spaces using
  these definitions
* Define the category of `X`-generated spaces and show that it is coreflective in `TopCat`
* Show that under suitable assumptions, the category of `X`-generated
  spaces is a closed cartesian monoidal category.

## References
* [Rainer M. Vogt, *Convenient categories of topological spaces
  for homotopy theory*][vogt-1971]
* [Martín Escardó, Jimmie Lawson and Alex Simpson, *Comparing Cartesian closed
  categories of (core) compactly generated spaces*][escardo-lawson-simpson-2004]

-/

@[expose] public section

universe v v' t u

open Topology

variable {ι : Type t} (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
  {Y : Type v} [tY : TopologicalSpace Y] {Z : Type v'} [TopologicalSpace Z]

namespace TopologicalSpace

/-- Given a family of topological spaces `X i`, the `X`-generated topology on
a topological space `Y` is the topology that is coinduced
by all continuous maps `X i → Y`. -/
@[implicit_reducible]
def generatedBy : TopologicalSpace Y :=
  ⨆ (i : ι) (f : C(X i, Y)), coinduced f inferInstance

/-- The `X`-generated topology is also coinduced by a single map out of a sigma type. -/
lemma generatedBy_eq_coinduced :
    generatedBy X (Y := Y) =
      coinduced (fun (x : (f : (i : ι) × C(X i, Y)) × X f.1) ↦ x.1.2 x.2) inferInstance := by
  rw [generatedBy, instTopologicalSpaceSigma, coinduced_iSup, iSup_sigma]
  rfl

end TopologicalSpace

variable {X}

namespace Topology

/-- Given a family of topological spaces `X i`, and a topological space `Y`,
this is a type synonym for `Y` which we endow with the `X`-generated topology. -/
@[nolint unusedArguments]
def WithGeneratedByTopology (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
    (Y : Type v) [TopologicalSpace Y] := Y

namespace WithGeneratedByTopology

/-- The obvious bijection `WithGeneratedByTopology X Y ≃ Y`, where
the source is endowed with the `X`-generated topology. See `continuous_equiv`
for the continuity of `equiv`. The inverse map `equiv.symm` is continuous
iff `Y` is `X`-generated, see `isGeneratedBy_iff`. -/
def equiv : WithGeneratedByTopology X Y ≃ Y := Equiv.refl _

instance {Y : Type v} [TopologicalSpace Y] :
    TopologicalSpace (WithGeneratedByTopology X Y) :=
  -- fast_instance% .generatedBy X (Y := Y) fails
  letI : TopologicalSpace Y := .generatedBy X (Y := Y)
  inferInstanceAs <| TopologicalSpace Y

set_option backward.isDefEq.respectTransparency false in
lemma isOpen_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsOpen U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsOpen (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isOpen_iSup_iff, isOpen_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in
lemma isClosed_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsClosed U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsClosed (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isClosed_iSup_iff, isClosed_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in

Target:
lemma continuous_from_iff (g : WithGeneratedByTopology X Y → Z) :
    Continuous g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous (g ∘ equiv.symm ∘ f : X i → Z) :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 1719088) 0) = ((fun x : Nat => x + 1719088) 0) by rfl]
  simp only [continuous_iSup_dom]
  exact forall_congr' (fun i ↦ forall_congr'
    (fun f ↦ (by rw [continuous_coinduced_dom]; simp [equiv, Equiv.refl])))

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:102:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 1719088) 0\nin the target expression\n  Continuous[_, inst✝] g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous[inst✝¹ i, inst✝] (g ∘ ⇑equiv.symm ∘ ⇑f)\n\nι : Type t\nX : ι → Type u\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ntY : TopologicalSpace Y\nZ : Type v'\ninst✝ : TopologicalSpace Z\ng : WithGeneratedByTopology X Y → Z\n⊢ Continuous[_, inst✝] g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous[inst✝¹ i, inst✝] (g ∘ ⇑equiv.symm ∘ ⇑f)", "returncode": 1}

Corrected proof:

[Assistant]
by
  simp only [continuous_iSup_dom]
  exact forall_congr' (fun i ↦ forall_congr'
    (fun f ↦ (by rw [continuous_coinduced_dom]; simp [equiv, Equiv.refl])))

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.Topology.ContinuousMap.Basic

Namespace:
Topology.WithGeneratedByTopology

Local safe context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Dagur Asgeirsson, Ben Eltschig
-/
/-!
# The topology that is generated by a family of topological spaces

Let `X : ι → Type u` be a family of topological spaces.
Let `Y` be a topological space. We introduce a type synonym
`WithGeneratedByTopology X Y` for `Y`. This type is endowed
with the `X`-generated topology, which is coinduced by
all continuous maps `X i → Y`. When the bijection
`WithGeneratedByTopology X Y ≃ Y` is a homeomorphism,
we say that `Y` is `X`-generated (typeclass `IsGeneratedBy X Y`).

## TODO (@joelriou)
* Redefine compactly generated spaces and delta generated spaces using
  these definitions
* Define the category of `X`-generated spaces and show that it is coreflective in `TopCat`
* Show that under suitable assumptions, the category of `X`-generated
  spaces is a closed cartesian monoidal category.

## References
* [Rainer M. Vogt, *Convenient categories of topological spaces
  for homotopy theory*][vogt-1971]
* [Martín Escardó, Jimmie Lawson and Alex Simpson, *Comparing Cartesian closed
  categories of (core) compactly generated spaces*][escardo-lawson-simpson-2004]

-/

@[expose] public section

universe v v' t u

open Topology

variable {ι : Type t} (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
  {Y : Type v} [tY : TopologicalSpace Y] {Z : Type v'} [TopologicalSpace Z]

namespace TopologicalSpace

/-- Given a family of topological spaces `X i`, the `X`-generated topology on
a topological space `Y` is the topology that is coinduced
by all continuous maps `X i → Y`. -/
@[implicit_reducible]
def generatedBy : TopologicalSpace Y :=
  ⨆ (i : ι) (f : C(X i, Y)), coinduced f inferInstance

/-- The `X`-generated topology is also coinduced by a single map out of a sigma type. -/
lemma generatedBy_eq_coinduced :
    generatedBy X (Y := Y) =
      coinduced (fun (x : (f : (i : ι) × C(X i, Y)) × X f.1) ↦ x.1.2 x.2) inferInstance := by
  rw [generatedBy, instTopologicalSpaceSigma, coinduced_iSup, iSup_sigma]
  rfl

end TopologicalSpace

variable {X}

namespace Topology

/-- Given a family of topological spaces `X i`, and a topological space `Y`,
this is a type synonym for `Y` which we endow with the `X`-generated topology. -/
@[nolint unusedArguments]
def WithGeneratedByTopology (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
    (Y : Type v) [TopologicalSpace Y] := Y

namespace WithGeneratedByTopology

/-- The obvious bijection `WithGeneratedByTopology X Y ≃ Y`, where
the source is endowed with the `X`-generated topology. See `continuous_equiv`
for the continuity of `equiv`. The inverse map `equiv.symm` is continuous
iff `Y` is `X`-generated, see `isGeneratedBy_iff`. -/
def equiv : WithGeneratedByTopology X Y ≃ Y := Equiv.refl _

instance {Y : Type v} [TopologicalSpace Y] :
    TopologicalSpace (WithGeneratedByTopology X Y) :=
  -- fast_instance% .generatedBy X (Y := Y) fails
  letI : TopologicalSpace Y := .generatedBy X (Y := Y)
  inferInstanceAs <| TopologicalSpace Y

set_option backward.isDefEq.respectTransparency false in
lemma isOpen_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsOpen U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsOpen (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isOpen_iSup_iff, isOpen_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in
lemma isClosed_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsClosed U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsClosed (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isClosed_iSup_iff, isClosed_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in

