-- Module: Mathlib.RingTheory.Smooth.Kaehler
Goal:
R : Type u_1
P : Type u_2
S : Type u_3
inst : CommRing R
inst_1 : CommRing P
inst_2 : CommRing S
inst_3 : Algebra R P
inst_4 : Algebra P S
inst_5 : Algebra R S
inst_6 : IsScalarTower R P S
hf' : HPow.hPow (α := Ideal P) (β := ℕ) (RingHom.ker (F := P →+* S) (algebraMap P S)) 2 = ⊥
hf : Surjective.{u_2 + 1, u_3 + 1} (α := P) (β := S) (⇑(algebraMap P S) : (a : P) → S)
P' : Type u_2 := P ⧸ HPow.hPow (α := Ideal P) (β := ℕ) (RingHom.ker (F := P →+* S) (algebraMap P S)) 2
h₁ : Surjective.{u_2 + 1, u_3 + 1} (α := P') (β := S) (⇑(algebraMap P' S) : (a : P') → S)
h₂ : HPow.hPow (α := Ideal P') (β := ℕ) (RingHom.ker (F := P' →+* S) (algebraMap P' S)) 2 = ⊥
e₁ : Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)) ≃ₗ[P]
  ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S)) :=
  Ideal.cotangentEquivIdeal (RingHom.ker (F := P →+* S) (algebraMap P S)) ≪≫ₗ
    LinearEquiv.restrictScalars P
      (LinearEquiv.ofEq
        (Ideal.cotangentIdeal (RingHom.ker (F := P →+* S) (AlgHom.toRingHom (IsScalarTower.toAlgHom R P S))))
        (RingHom.ker (F :=
          RingHom.{u_2, u_3}
            (P ⧸
              HPow.hPow (α := Ideal P) (β := ℕ)
                (RingHom.ker (F := P →+* S) (AlgHom.toRingHom (IsScalarTower.toAlgHom R P S))) 2)
            S)
          (AlgHom.toRingHom (AlgHom.kerSquareLift (IsScalarTower.toAlgHom R P S))))
        ⋯)
e₂ : S ⊗[P'] Ω[P'⁄R] ≃ₗ[P] S ⊗[P] Ω[P⁄R] := LinearEquiv.restrictScalars P (tensorKaehlerQuotKerSqEquiv R P S)
H :
  kerCotangentToTensor R P S =
    (↑e₂ : S ⊗[P'] Ω[P'⁄R] →ₗ[P] S ⊗[P] Ω[P⁄R]) ∘ₗ
      (↑P :
            {S M : Type u_2} →
              {M₂ : Type (max u_2 u_3)} →
                [inst : Semiring P] →
                  [inst_7 : Semiring S] →
                    [inst_8 : AddCommMonoid M] →
                      [inst_9 : AddCommMonoid M₂] →
                        [inst_10 : Module P M] →
                          [inst_11 : Module P M₂] →
                            [inst_12 : Module S M] →
                              [inst_13 : Module S M₂] →
                                [LinearMap.CompatibleSMul M M₂ P S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[P] M₂)
          (kerToTensor R P' S) ∘ₗ
        (↑e₁ : (RingHom.ker (algebraMap P S)).Cotangent →ₗ[P] ↥(RingHom.ker (algebraMap P' S)))
refine_1 :
  ∀
    (x :
      { l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)) //
        l ∘ₗ kerCotangentToTensor R P S = LinearMap.id })
    (l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)))
    (hl : l ∘ₗ kerCotangentToTensor R P S = LinearMap.id) (m : P') (x : S ⊗[P'] Ω[P'⁄R]),
    AddHom.toFun
        (↑((↑e₁ : (RingHom.ker (algebraMap P S)).Cotangent →ₗ[P] ↥(RingHom.ker (algebraMap P' S))) ∘ₗ
              l ∘ₗ (↑e₂ : S ⊗[P'] Ω[P'⁄R] →ₗ[P] S ⊗[P] Ω[P⁄R])) :
          S ⊗[P'] Ω[P'⁄R] →ₙ+ ↥(RingHom.ker (algebraMap P' S)))
        (m • x) =
      HSMul.hSMul (α := P') ((RingHom.id P' : (a : P') → P') m : P')
        (AddHom.toFun
          (↑((↑e₁ : (RingHom.ker (algebraMap P S)).Cotangent →ₗ[P] ↥(RingHom.ker (algebraMap P' S))) ∘ₗ
                l ∘ₗ (↑e₂ : S ⊗[P'] Ω[P'⁄R] →ₗ[P] S ⊗[P] Ω[P⁄R])) :
            S ⊗[P'] Ω[P'⁄R] →ₙ+ ↥(RingHom.ker (algebraMap P' S)))
          x)
refine_2 :
  ∀
    (x :
      { l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)) //
        l ∘ₗ kerCotangentToTensor R P S = LinearMap.id })
    (l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)))
    (hl : l ∘ₗ kerCotangentToTensor R P S = LinearMap.id),
