-- Module: Mathlib.LinearAlgebra.TensorProduct.Associator
Goal:
R : Type u_11
A : Type u_12
A' : Type u_13
B : Type u_14
B' : Type u_15
C : Type u_16
C' : Type u_17
inst : CommSemiring R
inst_1 : AddCommMonoid A'
inst_2 : AddCommMonoid B'
inst_3 : AddCommMonoid C'
inst_4 : Module R A'
inst_5 : Module R B'
inst_6 : Module R C'
eA : A ≃ A'
eB : B ≃ B'
eC : C ≃ C'
x : A ⊗[R] B ⊗[R] C
h :
  Eq (α := A ⊗[R] (B ⊗[R] C)) ((TensorProduct.assoc R A B C : (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C)) 0)
    ((TensorProduct.congr (TensorProduct.congr (linearEquiv R eA) (linearEquiv R eB)) (linearEquiv R eC) ≪≫ₗ
          (TensorProduct.assoc R A' B' C' ≪≫ₗ
            TensorProduct.congr (LinearEquiv.symm (linearEquiv R eA))
              (TensorProduct.congr (LinearEquiv.symm (linearEquiv R eB)) (LinearEquiv.symm (linearEquiv R eC)))) :
        (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C))
      0)
h₁ :
  ∀ (x y : A ⊗[R] B ⊗[R] C),
    Eq (α := A ⊗[R] (B ⊗[R] C))
        ((TensorProduct.assoc R A B C : (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C)) x : A ⊗[R] (B ⊗[R] C))
        ((TensorProduct.congr (TensorProduct.congr (linearEquiv R eA) (linearEquiv R eB)) (linearEquiv R eC) ≪≫ₗ
                (TensorProduct.assoc R A' B' C' ≪≫ₗ
                  TensorProduct.congr (LinearEquiv.symm (linearEquiv R eA))
                    (TensorProduct.congr (LinearEquiv.symm (linearEquiv R eB)) (LinearEquiv.symm (linearEquiv R eC)))) :
              (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C))
            x :
          A ⊗[R] (B ⊗[R] C)) →
      Eq (α := A ⊗[R] (B ⊗[R] C))
          ((TensorProduct.assoc R A B C : (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C)) y : A ⊗[R] (B ⊗[R] C))
          ((TensorProduct.congr (TensorProduct.congr (linearEquiv R eA) (linearEquiv R eB)) (linearEquiv R eC) ≪≫ₗ
                  (TensorProduct.assoc R A' B' C' ≪≫ₗ
                    TensorProduct.congr (LinearEquiv.symm (linearEquiv R eA))
                      (TensorProduct.congr (LinearEquiv.symm (linearEquiv R eB))
                        (LinearEquiv.symm (linearEquiv R eC)))) :
                (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C))
              y :
            A ⊗[R] (B ⊗[R] C)) →
        Eq (α := A ⊗[R] (B ⊗[R] C))
          ((TensorProduct.assoc R A B C : (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C)) (x + y) : A ⊗[R] (B ⊗[R] C))
          ((TensorProduct.congr (TensorProduct.congr (linearEquiv R eA) (linearEquiv R eB)) (linearEquiv R eC) ≪≫ₗ
                  (TensorProduct.assoc R A' B' C' ≪≫ₗ
                    TensorProduct.congr (LinearEquiv.symm (linearEquiv R eA))
                      (TensorProduct.congr (LinearEquiv.symm (linearEquiv R eB))
                        (LinearEquiv.symm (linearEquiv R eC)))) :
                (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C))
              (x + y) :
            A ⊗[R] (B ⊗[R] C))
h₂ :
  ∀ (x : A ⊗[R] B) (a : C),
    Eq (α := A ⊗[R] (B ⊗[R] C))
      ((TensorProduct.assoc R A B C : (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C)) (x ⊗ₜ[R] a) : A ⊗[R] (B ⊗[R] C))
      ((TensorProduct.congr (TensorProduct.congr (linearEquiv R eA) (linearEquiv R eB)) (linearEquiv R eC) ≪≫ₗ
              (TensorProduct.assoc R A' B' C' ≪≫ₗ
                TensorProduct.congr (LinearEquiv.symm (linearEquiv R eA))
                  (TensorProduct.congr (LinearEquiv.symm (linearEquiv R eB)) (LinearEquiv.symm (linearEquiv R eC)))) :
            (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C))
          (x ⊗ₜ[R] a) :
        A ⊗[R] (B ⊗[R] C))
⊢ Eq (α := A ⊗[R] (B ⊗[R] C))
    ((TensorProduct.assoc R A B C : (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C)) x : A ⊗[R] (B ⊗[R] C))
    ((TensorProduct.congr (TensorProduct.congr (linearEquiv R eA) (linearEquiv R eB)) (linearEquiv R eC) ≪≫ₗ
            (TensorProduct.assoc R A' B' C' ≪≫ₗ
              TensorProduct.congr (LinearEquiv.symm (linearEquiv R eA))
                (TensorProduct.congr (LinearEquiv.symm (linearEquiv R eB)) (LinearEquiv.symm (linearEquiv R eC)))) :
          (a : A ⊗[R] B ⊗[R] C) → A ⊗[R] (B ⊗[R] C))
        x :
      A ⊗[R] (B ⊗[R] C))

Tactic:
induction x with
  | zero => simp
  | add => simp [*]
  | tmul x a => induction x <;> simp [*, add_tmul]

Premises used: TensorProduct.add_tmul, TensorProduct.tmul