Target:
lemma continuous_from_iff (g : WithGeneratedByTopology X Y → Z) :
    Continuous g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous (g ∘ equiv.symm ∘ f : X i → Z) :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 1719088) 0) = ((fun x : Nat => x + 1719088) 0) by rfl]
  simp only [continuous_iSup_dom]
  exact forall_congr' (fun i ↦ forall_congr'
    (fun f ↦ (by rw [continuous_coinduced_dom]; simp [equiv, Equiv.refl])))

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:102:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 1719088) 0\nin the target expression\n  Continuous[_, inst✝] g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous[inst✝¹ i, inst✝] (g ∘ ⇑equiv.symm ∘ ⇑f)\n\nι : Type t\nX : ι → Type u\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ntY : TopologicalSpace Y\nZ : Type v'\ninst✝ : TopologicalSpace Z\ng : WithGeneratedByTopology X Y → Z\n⊢ Continuous[_, inst✝] g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous[inst✝¹ i, inst✝] (g ∘ ⇑equiv.symm ∘ ⇑f)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:102:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing7eccf61165.value`
<LEAN_FILE>:101:0: error: unsolved goals
ι : Type t
X : ι → Type u
inst✝¹ : (i : ι) → TopologicalSpace (X i)
Y : Type v
tY : TopologicalSpace Y
Z : Type v'
inst✝ : TopologicalSpace Z
g : WithGeneratedByTopology X Y → Z
⊢ Continuous[_, inst✝] g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous[inst✝¹ i, inst✝] (g ∘ ⇑equiv.symm ∘ ⇑f)

Response:
by
  simp only [continuous_iSup_dom]
  exact forall_congr' (fun i ↦ forall_congr'
    (fun f ↦ (by rw [continuous_coinduced_dom]; simp [equiv, Equiv.refl])))

[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.Topology.ContinuousMap.Basic

Namespace:
Topology.WithGeneratedByTopology

Local safe context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Dagur Asgeirsson, Ben Eltschig
-/
/-!
# The topology that is generated by a family of topological spaces

Let `X : ι → Type u` be a family of topological spaces.
Let `Y` be a topological space. We introduce a type synonym
`WithGeneratedByTopology X Y` for `Y`. This type is endowed
with the `X`-generated topology, which is coinduced by
all continuous maps `X i → Y`. When the bijection
`WithGeneratedByTopology X Y ≃ Y` is a homeomorphism,
we say that `Y` is `X`-generated (typeclass `IsGeneratedBy X Y`).

## TODO (@joelriou)
* Redefine compactly generated spaces and delta generated spaces using
  these definitions
* Define the category of `X`-generated spaces and show that it is coreflective in `TopCat`
* Show that under suitable assumptions, the category of `X`-generated
  spaces is a closed cartesian monoidal category.

## References
* [Rainer M. Vogt, *Convenient categories of topological spaces
  for homotopy theory*][vogt-1971]
* [Martín Escardó, Jimmie Lawson and Alex Simpson, *Comparing Cartesian closed
  categories of (core) compactly generated spaces*][escardo-lawson-simpson-2004]

-/

@[expose] public section

universe v v' t u

open Topology

variable {ι : Type t} (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
  {Y : Type v} [tY : TopologicalSpace Y] {Z : Type v'} [TopologicalSpace Z]

namespace TopologicalSpace

/-- Given a family of topological spaces `X i`, the `X`-generated topology on
a topological space `Y` is the topology that is coinduced
by all continuous maps `X i → Y`. -/
@[implicit_reducible]
def generatedBy : TopologicalSpace Y :=
  ⨆ (i : ι) (f : C(X i, Y)), coinduced f inferInstance

/-- The `X`-generated topology is also coinduced by a single map out of a sigma type. -/
lemma generatedBy_eq_coinduced :
    generatedBy X (Y := Y) =
      coinduced (fun (x : (f : (i : ι) × C(X i, Y)) × X f.1) ↦ x.1.2 x.2) inferInstance := by
  rw [generatedBy, instTopologicalSpaceSigma, coinduced_iSup, iSup_sigma]
  rfl

end TopologicalSpace

variable {X}

namespace Topology

/-- Given a family of topological spaces `X i`, and a topological space `Y`,
this is a type synonym for `Y` which we endow with the `X`-generated topology. -/
@[nolint unusedArguments]
def WithGeneratedByTopology (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
    (Y : Type v) [TopologicalSpace Y] := Y

namespace WithGeneratedByTopology

/-- The obvious bijection `WithGeneratedByTopology X Y ≃ Y`, where
the source is endowed with the `X`-generated topology. See `continuous_equiv`
for the continuity of `equiv`. The inverse map `equiv.symm` is continuous
iff `Y` is `X`-generated, see `isGeneratedBy_iff`. -/
def equiv : WithGeneratedByTopology X Y ≃ Y := Equiv.refl _

instance {Y : Type v} [TopologicalSpace Y] :
    TopologicalSpace (WithGeneratedByTopology X Y) :=
  -- fast_instance% .generatedBy X (Y := Y) fails
  letI : TopologicalSpace Y := .generatedBy X (Y := Y)
  inferInstanceAs <| TopologicalSpace Y

set_option backward.isDefEq.respectTransparency false in
lemma isOpen_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsOpen U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsOpen (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isOpen_iSup_iff, isOpen_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in
lemma isClosed_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsClosed U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsClosed (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isClosed_iSup_iff, isClosed_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in

Target:
lemma continuous_from_iff (g : WithGeneratedByTopology X Y → Z) :
    Continuous g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous (g ∘ equiv.symm ∘ f : X i → Z) :=

Failed candidate:
by
  have __repair_7eccf61165 := OProverMissing7eccf61165.value
  simp only [continuous_iSup_dom]
  exact forall_congr' (fun i ↦ forall_congr'
    (fun f ↦ (by rw [continuous_coinduced_dom]; simp [equiv, Equiv.refl])))

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:102:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing7eccf61165.value`\n<LEAN_FILE>:101:0: error: unsolved goals\nι : Type t\nX : ι → Type u\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ntY : TopologicalSpace Y\nZ : Type v'\ninst✝ : TopologicalSpace Z\ng : WithGeneratedByTopology X Y → Z\n⊢ Continuous[_, inst✝] g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous[inst✝¹ i, inst✝] (g ∘ ⇑equiv.symm ∘ ⇑f)", "returncode": 1}

Corrected proof:

[Assistant]
by
  simp only [continuous_iSup_dom]
  exact forall_congr' (fun i ↦ forall_congr'
    (fun f ↦ (by rw [continuous_coinduced_dom]; simp [equiv, Equiv.refl])))

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.Topology.ContinuousMap.Basic

Namespace:
Topology.WithGeneratedByTopology

Local safe context:
/-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Dagur Asgeirsson, Ben Eltschig
-/
/-!
# The topology that is generated by a family of topological spaces

Let `X : ι → Type u` be a family of topological spaces.
Let `Y` be a topological space. We introduce a type synonym
`WithGeneratedByTopology X Y` for `Y`. This type is endowed
with the `X`-generated topology, which is coinduced by
all continuous maps `X i → Y`. When the bijection
`WithGeneratedByTopology X Y ≃ Y` is a homeomorphism,
we say that `Y` is `X`-generated (typeclass `IsGeneratedBy X Y`).

