-- Module: Mathlib.RingTheory.Coalgebra.TensorProduct
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst : CommSemiring R
inst_1 : CommSemiring S
inst_2 : AddCommMonoid A
inst_3 : AddCommMonoid B
inst_4 : Algebra R S
inst_5 : Module R A
inst_6 : Module S A
inst_7 : Module R B
inst_8 : IsScalarTower R S A
inst_9 : Coalgebra R B
inst_10 : Coalgebra S A
x : A
y : B
F : TensorProduct.{u_1, u_3, u_4} R (A ⊗[S] (A ⊗[S] A)) (B ⊗[R] (B ⊗[R] B)) ≃ₗ[S]
  A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)) :=
  (AlgebraTensorModule.tensorTensorTensorComm R S R S A (A ⊗[S] A) B (B ⊗[R] B) :
      TensorProduct.{u_1, u_3, u_4} R (A ⊗[S] (A ⊗[S] A)) (B ⊗[R] (B ⊗[R] B)) ≃ₗ[S]
        A ⊗[R] B ⊗[S] TensorProduct.{u_1, u_3, u_4} R (A ⊗[S] A) (B ⊗[R] B)) ≪≫ₗ
    (AlgebraTensorModule.congr (LinearEquiv.refl S (A ⊗[R] B))
        (AlgebraTensorModule.tensorTensorTensorComm R S R S A A B B) :
      A ⊗[R] B ⊗[S] TensorProduct.{u_1, u_3, u_4} R (A ⊗[S] A) (B ⊗[R] B) ≃ₗ[S]
        A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
F' : TensorProduct.{u_1, u_3, u_4} R (A ⊗[S] (A ⊗[S] A)) (B ⊗[R] (B ⊗[R] B)) →ₗ[S]
  A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)) :=
  mapOfCompatibleSMul S S S (A ⊗[R] B) (A ⊗[R] B ⊗[S] (A ⊗[R] B)) ∘ₗ
    map LinearMap.id (mapOfCompatibleSMul S S S (A ⊗[R] B) (A ⊗[R] B)) ∘ₗ
      (↑F : A ⊗[S] (A ⊗[S] A) ⊗[R] (B ⊗[R] (B ⊗[R] B)) →ₗ[S] A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
x₁ x₂ : A
y₁ y₂ : B
x₁₁ x₁₂ : A
h :
  ∀ (y₁₁ y₁₂ : B),
    Eq (α := A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
      ((TensorProduct.assoc S (A ⊗[R] B) (A ⊗[R] B) (A ⊗[R] B) :
            (a : A ⊗[R] B ⊗[S] (A ⊗[R] B) ⊗[S] (A ⊗[R] B)) → A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
          (((AlgebraTensorModule.tensorTensorTensorComm R S R S A A B B :
                  TensorProduct.{u_1, u_3, u_4} R (A ⊗[S] A) (B ⊗[R] B) ≃ₗ[S] A ⊗[R] B ⊗[S] (A ⊗[R] B)) :
                (a : A ⊗[S] A ⊗[R] (B ⊗[R] B)) → A ⊗[R] B ⊗[S] (A ⊗[R] B))
              (x₁₁ ⊗ₜ[S] x₁₂ ⊗ₜ[R] (y₁₁ ⊗ₜ[R] y₁₂)) ⊗ₜ[S]
            (x₂ ⊗ₜ[R] y₂)) :
        A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
      ((F : (a : A ⊗[S] (A ⊗[S] A) ⊗[R] (B ⊗[R] (B ⊗[R] B))) → A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
          ((TensorProduct.assoc S A A A : (a : A ⊗[S] A ⊗[S] A) → A ⊗[S] (A ⊗[S] A)) (x₁₁ ⊗ₜ[S] x₁₂ ⊗ₜ[S] x₂) ⊗ₜ[R]
            (TensorProduct.assoc R B B B : (a : B ⊗[R] B ⊗[R] B) → B ⊗[R] (B ⊗[R] B)) (y₁₁ ⊗ₜ[R] y₁₂ ⊗ₜ[R] y₂)) :
        A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
⊢ Eq (α := A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
    ((TensorProduct.assoc S (A ⊗[R] B) (A ⊗[R] B) (A ⊗[R] B) :
          (a : A ⊗[R] B ⊗[S] (A ⊗[R] B) ⊗[S] (A ⊗[R] B)) → A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
        (((AlgebraTensorModule.tensorTensorTensorComm R S R S A A B B :
                TensorProduct.{u_1, u_3, u_4} R (A ⊗[S] A) (B ⊗[R] B) ≃ₗ[S] A ⊗[R] B ⊗[S] (A ⊗[R] B)) :
              (a : A ⊗[S] A ⊗[R] (B ⊗[R] B)) → A ⊗[R] B ⊗[S] (A ⊗[R] B))
            (x₁₁ ⊗ₜ[S] x₁₂ ⊗ₜ[R] (comul : (a : B) → B ⊗[R] B) y₁) ⊗ₜ[S]
          (x₂ ⊗ₜ[R] y₂)) :
      A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
    ((F : (a : A ⊗[S] (A ⊗[S] A) ⊗[R] (B ⊗[R] (B ⊗[R] B))) → A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))
        ((TensorProduct.assoc S A A A : (a : A ⊗[S] A ⊗[S] A) → A ⊗[S] (A ⊗[S] A)) (x₁₁ ⊗ₜ[S] x₁₂ ⊗ₜ[S] x₂) ⊗ₜ[R]
          (TensorProduct.assoc R B B B : (a : B ⊗[R] B ⊗[R] B) → B ⊗[R] (B ⊗[R] B))
            ((comul : (a : B) → B ⊗[R] B) y₁ ⊗ₜ[R] y₂)) :
      A ⊗[R] B ⊗[S] (A ⊗[R] B ⊗[S] (A ⊗[R] B)))

Tactic:
hopf_tensor_induction comul (R := R) y₁ with y₁₁ y₁₂

Premises used: CoalgebraStruct.comul, DFunLike.coe