Module: Mathlib.RingTheory.Coalgebra.TensorProduct

Goal state before:
pp: case e'_2
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid A
inst✝⁷ : AddCommMonoid B
inst✝⁶ : Algebra R S
inst✝⁵ : Module R A
inst✝⁴ : Module S A
inst✝³ : Module R B
inst✝² : IsScalarTower R S A
inst✝¹ : Coalgebra R B
inst✝ : Coalgebra S A
x : A
y : B
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)) :=
  AlgebraTensorModule.tensorTensorTensorComm R S R S A (A ⊗[S] A) B (B ⊗[R] B) ≪≫ₗ
    AlgebraTensorModule.congr (LinearEquiv.refl S (A ⊗[R] B))
      (AlgebraTensorModule.tensorTensorTensorComm R S R S A A B B)
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)) :=
  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
⊢ ((AlgebraTensorModule.curry
          (↑(TensorProduct.assoc S (A ⊗[R] B) (A ⊗[R] B) (A ⊗[R] B)) ∘ₗ LinearMap.rTensor (A ⊗[R] B) comul ∘ₗ comul))
        x)
      y =
    F
      ((TensorProduct.assoc S A A A) ((LinearMap.rTensor A comul) (comul x)) ⊗ₜ[R]
        (TensorProduct.assoc R B B B) ((LinearMap.rTensor B comul) (comul y)))

pp: case e'_3
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid A
inst✝⁷ : AddCommMonoid B
inst✝⁶ : Algebra R S
inst✝⁵ : Module R A
inst✝⁴ : Module S A
inst✝³ : Module R B
inst✝² : IsScalarTower R S A
inst✝¹ : Coalgebra R B
inst✝ : Coalgebra S A
x : A
y : B
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)) :=
  AlgebraTensorModule.tensorTensorTensorComm R S R S A (A ⊗[S] A) B (B ⊗[R] B) ≪≫ₗ
    AlgebraTensorModule.congr (LinearEquiv.refl S (A ⊗[R] B))
      (AlgebraTensorModule.tensorTensorTensorComm R S R S A A B B)
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)) :=
  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
⊢ ((AlgebraTensorModule.curry (LinearMap.lTensor (A ⊗[R] B) comul ∘ₗ comul)) x) y =
    F ((LinearMap.lTensor A comul) (comul x) ⊗ₜ[R] (LinearMap.lTensor B comul) (comul y))

Tactic:
· dsimp
    hopf_tensor_induction comul (R := S) x with x₁ x₂
    hopf_tensor_induction comul (R := R) y with y₁ y₂
    dsimp
    hopf_tensor_induction comul (R := S) x₁ with x₁₁ x₁₂
    hopf_tensor_induction comul (R := R) y₁ with y₁₁ y₁₂
    rfl

Goal state after:
case e'_3
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : AddCommMonoid A
inst✝⁷ : AddCommMonoid B
inst✝⁶ : Algebra R S
inst✝⁵ : Module R A
inst✝⁴ : Module S A
inst✝³ : Module R B
inst✝² : IsScalarTower R S A
inst✝¹ : Coalgebra R B
inst✝ : Coalgebra S A
x : A
y : B
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)) :=
  AlgebraTensorModule.tensorTensorTensorComm R S R S A (A ⊗[S] A) B (B ⊗[R] B) ≪≫ₗ
    AlgebraTensorModule.congr (LinearEquiv.refl S (A ⊗[R] B))
      (AlgebraTensorModule.tensorTensorTensorComm R S R S A A B B)
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)) :=
  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
⊢ ((AlgebraTensorModule.curry (LinearMap.lTensor (A ⊗[R] B) comul ∘ₗ comul)) x) y =
    F ((LinearMap.lTensor A comul) (comul x) ⊗ₜ[R] (LinearMap.lTensor B comul) (comul y))