## TODO (@joelriou)
* Redefine compactly generated spaces and delta generated spaces using
  these definitions
* Define the category of `X`-generated spaces and show that it is coreflective in `TopCat`
* Show that under suitable assumptions, the category of `X`-generated
  spaces is a closed cartesian monoidal category.

## References
* [Rainer M. Vogt, *Convenient categories of topological spaces
  for homotopy theory*][vogt-1971]
* [Martín Escardó, Jimmie Lawson and Alex Simpson, *Comparing Cartesian closed
  categories of (core) compactly generated spaces*][escardo-lawson-simpson-2004]

-/

@[expose] public section

universe v v' t u

open Topology

variable {ι : Type t} (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
  {Y : Type v} [tY : TopologicalSpace Y] {Z : Type v'} [TopologicalSpace Z]

namespace TopologicalSpace

/-- Given a family of topological spaces `X i`, the `X`-generated topology on
a topological space `Y` is the topology that is coinduced
by all continuous maps `X i → Y`. -/
@[implicit_reducible]
def generatedBy : TopologicalSpace Y :=
  ⨆ (i : ι) (f : C(X i, Y)), coinduced f inferInstance

/-- The `X`-generated topology is also coinduced by a single map out of a sigma type. -/
lemma generatedBy_eq_coinduced :
    generatedBy X (Y := Y) =
      coinduced (fun (x : (f : (i : ι) × C(X i, Y)) × X f.1) ↦ x.1.2 x.2) inferInstance := by
  rw [generatedBy, instTopologicalSpaceSigma, coinduced_iSup, iSup_sigma]
  rfl

end TopologicalSpace

variable {X}

namespace Topology

/-- Given a family of topological spaces `X i`, and a topological space `Y`,
this is a type synonym for `Y` which we endow with the `X`-generated topology. -/
@[nolint unusedArguments]
def WithGeneratedByTopology (X : ι → Type u) [∀ i, TopologicalSpace (X i)]
    (Y : Type v) [TopologicalSpace Y] := Y

namespace WithGeneratedByTopology

/-- The obvious bijection `WithGeneratedByTopology X Y ≃ Y`, where
the source is endowed with the `X`-generated topology. See `continuous_equiv`
for the continuity of `equiv`. The inverse map `equiv.symm` is continuous
iff `Y` is `X`-generated, see `isGeneratedBy_iff`. -/
def equiv : WithGeneratedByTopology X Y ≃ Y := Equiv.refl _

instance {Y : Type v} [TopologicalSpace Y] :
    TopologicalSpace (WithGeneratedByTopology X Y) :=
  -- fast_instance% .generatedBy X (Y := Y) fails
  letI : TopologicalSpace Y := .generatedBy X (Y := Y)
  inferInstanceAs <| TopologicalSpace Y

set_option backward.isDefEq.respectTransparency false in
lemma isOpen_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsOpen U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsOpen (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isOpen_iSup_iff, isOpen_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in
lemma isClosed_iff {U : Set (WithGeneratedByTopology X Y)} :
    IsClosed U ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), IsClosed (f ⁻¹' equiv.symm ⁻¹' U) := by
  simp [isClosed_iSup_iff, isClosed_coinduced, equiv, Equiv.refl]

set_option backward.isDefEq.respectTransparency false in

Target:
lemma continuous_from_iff (g : WithGeneratedByTopology X Y → Z) :
    Continuous g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous (g ∘ equiv.symm ∘ f : X i → Z) :=

Failed candidate:
by
  have __repair_7eccf61165 := OProverMissing7eccf61165.value
  simp only [continuous_iSup_dom]
  exact forall_congr' (fun i ↦ forall_congr'
    (fun f ↦ (by rw [continuous_coinduced_dom]; simp [equiv, Equiv.refl])))

Structured Lean error:
{"kind": "unknown_identifier_namespace_import", "message": "<LEAN_FILE>:102:30: error(lean.unknownIdentifier): Unknown identifier `OProverMissing7eccf61165.value`\n<LEAN_FILE>:101:0: error: unsolved goals\nι : Type t\nX : ι → Type u\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ntY : TopologicalSpace Y\nZ : Type v'\ninst✝ : TopologicalSpace Z\ng : WithGeneratedByTopology X Y → Z\n⊢ Continuous[_, inst✝] g ↔ ∀ ⦃i : ι⦄ (f : C(X i, Y)), Continuous[inst✝¹ i, inst✝] (g ∘ ⇑equiv.symm ∘ ⇑f)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:271:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression
  True.intro
of type `True`
<LEAN_FILE>:270:0: error: unsolved goals
X : Compactum
A : Set X.A
⊢ IsClosed (cl A)

Response:
by
  rw [isClosed_iff]
  intro F hF
  exact cl_cl _ ⟨F, hF, 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.CategoryTheory.Monad.Types
public import Mathlib.CategoryTheory.Monad.Limits
public import Mathlib.CategoryTheory.Equivalence
public import Mathlib.Topology.Category.CompHaus.Basic
public import Mathlib.Topology.Category.Profinite.Basic
public import Mathlib.Data.Set.Constructions

Namespace:
Compactum

Local safe context:
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
/-!

# Compacta and Compact Hausdorff Spaces

Recall that, given a monad `M` on `Type*`, an *algebra* for `M` consists of the following data:
- A type `X : Type*`
- A "structure" map `M X → X`.

This data must also satisfy a distributivity and unit axiom, and algebras for `M` form a category
in an evident way.

See the file `Mathlib/CategoryTheory/Monad/Algebra.lean` for a general version, as well as the
following link.
https://ncatlab.org/nlab/show/monad

This file proves the equivalence between the category of *compact Hausdorff topological spaces*
and the category of algebras for the *ultrafilter monad*.

## Notation:

Here are the main objects introduced in this file.
- `Compactum` is the type of compacta, which we define as algebras for the ultrafilter monad.
- `compactumToCompHaus` is the functor `Compactum ⥤ CompHaus`. Here `CompHaus` is the usual
  category of compact Hausdorff spaces.
- `compactumToCompHaus.isEquivalence` is a term of type `IsEquivalence compactumToCompHaus`.

The proof of this equivalence is a bit technical. But the idea is quite simply that the structure
map `Ultrafilter X → X` for an algebra `X` of the ultrafilter monad should be considered as the map
sending an ultrafilter to its limit in `X`. The topology on `X` is then defined by mimicking the
characterization of open sets in terms of ultrafilters.