    {
          toAddHom :=
            (↑((↑e₁ : (RingHom.ker (algebraMap P S)).Cotangent →ₗ[P] ↥(RingHom.ker (algebraMap P' S))) ∘ₗ
                  l ∘ₗ (↑e₂ : S ⊗[P'] Ω[P'⁄R] →ₗ[P] S ⊗[P] Ω[P⁄R])) :
              S ⊗[P'] Ω[P'⁄R] →ₙ+ ↥(RingHom.ker (algebraMap P' S))),
          map_smul' := ⋯ } ∘ₗ
        kerToTensor R P' S =
      LinearMap.id
refine_3 :
  ∀
    (x :
      { l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S)) //
        l ∘ₗ kerToTensor R P' S = LinearMap.id })
    (l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S)))
    (hl : l ∘ₗ kerToTensor R P' S = LinearMap.id),
    ((↑e₁.symm : ↥(RingHom.ker (algebraMap P' S)) →ₗ[P] (RingHom.ker (algebraMap P S)).Cotangent) ∘ₗ
          (↑P :
                {S : Type u_2} →
                  {M : Type (max u_2 u_3)} →
                    {M₂ : Type u_2} →
                      [inst : Semiring P] →
                        [inst_7 : Semiring S] →
                          [inst_8 : AddCommMonoid M] →
                            [inst_9 : AddCommMonoid M₂] →
                              [inst_10 : Module P M] →
                                [inst_11 : Module P M₂] →
                                  [inst_12 : Module S M] →
                                    [inst_13 : Module S M₂] →
                                      [LinearMap.CompatibleSMul M M₂ P S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[P] M₂)
              l ∘ₗ
            (↑e₂.symm : S ⊗[P] Ω[P⁄R] →ₗ[P] S ⊗[P'] Ω[P'⁄R])) ∘ₗ
        kerCotangentToTensor R P S =
      LinearMap.id
refine_4 :
  Function.LeftInverse
    (fun
        (x :
          { l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S)) //
            l ∘ₗ kerToTensor R P' S = LinearMap.id }) ↦
      match (motive :=
        { l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S)) //
            l ∘ₗ kerToTensor R P' S = LinearMap.id } →
          { l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)) //
            l ∘ₗ kerCotangentToTensor R P S = LinearMap.id })
        x with
      | ⟨l, hl⟩ =>
        Subtype.mk (p := fun (l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S))) ↦
          l ∘ₗ kerCotangentToTensor R P S = LinearMap.id)
          ((↑e₁.symm : ↥(RingHom.ker (algebraMap P' S)) →ₗ[P] (RingHom.ker (algebraMap P S)).Cotangent) ∘ₗ
            (↑P :
                  {S : Type u_2} →
                    {M : Type (max u_2 u_3)} →
                      {M₂ : Type u_2} →
                        [inst : Semiring P] →
                          [inst_7 : Semiring S] →
                            [inst_8 : AddCommMonoid M] →
                              [inst_9 : AddCommMonoid M₂] →
                                [inst_10 : Module P M] →
                                  [inst_11 : Module P M₂] →
                                    [inst_12 : Module S M] →
                                      [inst_13 : Module S M₂] →
                                        [LinearMap.CompatibleSMul M M₂ P S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[P] M₂)
                l ∘ₗ
              (↑e₂.symm : S ⊗[P] Ω[P⁄R] →ₗ[P] S ⊗[P'] Ω[P'⁄R]))
          ⋯)
    fun
      (x :
        { l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)) //
          l ∘ₗ kerCotangentToTensor R P S = LinearMap.id }) ↦
    match (motive :=
      { l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)) //
          l ∘ₗ kerCotangentToTensor R P S = LinearMap.id } →
        { l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S)) //