Any `X : Compactum` is endowed with a coercion to `Type*`, as well as the following instances:
- `TopologicalSpace X`.
- `CompactSpace X`.
- `T2Space X`.

Any morphism `f : X ⟶ Y` of is endowed with a coercion to a function `X → Y`, which is shown to
be continuous in `continuous_of_hom`.

The function `Compactum.ofTopologicalSpace` can be used to construct a `Compactum` from a
topological space which satisfies `CompactSpace` and `T2Space`.

We also add wrappers around structures which already exist. Here are the main ones, all in the
`Compactum` namespace:

- `forget : Compactum ⥤ Type*` is the forgetful functor, which induces a `ConcreteCategory`
  instance for `Compactum`.
- `free : Type* ⥤ Compactum` is the left adjoint to `forget`, and the adjunction is in `adj`.
- `str : Ultrafilter X → X` is the structure map for `X : Compactum`.
  The notation `X.str` is preferred.
- `join : Ultrafilter (Ultrafilter X) → Ultrafilter X` is the monadic join for `X : Compactum`.
  Again, the notation `X.join` is preferred.
- `incl : X → Ultrafilter X` is the unit for `X : Compactum`. The notation `X.incl` is preferred.

## References

- E. Manes, Algebraic Theories, Graduate Texts in Mathematics 26, Springer-Verlag, 1976.
- https://ncatlab.org/nlab/show/ultrafilter

-/

@[expose] public section

universe u

open CategoryTheory Filter Ultrafilter TopologicalSpace CategoryTheory.Limits FiniteInter
open scoped Topology

local notation "β" => ofTypeMonad Ultrafilter

/-- The type `Compactum` of Compacta, defined as algebras for the ultrafilter monad. -/
def Compactum :=
  Monad.Algebra β deriving Category, Inhabited

namespace Compactum

/-- The forgetful functor to TypeCat -/
def forget : Compactum ⥤ Type _ :=
  Monad.forget _

instance : forget.Faithful :=
  show (Monad.forget _).Faithful from inferInstance

noncomputable instance : CreatesLimits forget :=
  show CreatesLimits <| Monad.forget _ from inferInstance

/-- The "free" Compactum functor. -/
def free : Type _ ⥤ Compactum :=
  Monad.free _

/-- The adjunction between `free` and `forget`. -/
def adj : free ⊣ forget :=
  Monad.adj _

instance : CoeSort Compactum Type* :=
  ⟨fun X => X.A⟩

instance {X Y : Compactum} : FunLike (X ⟶ Y) X Y where
  coe f := f.f
  coe_injective _ _ h := (Monad.forget_faithful β).map_injective (by aesop)

-- Basic instances
instance : ConcreteCategory Compactum (· ⟶ ·) where
  hom f := f
  ofHom f := f

instance : HasLimits Compactum :=
  hasLimits_of_hasLimits_createsLimits forget

/-- The structure map for a compactum, essentially sending an ultrafilter to its limit. -/
def str (X : Compactum) : Ultrafilter X → X :=
  X.a

/-- The monadic join. -/
def join (X : Compactum) : Ultrafilter (Ultrafilter X) → Ultrafilter X :=
  (β).μ.app _

/-- The inclusion of `X` into `Ultrafilter X`. -/
def incl (X : Compactum) : X → Ultrafilter X :=
  (β).η.app _

@[simp]
theorem str_incl (X : Compactum) (x : X) : X.str (X.incl x) = x := by
  change ((β).η.app _ ≫ X.a) _ = _
  rw [Monad.Algebra.unit]
  rfl

@[simp]
theorem str_hom_commute (X Y : Compactum) (f : X ⟶ Y) (xs : Ultrafilter X) :
    f (X.str xs) = Y.str (map f xs) := by
  change (X.a ≫ f.f) _ = _
  rw [← f.h]
  rfl

@[simp]
theorem join_distrib (X : Compactum) (uux : Ultrafilter (Ultrafilter X)) :
    X.str (X.join uux) = X.str (map X.str uux) := by
  change ((β).μ.app _ ≫ X.a) _ = _
  rw [Monad.Algebra.assoc]
  rfl

instance {X : Compactum} : TopologicalSpace X where
  IsOpen U := ∀ F : Ultrafilter X, X.str F ∈ U → U ∈ F
  isOpen_univ _ _ := Filter.univ_sets _
  isOpen_inter _ _ h3 h4 _ h6 := Filter.inter_sets _ (h3 _ h6.1) (h4 _ h6.2)
  isOpen_sUnion := fun _ h1 _ ⟨T, hT, h2⟩ =>
    mem_of_superset (h1 T hT _ h2) (Set.subset_sUnion_of_mem hT)

theorem isClosed_iff {X : Compactum} (S : Set X) :
    IsClosed S ↔ ∀ F : Ultrafilter X, S ∈ F → X.str F ∈ S := by
  rw [← isOpen_compl_iff]
  constructor
  · intro cond F h
    by_contra c
    specialize cond F c
    rw [compl_mem_iff_notMem] at cond
    contradiction
  · intro h1 F h2
    specialize h1 F
    rcases F.mem_or_compl_mem S with h | h
    exacts [absurd (h1 h) h2, h]

instance {X : Compactum} : CompactSpace X := by
  constructor
  rw [isCompact_iff_ultrafilter_le_nhds]
  intro F _
  refine ⟨X.str F, by tauto, ?_⟩
  rw [le_nhds_iff]
  intro S h1 h2
  exact h2 F h1

/-- A local definition used only in the proofs. -/
private def basic {X : Compactum} (A : Set X) : Set (Ultrafilter X) :=
  { F | A ∈ F }

set_option backward.privateInPublic true in
/-- A local definition used only in the proofs. -/
private def cl {X : Compactum} (A : Set X) : Set X :=
  X.str '' basic A

private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · intro hG
    constructor <;> filter_upwards [hG] with _
    exacts [And.left, And.right]
  · rintro ⟨h1, h2⟩
    exact inter_mem h1 h2

private theorem subset_cl {X : Compactum} (A : Set X) : A ⊆ cl A := fun a ha =>
  ⟨X.incl a, ha, by simp⟩

private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine ⟨P, hq2, ?_⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      rcases this P hP with h | h
      · exact claim2 _ h
      · rcases h with ⟨Q, hQ, rfl⟩
        exact claim3 _ hQ
    intro P hP
    exact claim5.finiteInterClosure_insert _ hP
  intro T hT
  -- Suffices to show that the intersection of the T's is contained in C2.
  suffices ⋂₀ ι T ∈ C2 by exact claim6 _ this
  -- Finish
  apply claim4.finiteInter_mem T
  intro t ht
  exact finiteInterClosure.basic (@hT t ht)

set_option backward.privateInPublic true in
set_option backward.privateInPublic.warn false in

Target:
theorem isClosed_cl {X : Compactum} (A : Set X) : IsClosed (cl A) :=

Failed candidate:
by
  have __repair_f0dc642b89 := True.intro.nonexistentProjection
  rw [isClosed_iff]
  intro F hF
  exact cl_cl _ ⟨F, hF, rfl⟩

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:271:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:270:0: error: unsolved goals\nX : Compactum\nA : Set X.A\n⊢ IsClosed (cl A)", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [isClosed_iff]
  intro F hF
  exact cl_cl _ ⟨F, hF, 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.CategoryTheory.Monad.Types
public import Mathlib.CategoryTheory.Monad.Limits
public import Mathlib.CategoryTheory.Equivalence
public import Mathlib.Topology.Category.CompHaus.Basic
public import Mathlib.Topology.Category.Profinite.Basic
public import Mathlib.Data.Set.Constructions

Namespace:
Compactum

Local safe context:
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
/-!

# Compacta and Compact Hausdorff Spaces

Recall that, given a monad `M` on `Type*`, an *algebra* for `M` consists of the following data:
- A type `X : Type*`
- A "structure" map `M X → X`.

This data must also satisfy a distributivity and unit axiom, and algebras for `M` form a category
in an evident way.

See the file `Mathlib/CategoryTheory/Monad/Algebra.lean` for a general version, as well as the
following link.
https://ncatlab.org/nlab/show/monad

This file proves the equivalence between the category of *compact Hausdorff topological spaces*
and the category of algebras for the *ultrafilter monad*.

## Notation:

Here are the main objects introduced in this file.
- `Compactum` is the type of compacta, which we define as algebras for the ultrafilter monad.
- `compactumToCompHaus` is the functor `Compactum ⥤ CompHaus`. Here `CompHaus` is the usual
  category of compact Hausdorff spaces.
- `compactumToCompHaus.isEquivalence` is a term of type `IsEquivalence compactumToCompHaus`.

The proof of this equivalence is a bit technical. But the idea is quite simply that the structure
map `Ultrafilter X → X` for an algebra `X` of the ultrafilter monad should be considered as the map
sending an ultrafilter to its limit in `X`. The topology on `X` is then defined by mimicking the
characterization of open sets in terms of ultrafilters.

Any `X : Compactum` is endowed with a coercion to `Type*`, as well as the following instances:
- `TopologicalSpace X`.
- `CompactSpace X`.
- `T2Space X`.

Any morphism `f : X ⟶ Y` of is endowed with a coercion to a function `X → Y`, which is shown to
be continuous in `continuous_of_hom`.

The function `Compactum.ofTopologicalSpace` can be used to construct a `Compactum` from a
topological space which satisfies `CompactSpace` and `T2Space`.

We also add wrappers around structures which already exist. Here are the main ones, all in the
`Compactum` namespace:

- `forget : Compactum ⥤ Type*` is the forgetful functor, which induces a `ConcreteCategory`
  instance for `Compactum`.
- `free : Type* ⥤ Compactum` is the left adjoint to `forget`, and the adjunction is in `adj`.
- `str : Ultrafilter X → X` is the structure map for `X : Compactum`.
  The notation `X.str` is preferred.
- `join : Ultrafilter (Ultrafilter X) → Ultrafilter X` is the monadic join for `X : Compactum`.
  Again, the notation `X.join` is preferred.
- `incl : X → Ultrafilter X` is the unit for `X : Compactum`. The notation `X.incl` is preferred.

## References

- E. Manes, Algebraic Theories, Graduate Texts in Mathematics 26, Springer-Verlag, 1976.
- https://ncatlab.org/nlab/show/ultrafilter

-/

@[expose] public section

universe u

open CategoryTheory Filter Ultrafilter TopologicalSpace CategoryTheory.Limits FiniteInter
open scoped Topology

local notation "β" => ofTypeMonad Ultrafilter

/-- The type `Compactum` of Compacta, defined as algebras for the ultrafilter monad. -/
def Compactum :=
  Monad.Algebra β deriving Category, Inhabited

namespace Compactum

/-- The forgetful functor to TypeCat -/
def forget : Compactum ⥤ Type _ :=
  Monad.forget _

instance : forget.Faithful :=
  show (Monad.forget _).Faithful from inferInstance

noncomputable instance : CreatesLimits forget :=
  show CreatesLimits <| Monad.forget _ from inferInstance

/-- The "free" Compactum functor. -/
def free : Type _ ⥤ Compactum :=
  Monad.free _

/-- The adjunction between `free` and `forget`. -/
def adj : free ⊣ forget :=
  Monad.adj _

instance : CoeSort Compactum Type* :=
  ⟨fun X => X.A⟩

instance {X Y : Compactum} : FunLike (X ⟶ Y) X Y where
  coe f := f.f
  coe_injective _ _ h := (Monad.forget_faithful β).map_injective (by aesop)

-- Basic instances
instance : ConcreteCategory Compactum (· ⟶ ·) where
  hom f := f
  ofHom f := f

instance : HasLimits Compactum :=
  hasLimits_of_hasLimits_createsLimits forget

/-- The structure map for a compactum, essentially sending an ultrafilter to its limit. -/
def str (X : Compactum) : Ultrafilter X → X :=
  X.a

/-- The monadic join. -/
def join (X : Compactum) : Ultrafilter (Ultrafilter X) → Ultrafilter X :=
  (β).μ.app _

/-- The inclusion of `X` into `Ultrafilter X`. -/
def incl (X : Compactum) : X → Ultrafilter X :=
  (β).η.app _

@[simp]
theorem str_incl (X : Compactum) (x : X) : X.str (X.incl x) = x := by
  change ((β).η.app _ ≫ X.a) _ = _
  rw [Monad.Algebra.unit]
  rfl

@[simp]
theorem str_hom_commute (X Y : Compactum) (f : X ⟶ Y) (xs : Ultrafilter X) :
    f (X.str xs) = Y.str (map f xs) := by
  change (X.a ≫ f.f) _ = _
  rw [← f.h]
  rfl

@[simp]
theorem join_distrib (X : Compactum) (uux : Ultrafilter (Ultrafilter X)) :
    X.str (X.join uux) = X.str (map X.str uux) := by
  change ((β).μ.app _ ≫ X.a) _ = _
  rw [Monad.Algebra.assoc]
  rfl

instance {X : Compactum} : TopologicalSpace X where
  IsOpen U := ∀ F : Ultrafilter X, X.str F ∈ U → U ∈ F
  isOpen_univ _ _ := Filter.univ_sets _
  isOpen_inter _ _ h3 h4 _ h6 := Filter.inter_sets _ (h3 _ h6.1) (h4 _ h6.2)
  isOpen_sUnion := fun _ h1 _ ⟨T, hT, h2⟩ =>
    mem_of_superset (h1 T hT _ h2) (Set.subset_sUnion_of_mem hT)

theorem isClosed_iff {X : Compactum} (S : Set X) :
    IsClosed S ↔ ∀ F : Ultrafilter X, S ∈ F → X.str F ∈ S := by
  rw [← isOpen_compl_iff]
  constructor
  · intro cond F h
    by_contra c
    specialize cond F c
    rw [compl_mem_iff_notMem] at cond
    contradiction
  · intro h1 F h2
    specialize h1 F
    rcases F.mem_or_compl_mem S with h | h
    exacts [absurd (h1 h) h2, h]

instance {X : Compactum} : CompactSpace X := by
  constructor
  rw [isCompact_iff_ultrafilter_le_nhds]
  intro F _
  refine ⟨X.str F, by tauto, ?_⟩
  rw [le_nhds_iff]
  intro S h1 h2
  exact h2 F h1