          l ∘ₗ kerToTensor R P' S = LinearMap.id })
      x with
    | ⟨l, hl⟩ =>
      Subtype.mk (p := fun (l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S))) ↦
        l ∘ₗ kerToTensor R P' S = LinearMap.id)
        {
          toAddHom :=
            (↑((↑e₁ : (RingHom.ker (algebraMap P S)).Cotangent →ₗ[P] ↥(RingHom.ker (algebraMap P' S))) ∘ₗ
                  l ∘ₗ (↑e₂ : S ⊗[P'] Ω[P'⁄R] →ₗ[P] S ⊗[P] Ω[P⁄R])) :
              S ⊗[P'] Ω[P'⁄R] →ₙ+ ↥(RingHom.ker (algebraMap P' S))),
          map_smul' := ⋯ }
        ⋯
refine_5 :
  Function.RightInverse
    (fun
        (x :
          { l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S)) //
            l ∘ₗ kerToTensor R P' S = LinearMap.id }) ↦
      match (motive :=
        { l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S)) //
            l ∘ₗ kerToTensor R P' S = LinearMap.id } →
          { l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)) //
            l ∘ₗ kerCotangentToTensor R P S = LinearMap.id })
        x with
      | ⟨l, hl⟩ =>
        Subtype.mk (p := fun (l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S))) ↦
          l ∘ₗ kerCotangentToTensor R P S = LinearMap.id)
          ((↑e₁.symm : ↥(RingHom.ker (algebraMap P' S)) →ₗ[P] (RingHom.ker (algebraMap P S)).Cotangent) ∘ₗ
            (↑P :
                  {S : Type u_2} →
                    {M : Type (max u_2 u_3)} →
                      {M₂ : Type u_2} →
                        [inst : Semiring P] →
                          [inst_7 : Semiring S] →
                            [inst_8 : AddCommMonoid M] →
                              [inst_9 : AddCommMonoid M₂] →
                                [inst_10 : Module P M] →
                                  [inst_11 : Module P M₂] →
                                    [inst_12 : Module S M] →
                                      [inst_13 : Module S M₂] →
                                        [LinearMap.CompatibleSMul M M₂ P S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[P] M₂)
                l ∘ₗ
              (↑e₂.symm : S ⊗[P] Ω[P⁄R] →ₗ[P] S ⊗[P'] Ω[P'⁄R]))
          ⋯)
    fun
      (x :
        { l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)) //
          l ∘ₗ kerCotangentToTensor R P S = LinearMap.id }) ↦
    match (motive :=
      { l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)) //
          l ∘ₗ kerCotangentToTensor R P S = LinearMap.id } →
        { l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S)) //
          l ∘ₗ kerToTensor R P' S = LinearMap.id })
      x with
    | ⟨l, hl⟩ =>
      Subtype.mk (p := fun (l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S))) ↦
        l ∘ₗ kerToTensor R P' S = LinearMap.id)
        {
          toAddHom :=
            (↑((↑e₁ : (RingHom.ker (algebraMap P S)).Cotangent →ₗ[P] ↥(RingHom.ker (algebraMap P' S))) ∘ₗ
                  l ∘ₗ (↑e₂ : S ⊗[P'] Ω[P'⁄R] →ₗ[P] S ⊗[P] Ω[P⁄R])) :
              S ⊗[P'] Ω[P'⁄R] →ₙ+ ↥(RingHom.ker (algebraMap P' S))),
          map_smul' := ⋯ }
        ⋯
⊢ { l : S ⊗[P] Ω[P⁄R] →ₗ[P] Ideal.Cotangent (RingHom.ker (F := P →+* S) (algebraMap P S)) //
      l ∘ₗ kerCotangentToTensor R P S = LinearMap.id } ≃
    { l : S ⊗[P'] Ω[P'⁄R] →ₗ[P'] ↥(RingHom.ker (F := P' →+* S) (algebraMap P' S)) //
      l ∘ₗ kerToTensor R P' S = LinearMap.id }

Tactic:
refine ⟨fun ⟨l, hl⟩ ↦ ⟨⟨e₁.toLinearMap ∘ₗ l ∘ₗ e₂.toLinearMap, ?_⟩, ?_⟩,
    fun ⟨l, hl⟩ ↦ ⟨e₁.symm.toLinearMap ∘ₗ l.restrictScalars P ∘ₗ e₂.symm.toLinearMap, ?_⟩, ?_, ?_⟩