/-- A local definition used only in the proofs. -/
private def basic {X : Compactum} (A : Set X) : Set (Ultrafilter X) :=
  { F | A ∈ F }

set_option backward.privateInPublic true in
/-- A local definition used only in the proofs. -/
private def cl {X : Compactum} (A : Set X) : Set X :=
  X.str '' basic A

private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · intro hG
    constructor <;> filter_upwards [hG] with _
    exacts [And.left, And.right]
  · rintro ⟨h1, h2⟩
    exact inter_mem h1 h2

private theorem subset_cl {X : Compactum} (A : Set X) : A ⊆ cl A := fun a ha =>
  ⟨X.incl a, ha, by simp⟩

private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine ⟨P, hq2, ?_⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      rcases this P hP with h | h
      · exact claim2 _ h
      · rcases h with ⟨Q, hQ, rfl⟩
        exact claim3 _ hQ
    intro P hP
    exact claim5.finiteInterClosure_insert _ hP
  intro T hT
  -- Suffices to show that the intersection of the T's is contained in C2.
  suffices ⋂₀ ι T ∈ C2 by exact claim6 _ this
  -- Finish
  apply claim4.finiteInter_mem T
  intro t ht
  exact finiteInterClosure.basic (@hT t ht)

set_option backward.privateInPublic true in
set_option backward.privateInPublic.warn false in

Target:
theorem isClosed_cl {X : Compactum} (A : Set X) : IsClosed (cl A) :=

Failed candidate:
by
  have __repair_f0dc642b89 := True.intro.nonexistentProjection
  rw [isClosed_iff]
  intro F hF
  exact cl_cl _ ⟨F, hF, rfl⟩

Structured Lean error:
{"kind": "invalid_projection", "message": "<LEAN_FILE>:271:41: error(lean.invalidField): Invalid field `nonexistentProjection`: The environment does not contain `True.nonexistentProjection`, so it is not possible to project the field `nonexistentProjection` from an expression\n  True.intro\nof type `True`\n<LEAN_FILE>:270:0: error: unsolved goals\nX : Compactum\nA : Set X.A\n⊢ IsClosed (cl A)", "returncode": 1}

Corrected proof:


<LEAN_FILE>:125:2: warning: declaration uses `sorry`
<LEAN_FILE>:129:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern
  (fun x => x + 17288228) 0
in the target expression
  S.rightHom (S.inl n) = 1

N : Type u_1
E : Type u_2
G : Type u_3
inst✝² : Group N
inst✝¹ : Group E
inst✝ : Group G
S : GroupExtension N E G
n : N
⊢ S.rightHom (S.inl n) = 1

Response:
by
  rw [← MonoidHom.mem_ker, ← S.range_inl_eq_ker_rightHom, MonoidHom.mem_range]
  exact exists_apply_eq_apply S.inl n

[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.GroupTheory.GroupAction.ConjAct
public import Mathlib.GroupTheory.SemidirectProduct

Namespace:
GroupExtension

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

This file defines extensions of multiplicative and additive groups and their associated structures
such as splittings and equivalences.

## Main definitions

- `(Add?)GroupExtension N E G`: structure for extensions of `G` by `N` as short exact sequences
  `1 → N → E → G → 1` (`0 → N → E → G → 0` for additive groups)
- `(Add?)GroupExtension.Equiv S S'`: structure for equivalences of two group extensions `S` and `S'`
  as specific homomorphisms `E → E'` such that each diagram below is commutative

```text
For multiplicative groups:
      ↗ E  ↘
1 → N   ↓    G → 1
      ↘ E' ↗

For additive groups:
      ↗ E  ↘
0 → N   ↓    G → 0
      ↘ E' ↗
```

- `(Add?)GroupExtension.Section S`: structure for right inverses to `rightHom` of a group extension
  `S` of `G` by `N`
- `(Add?)GroupExtension.Splitting S`: structure for section homomorphisms of a group extension `S`
  of `G` by `N`
- `SemidirectProduct.toGroupExtension φ`: the multiplicative group extension associated to the
  semidirect product coming from `φ : G →* MulAut N`, `1 → N → N ⋊[φ] G → G → 1`

## TODO

If `N` is abelian,

- there is a bijection between `N`-conjugacy classes of
  `(SemidirectProduct.toGroupExtension φ).Splitting` and `groupCohomology.H1`
  (which will be available in the planned file `Mathlib/GroupTheory/GroupExtension/Abelian.lean` to
  be added in a later PR).
- there is a bijection between equivalence classes of group extensions and `groupCohomology.H2`
  (which is also stated as a TODO in `Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean`).
-/

@[expose] public section

variable (N E G : Type*)

/-- `AddGroupExtension N E G` is a short exact sequence of additive groups `0 → N → E → G → 0`. -/
structure AddGroupExtension [AddGroup N] [AddGroup E] [AddGroup G] where
  /-- The inclusion homomorphism `N →+ E` -/
  inl : N →+ E
  /-- The projection homomorphism `E →+ G` -/
  rightHom : E →+ G
  /-- The inclusion map is injective. -/
  inl_injective : Function.Injective inl
  /-- The range of the inclusion map is equal to the kernel of the projection map. -/
  range_inl_eq_ker_rightHom : inl.range = rightHom.ker
  /-- The projection map is surjective. -/
  rightHom_surjective : Function.Surjective rightHom

/-- `GroupExtension N E G` is a short exact sequence of groups `1 → N → E → G → 1`. -/
@[to_additive]
structure GroupExtension [Group N] [Group E] [Group G] where
  /-- The inclusion homomorphism `N →* E` -/
  inl : N →* E
  /-- The projection homomorphism `E →* G` -/
  rightHom : E →* G
  /-- The inclusion map is injective. -/
  inl_injective : Function.Injective inl
  /-- The range of the inclusion map is equal to the kernel of the projection map. -/
  range_inl_eq_ker_rightHom : inl.range = rightHom.ker
  /-- The projection map is surjective. -/
  rightHom_surjective : Function.Surjective rightHom

variable {N E G}

namespace AddGroupExtension

variable [AddGroup N] [AddGroup E] [AddGroup G] (S : AddGroupExtension N E G)

/-- `AddGroupExtension`s are equivalent iff there is an isomorphism making a commuting diagram.
  Use `AddGroupExtension.Equiv.ofMonoidHom` in `Mathlib/GroupTheory/GroupExtension/Basic.lean` to
  construct an equivalence without providing the inverse map. -/
structure Equiv {E' : Type*} [AddGroup E'] (S' : AddGroupExtension N E' G) extends E ≃+ E' where
  /-- The left-hand side of the diagram commutes. -/
  inl_comm : toAddEquiv ∘ S.inl = S'.inl
  /-- The right-hand side of the diagram commutes. -/
  rightHom_comm : S'.rightHom ∘ toAddEquiv = S.rightHom

/-- `Section` of an additive group extension is a right inverse to `S.rightHom`. -/
structure Section where
  /-- The underlying function -/
  toFun : G → E
  /-- `Section` is a right inverse to `S.rightHom` -/
  rightInverse_rightHom : Function.RightInverse toFun S.rightHom

/-- `Splitting` of an additive group extension is a section homomorphism. -/
structure Splitting extends G →+ E, S.Section

/-- A splitting of an additive group extension as a (set-theoretic) section. -/
add_decl_doc Splitting.toSection

end AddGroupExtension

namespace GroupExtension

variable [Group N] [Group E] [Group G] (S : GroupExtension N E G)

/-- The range of the inclusion map is a normal subgroup. -/
@[to_additive /-- The range of the inclusion map is a normal additive subgroup. -/]
instance normal_inl_range : S.inl.range.Normal :=
  S.range_inl_eq_ker_rightHom ▸ S.rightHom.normal_ker

@[to_additive (attr := simp)]

Target:
theorem rightHom_inl (n : N) : S.rightHom (S.inl n) = 1 :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 17288228) 0) = ((fun x : Nat => x + 17288228) 0) by rfl]
  rw [← MonoidHom.mem_ker, ← S.range_inl_eq_ker_rightHom, MonoidHom.mem_range]
  exact exists_apply_eq_apply S.inl n

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:125:2: warning: declaration uses `sorry`\n<LEAN_FILE>:129:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 17288228) 0\nin the target expression\n  S.rightHom (S.inl n) = 1\n\nN : Type u_1\nE : Type u_2\nG : Type u_3\ninst✝² : Group N\ninst✝¹ : Group E\ninst✝ : Group G\nS : GroupExtension N E G\nn : N\n⊢ S.rightHom (S.inl n) = 1", "returncode": 1}

Corrected proof:

[Assistant]
by
  rw [← MonoidHom.mem_ker, ← S.range_inl_eq_ker_rightHom, MonoidHom.mem_range]
  exact exists_apply_eq_apply S.inl n

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.GroupTheory.GroupAction.ConjAct
public import Mathlib.GroupTheory.SemidirectProduct

Namespace:
GroupExtension

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

This file defines extensions of multiplicative and additive groups and their associated structures
such as splittings and equivalences.

## Main definitions

- `(Add?)GroupExtension N E G`: structure for extensions of `G` by `N` as short exact sequences
  `1 → N → E → G → 1` (`0 → N → E → G → 0` for additive groups)
- `(Add?)GroupExtension.Equiv S S'`: structure for equivalences of two group extensions `S` and `S'`
  as specific homomorphisms `E → E'` such that each diagram below is commutative

```text
For multiplicative groups:
      ↗ E  ↘
1 → N   ↓    G → 1
      ↘ E' ↗

For additive groups:
      ↗ E  ↘
0 → N   ↓    G → 0
      ↘ E' ↗
```

- `(Add?)GroupExtension.Section S`: structure for right inverses to `rightHom` of a group extension
  `S` of `G` by `N`
- `(Add?)GroupExtension.Splitting S`: structure for section homomorphisms of a group extension `S`
  of `G` by `N`
- `SemidirectProduct.toGroupExtension φ`: the multiplicative group extension associated to the
  semidirect product coming from `φ : G →* MulAut N`, `1 → N → N ⋊[φ] G → G → 1`

## TODO

If `N` is abelian,

- there is a bijection between `N`-conjugacy classes of
  `(SemidirectProduct.toGroupExtension φ).Splitting` and `groupCohomology.H1`
  (which will be available in the planned file `Mathlib/GroupTheory/GroupExtension/Abelian.lean` to
  be added in a later PR).
- there is a bijection between equivalence classes of group extensions and `groupCohomology.H2`
  (which is also stated as a TODO in `Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean`).
-/

@[expose] public section

variable (N E G : Type*)

/-- `AddGroupExtension N E G` is a short exact sequence of additive groups `0 → N → E → G → 0`. -/
structure AddGroupExtension [AddGroup N] [AddGroup E] [AddGroup G] where
  /-- The inclusion homomorphism `N →+ E` -/
  inl : N →+ E
  /-- The projection homomorphism `E →+ G` -/
  rightHom : E →+ G
  /-- The inclusion map is injective. -/
  inl_injective : Function.Injective inl
  /-- The range of the inclusion map is equal to the kernel of the projection map. -/
  range_inl_eq_ker_rightHom : inl.range = rightHom.ker
  /-- The projection map is surjective. -/
  rightHom_surjective : Function.Surjective rightHom

/-- `GroupExtension N E G` is a short exact sequence of groups `1 → N → E → G → 1`. -/
@[to_additive]
structure GroupExtension [Group N] [Group E] [Group G] where
  /-- The inclusion homomorphism `N →* E` -/
  inl : N →* E
  /-- The projection homomorphism `E →* G` -/
  rightHom : E →* G
  /-- The inclusion map is injective. -/
  inl_injective : Function.Injective inl
  /-- The range of the inclusion map is equal to the kernel of the projection map. -/
  range_inl_eq_ker_rightHom : inl.range = rightHom.ker
  /-- The projection map is surjective. -/
  rightHom_surjective : Function.Surjective rightHom

variable {N E G}

namespace AddGroupExtension

variable [AddGroup N] [AddGroup E] [AddGroup G] (S : AddGroupExtension N E G)

/-- `AddGroupExtension`s are equivalent iff there is an isomorphism making a commuting diagram.
  Use `AddGroupExtension.Equiv.ofMonoidHom` in `Mathlib/GroupTheory/GroupExtension/Basic.lean` to
  construct an equivalence without providing the inverse map. -/
structure Equiv {E' : Type*} [AddGroup E'] (S' : AddGroupExtension N E' G) extends E ≃+ E' where
  /-- The left-hand side of the diagram commutes. -/
  inl_comm : toAddEquiv ∘ S.inl = S'.inl
  /-- The right-hand side of the diagram commutes. -/
  rightHom_comm : S'.rightHom ∘ toAddEquiv = S.rightHom

/-- `Section` of an additive group extension is a right inverse to `S.rightHom`. -/
structure Section where
  /-- The underlying function -/
  toFun : G → E
  /-- `Section` is a right inverse to `S.rightHom` -/
  rightInverse_rightHom : Function.RightInverse toFun S.rightHom

/-- `Splitting` of an additive group extension is a section homomorphism. -/
structure Splitting extends G →+ E, S.Section

/-- A splitting of an additive group extension as a (set-theoretic) section. -/
add_decl_doc Splitting.toSection

end AddGroupExtension

namespace GroupExtension

variable [Group N] [Group E] [Group G] (S : GroupExtension N E G)

/-- The range of the inclusion map is a normal subgroup. -/
@[to_additive /-- The range of the inclusion map is a normal additive subgroup. -/]
instance normal_inl_range : S.inl.range.Normal :=
  S.range_inl_eq_ker_rightHom ▸ S.rightHom.normal_ker

@[to_additive (attr := simp)]

Target:
theorem rightHom_inl (n : N) : S.rightHom (S.inl n) = 1 :=

Failed candidate:
by
  rw [show ((fun x : Nat => x + 17288228) 0) = ((fun x : Nat => x + 17288228) 0) by rfl]
  rw [← MonoidHom.mem_ker, ← S.range_inl_eq_ker_rightHom, MonoidHom.mem_range]
  exact exists_apply_eq_apply S.inl n

Structured Lean error:
{"kind": "rewrite_no_match", "message": "<LEAN_FILE>:125:2: warning: declaration uses `sorry`\n<LEAN_FILE>:129:6: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  (fun x => x + 17288228) 0\nin the target expression\n  S.rightHom (S.inl n) = 1\n\nN : Type u_1\nE : Type u_2\nG : Type u_3\ninst✝² : Group N\ninst✝¹ : Group E\ninst✝ : Group G\nS : GroupExtension N E G\nn : N\n⊢ S.rightHom (S.inl n) = 1", "returncode": 1}

Corrected proof:


Response:
by
  simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]

[User]
Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.

Imports:
public import Mathlib.Order.Filter.Ultrafilter.Basic
public import Mathlib.Topology.Continuous

Namespace:
(root)

Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
/-! # Characterization of basic topological properties in terms of ultrafilters -/

public section

open Set Filter Topology

universe u v w x

variable {X : Type u} {Y : Type v} {ι : Sort w} {α β : Type*} {x : X} {s s₁ s₂ t : Set X}
    {p p₁ p₂ : X → Prop} [TopologicalSpace X] [TopologicalSpace Y] {F : Filter α} {u : α → X}

theorem Ultrafilter.clusterPt_iff {f : Ultrafilter X} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
  ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩

theorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f ↔
    ∃ u : Ultrafilter X, u ≤ f ∧ u ≤ 𝓝 x := by
  simp_rw [ClusterPt, ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem mapClusterPt_iff_ultrafilter :
    MapClusterPt x F u ↔ ∃ U : Ultrafilter α, U ≤ F ∧ Tendsto u U (𝓝 x) := by
  simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, tendsto_iff_comap,
    ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem isOpen_iff_ultrafilter :
    IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l := by
  simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]

/-- `x` belongs to the closure of `s` if and only if some ultrafilter
  supported on `s` converges to `x`. -/

Target:
theorem mem_closure_iff_ultrafilter :
    x ∈ closure s ↔ ∃ u : Ultrafilter X, s ∈ u ∧ ↑u ≤ 𝓝 x :=

Proof body:

[Assistant]
by
  simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]

Prompt:
Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.

Imports:
public import Mathlib.Order.Filter.Ultrafilter.Basic
public import Mathlib.Topology.Continuous

Namespace:
(root)

Local context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
/-! # Characterization of basic topological properties in terms of ultrafilters -/

public section

open Set Filter Topology

universe u v w x

variable {X : Type u} {Y : Type v} {ι : Sort w} {α β : Type*} {x : X} {s s₁ s₂ t : Set X}
    {p p₁ p₂ : X → Prop} [TopologicalSpace X] [TopologicalSpace Y] {F : Filter α} {u : α → X}

theorem Ultrafilter.clusterPt_iff {f : Ultrafilter X} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
  ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩

theorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f ↔
    ∃ u : Ultrafilter X, u ≤ f ∧ u ≤ 𝓝 x := by
  simp_rw [ClusterPt, ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem mapClusterPt_iff_ultrafilter :
    MapClusterPt x F u ↔ ∃ U : Ultrafilter α, U ≤ F ∧ Tendsto u U (𝓝 x) := by
  simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, tendsto_iff_comap,
    ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem isOpen_iff_ultrafilter :
    IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l := by
  simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]

/-- `x` belongs to the closure of `s` if and only if some ultrafilter
  supported on `s` converges to `x`. -/

Target:
theorem mem_closure_iff_ultrafilter :
    x ∈ closure s ↔ ∃ u : Ultrafilter X, s ∈ u ∧ ↑u ≤ 𝓝 x :=

Proof body:


<LEAN_FILE>:44:42: 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
  simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]

[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.Order.Filter.Ultrafilter.Basic
public import Mathlib.Topology.Continuous

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
/-! # Characterization of basic topological properties in terms of ultrafilters -/

public section

open Set Filter Topology

universe u v w x

variable {X : Type u} {Y : Type v} {ι : Sort w} {α β : Type*} {x : X} {s s₁ s₂ t : Set X}
    {p p₁ p₂ : X → Prop} [TopologicalSpace X] [TopologicalSpace Y] {F : Filter α} {u : α → X}

theorem Ultrafilter.clusterPt_iff {f : Ultrafilter X} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
  ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩

theorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f ↔
    ∃ u : Ultrafilter X, u ≤ f ∧ u ≤ 𝓝 x := by
  simp_rw [ClusterPt, ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem mapClusterPt_iff_ultrafilter :
    MapClusterPt x F u ↔ ∃ U : Ultrafilter α, U ≤ F ∧ Tendsto u U (𝓝 x) := by
  simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, tendsto_iff_comap,
    ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem isOpen_iff_ultrafilter :
    IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l := by
  simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]

/-- `x` belongs to the closure of `s` if and only if some ultrafilter
  supported on `s` converges to `x`. -/

Target:
theorem mem_closure_iff_ultrafilter :
    x ∈ closure s ↔ ∃ u : Ultrafilter X, s ∈ u ∧ ↑u ≤ 𝓝 x :=

Failed candidate:
by
  let __repair_fc8b5dd0c0 : Fact False := inferInstance
  simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:44:42: 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
  simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]

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.Order.Filter.Ultrafilter.Basic
public import Mathlib.Topology.Continuous

Namespace:
(root)

Local safe context:
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
/-! # Characterization of basic topological properties in terms of ultrafilters -/

public section

open Set Filter Topology

universe u v w x

variable {X : Type u} {Y : Type v} {ι : Sort w} {α β : Type*} {x : X} {s s₁ s₂ t : Set X}
    {p p₁ p₂ : X → Prop} [TopologicalSpace X] [TopologicalSpace Y] {F : Filter α} {u : α → X}

theorem Ultrafilter.clusterPt_iff {f : Ultrafilter X} : ClusterPt x f ↔ ↑f ≤ 𝓝 x :=
  ⟨f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h⟩

theorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f ↔
    ∃ u : Ultrafilter X, u ≤ f ∧ u ≤ 𝓝 x := by
  simp_rw [ClusterPt, ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem mapClusterPt_iff_ultrafilter :
    MapClusterPt x F u ↔ ∃ U : Ultrafilter α, U ≤ F ∧ Tendsto u U (𝓝 x) := by
  simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, tendsto_iff_comap,
    ← le_inf_iff, exists_ultrafilter_iff, inf_comm]

theorem isOpen_iff_ultrafilter :
    IsOpen s ↔ ∀ x ∈ s, ∀ (l : Ultrafilter X), ↑l ≤ 𝓝 x → s ∈ l := by
  simp_rw [isOpen_iff_mem_nhds, ← mem_iff_ultrafilter]

/-- `x` belongs to the closure of `s` if and only if some ultrafilter
  supported on `s` converges to `x`. -/

Target:
theorem mem_closure_iff_ultrafilter :
    x ∈ closure s ↔ ∃ u : Ultrafilter X, s ∈ u ∧ ↑u ≤ 𝓝 x :=

Failed candidate:
by
  let __repair_fc8b5dd0c0 : Fact False := inferInstance
  simp [closure_eq_cluster_pts, ClusterPt, ← exists_ultrafilter_iff, and_comm]

Structured Lean error:
{"kind": "typeclass_synthesis", "message": "<LEAN_FILE>:44:42: 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: