awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
⊢ Eq (α := A)
    ((LinearMap.mul' R A : (a : A ⊗[R] A) → A)
        ((TensorProduct.map (LinearMap.mul' R A)
                (LinearMap.mul' R A ∘ₗ
                  TensorProduct.map (antipode R) (antipode R) ∘ₗ
                    (↑(TensorProduct.comm R A A) : A ⊗[R] A →ₗ[R] A ⊗[R] A)) :
              (a : A ⊗[R] A ⊗[R] (A ⊗[R] A)) → A ⊗[R] A)
            (((TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R R R R A A A A :
                  A ⊗[R] A ⊗[R] (A ⊗[R] A) ≃ₗ[R] A ⊗[R] A ⊗[R] (A ⊗[R] A)) :
                (a : A ⊗[R] A ⊗[R] (A ⊗[R] A)) → A ⊗[R] A ⊗[R] (A ⊗[R] A))
              ((∑ i ∈ ℛx.index, ℛx.left i ⊗ₜ[R] ℛx.right i) ⊗ₜ[R] ∑ i ∈ ℛy.index, ℛy.left i ⊗ₜ[R] ℛy.right i)) :
          A ⊗[R] A) :
      A)
    ((algebraMap R A : (a : R) → A) ((counit : (a : A ⊗[R] A) → R) (x ⊗ₜ[R] y) : R) : A)
have refine_2 : Eq (α := A)
  (∑ x_1 ∈ ℛy.index,
      ∑ x_2 ∈ ℛx.index,
        HMul.hMul (β := A) (ℛx.left x_2 * ℛy.left x_1)
          ((LinearMap.mul' R A : (a : A ⊗[R] A) → A)
              ((TensorProduct.map (antipode R) (antipode R) : (a : A ⊗[R] A) → A ⊗[R] A)
                  (((↑(TensorProduct.comm R A A) : A ⊗[R] A →ₗ[R] A ⊗[R] A) : (a : A ⊗[R] A) → A ⊗[R] A)
                    (ℛx.right x_2 ⊗ₜ[R] ℛy.right x_1)) :
                A ⊗[R] A) :
            A) :
    A)
  ((algebraMap R A : (a : R) → A) ((counit : (a : A ⊗[R] A) → R) (x ⊗ₜ[R] y) : R) : A)
Tactic: simp only [TensorProduct.sum_tmul, TensorProduct.tmul_sum, map_sum,
      TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_tmul, TensorProduct.map_tmul,
      LinearMap.mul'_apply, LinearMap.comp_apply]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
⊢ Eq (α := A)
    (∑ x_1 ∈ ℛy.index,
        ∑ x_2 ∈ ℛx.index,
          HMul.hMul (β := A) (ℛx.left x_2 * ℛy.left x_1)
            ((LinearMap.mul' R A : (a : A ⊗[R] A) → A)
                ((TensorProduct.map (antipode R) (antipode R) : (a : A ⊗[R] A) → A ⊗[R] A)
                    (((↑(TensorProduct.comm R A A) : A ⊗[R] A →ₗ[R] A ⊗[R] A) : (a : A ⊗[R] A) → A ⊗[R] A)
                      (ℛx.right x_2 ⊗ₜ[R] ℛy.right x_1)) :
                  A ⊗[R] A) :
              A) :
      A)
    ((algebraMap R A : (a : R) → A) ((counit : (a : A ⊗[R] A) → R) (x ⊗ₜ[R] y) : R) : A)
have refine_2 : Eq (α := A)
  (∑ y_1 ∈ ℛx.index,
      ∑ x_1 ∈ ℛy.index,
        HMul.hMul (β := A) (ℛx.left y_1 * ℛy.left x_1)
          ((LinearMap.mul' R A : (a : A ⊗[R] A) → A)
              ((TensorProduct.map (antipode R) (antipode R) : (a : A ⊗[R] A) → A ⊗[R] A)
                  (((↑(TensorProduct.comm R A A) : A ⊗[R] A →ₗ[R] A ⊗[R] A) : (a : A ⊗[R] A) → A ⊗[R] A)
                    (ℛx.right y_1 ⊗ₜ[R] ℛy.right x_1)) :
                A ⊗[R] A) :
            A) :
    A)
  ((algebraMap R A : (a : R) → A) ((counit : (a : A ⊗[R] A) → R) (x ⊗ₜ[R] y) : R) : A)
Tactic: rw [Finset.sum_comm]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
⊢ Eq (α := A)
    (∑ y_1 ∈ ℛx.index,
        ∑ x_1 ∈ ℛy.index,
          HMul.hMul (β := A) (ℛx.left y_1 * ℛy.left x_1)
            ((LinearMap.mul' R A : (a : A ⊗[R] A) → A)
                ((TensorProduct.map (antipode R) (antipode R) : (a : A ⊗[R] A) → A ⊗[R] A)
                    (((↑(TensorProduct.comm R A A) : A ⊗[R] A →ₗ[R] A ⊗[R] A) : (a : A ⊗[R] A) → A ⊗[R] A)
                      (ℛx.right y_1 ⊗ₜ[R] ℛy.right x_1)) :
                  A ⊗[R] A) :
              A) :
      A)
    ((algebraMap R A : (a : R) → A) ((counit : (a : A ⊗[R] A) → R) (x ⊗ₜ[R] y) : R) : A)
have refine_2 : ∑ y_1 ∈ ℛx.index,
    ∑ x_1 ∈ ℛy.index,
      HMul.hMul (β := A) (ℛx.left y_1 * ℛy.left x_1)
        ((LinearMap.mul' R A : (a : A ⊗[R] A) → A)
            ((TensorProduct.map (antipode R) (antipode R) : (a : A ⊗[R] A) → A ⊗[R] A)
                (((↑(TensorProduct.comm R A A) : A ⊗[R] A →ₗ[R] A ⊗[R] A) : (a : A ⊗[R] A) → A ⊗[R] A)
                  (ℛx.right y_1 ⊗ₜ[R] ℛy.right x_1)) :
              A ⊗[R] A) :
          A) =
  HSMul.hSMul (β := A) (HSMul.hSMul (α := R) (β := R) ((counit : (a : A) → R) y : R) ((counit : (a : A) → R) x : R)) 1
Tactic: simp only [TensorProduct.counit_tmul, Algebra.algebraMap_eq_smul_one]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
⊢ Eq (α := A)
    (∑ i ∈ ℛx.index,
      ℛx.left i *
        HMul.hMul (α := A) (β := A)
          (∑ j ∈ ℛy.index, HMul.hMul (β := A) (ℛy.left j) ((antipode R : (a : A) → A) (ℛy.right j) : A) : A)
          ((antipode R : (a : A) → A) (ℛx.right i) : A))
    (∑ i ∈ ℛx.index,
      ℛx.left i *
        HMul.hMul (β := A) (HSMul.hSMul.{u, v, v} (α := R) (β := A) ((counit : (a : A) → R) y : R) 1)
          ((antipode R : (a : A) → A) (ℛx.right i) : A))
have rw : Eq (α := A)
  (∑ i ∈ ℛx.index,
    ℛx.left i *
      HMul.hMul (β := A) (HSMul.hSMul.{u, v, v} (α := R) (β := A) ((counit : (a : A) → R) y : R) 1)
        ((antipode R : (a : A) → A) (ℛx.right i) : A))
  (∑ i ∈ ℛx.index,
    ℛx.left i *
      HMul.hMul (β := A) (HSMul.hSMul.{u, v, v} (α := R) (β := A) ((counit : (a : A) → R) y : R) 1)
        ((antipode R : (a : A) → A) (ℛx.right i) : A))
Tactic: rw [sum_mul_antipode_eq_smul ℛy]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
⊢ Eq (α := A)
    (∑ i ∈ ℛx.index,
      ℛx.left i *
        HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
          ((antipode R : (a : A) → A) (ℛx.right i) : A))
    (∑ i ∈ ℛx.index,
      HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
        (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A)))
have e_f : Eq.{v + 1} (α := ℛx.ι → A)
  (fun (i : ℛx.ι) ↦
    ℛx.left i *
      HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
        ((antipode R : (a : A) → A) (ℛx.right i) : A))
  fun (i : ℛx.ι) ↦
  HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
    (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A))
Tactic: congr 1
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
⊢ Eq.{v + 1} (α := ℛx.ι → A)
    (fun (i : ℛx.ι) ↦
      ℛx.left i *
        HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
          ((antipode R : (a : A) → A) (ℛx.right i) : A))
    fun (i : ℛx.ι) ↦
    HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
      (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A))
have e_f : ∀ (i : ℛx.ι),
  ℛx.left i *
      HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
        ((antipode R : (a : A) → A) (ℛx.right i) : A) =
    HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
      (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A))
Tactic: ext i
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
i : ℛx.ι
⊢ ℛx.left i *
      HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
        ((antipode R : (a : A) → A) (ℛx.right i) : A) =
    HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
      (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A))
have e_f : HMul.hMul (β := A) (HMul.hMul (β := A) (ℛx.left i) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A))
    ((antipode R : (a : A) → A) (ℛx.right i) : A) =
  HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
    (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A))
Tactic: rw [← mul_assoc, ← mul_assoc, Algebra.commutes]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
i : ℛx.ι
e_f :
  HMul.hMul (β := A)
      (HMul.hMul (β := A) (ℛx.left i) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A))
      ((antipode R : (a : A) → A) (ℛx.right i) : A) =
    HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
      (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A))
⊢ ℛx.left i *
      HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
        ((antipode R : (a : A) → A) (ℛx.right i) : A) =
    HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
      (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A))
have e_f₁ : HMul.hMul (β := A) (HMul.hMul (β := A) (ℛx.left i) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A))
    ((antipode R : (a : A) → A) (ℛx.right i) : A) =
  HMul.hMul (β := A)
    (HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A) (ℛx.left i))
    ((antipode R : (a : A) → A) (ℛx.right i) : A)
Tactic: rw [← mul_assoc, ← mul_assoc, Algebra.commutes]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
i : ℛx.ι
e_f :
  HMul.hMul (β := A)
      (HMul.hMul (β := A) (ℛx.left i) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A))
      ((antipode R : (a : A) → A) (ℛx.right i) : A) =
    HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
      (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A))
e_f₁ :
  HMul.hMul (β := A)
      (HMul.hMul (β := A) (ℛx.left i) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A))
      ((antipode R : (a : A) → A) (ℛx.right i) : A) =
    HMul.hMul (β := A)
      (HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A) (ℛx.left i))
      ((antipode R : (a : A) → A) (ℛx.right i) : A)
⊢ ℛx.left i *
      HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
        ((antipode R : (a : A) → A) (ℛx.right i) : A) =
    HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
      (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A))
have e_f₂ : HMul.hMul (β := A) (HMul.hMul (β := A) (ℛx.left i) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A))
    ((antipode R : (a : A) → A) (ℛx.right i) : A) =
  HMul.hMul (β := A)
    (HMul.hMul (β := A) (ℛx.left i) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A))
    ((antipode R : (a : A) → A) (ℛx.right i) : A)
Tactic: rw [← mul_assoc, ← mul_assoc, Algebra.commutes]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
⊢ ∑ i ∈ ℛx.index,
      HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
        (HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A)) =
    HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
      (∑ i ∈ ℛx.index, HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A) : A)
have rw : HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
    (∑ i ∈ ℛx.index, HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A) : A) =
  HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
    (∑ i ∈ ℛx.index, HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A) : A)
Tactic: rw [← Finset.mul_sum]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
⊢ HMul.hMul (α := A) (β := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
      (∑ i ∈ ℛx.index, HMul.hMul (β := A) (ℛx.left i) ((antipode R : (a : A) → A) (ℛx.right i) : A) : A) =
    HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
      (HSMul.hSMul.{u, v, v} (α := R) (β := A) ((counit : (a : A) → R) x : R) 1)
have rw : HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
    (HSMul.hSMul.{u, v, v} (α := R) (β := A) ((counit : (a : A) → R) x : R) 1) =
  HMul.hMul (α := A) ((algebraMap R A : (a : R) → A) ((counit : (a : A) → R) y : R) : A)
    (HSMul.hSMul.{u, v, v} (α := R) (β := A) ((counit : (a : A) → R) x : R) 1)
Tactic: rw [sum_mul_antipode_eq_smul ℛx]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
⊢ ∑ y_1 ∈ ℛx.index,
      ∑ x_1 ∈ ℛy.index,
        HMul.hMul (β := A) (ℛx.left y_1 * ℛy.left x_1)
          ((LinearMap.mul' R A : (a : A ⊗[R] A) → A)
              ((TensorProduct.map (antipode R) (antipode R) : (a : A ⊗[R] A) → A ⊗[R] A)
                  (((↑(TensorProduct.comm R A A) : A ⊗[R] A →ₗ[R] A ⊗[R] A) : (a : A ⊗[R] A) → A ⊗[R] A)
                    (ℛx.right y_1 ⊗ₜ[R] ℛy.right x_1)) :
                A ⊗[R] A) :
            A) =
    HSMul.hSMul (β := A) (HSMul.hSMul (α := R) (β := R) ((counit : (a : A) → R) y : R) ((counit : (a : A) → R) x : R)) 1
have calc : Eq (α := A)
  (∑ i ∈ ℛx.index,
    ∑ j ∈ ℛy.index,
      ℛx.left i * ℛy.left j *
        HMul.hMul (α := A) (β := A) ((antipode R : (a : A) → A) (ℛy.right j) : A)
          ((antipode R : (a : A) → A) (ℛx.right i) : A))
  (∑ i ∈ ℛx.index,
    ∑ j ∈ ℛy.index,
      ℛx.left i *
        HMul.hMul (β := A) (HMul.hMul (β := A) (ℛy.left j) ((antipode R : (a : A) → A) (ℛy.right j) : A))
          ((antipode R : (a : A) → A) (ℛx.right i) : A))
Tactic: calc ∑ i ∈ ℛx.index, ∑ j ∈ ℛy.index,
        (ℛx.left i * ℛy.left j) * (antipode R (ℛy.right j) * antipode R (ℛx.right i))
      _ = ∑ i ∈ ℛx.index, ∑ j ∈ ℛy.index,
          ℛx.left i * (ℛy.left j * antipode R (ℛy.right j) * antipode R (ℛx.right i)) := by
        simp [mul_assoc]
      _ = ∑ i ∈ ℛx.index, ℛx.left i *
          ((∑ j ∈ ℛy.index, ℛy.left j * antipode R (ℛy.right j)) * antipode R (ℛx.right i)) := by
        simp [Finset.sum_mul, Finset.mul_sum]
      _ = ∑ i ∈ ℛx.index, ℛx.left i *
          (counit y • 1 * antipode R (ℛx.right i)) := by
        rw [sum_mul_antipode_eq_smul ℛy]
      _ = ∑ i ∈ ℛx.index, ℛx.left i *
          (algebraMap R A (counit y) * antipode R (ℛx.right i)) := by
        simp [Algebra.smul_def]
      _ = ∑ i ∈ ℛx.index, algebraMap R A (counit y) * (ℛx.left i * antipode R (ℛx.right i)) := by
        congr 1; ext i; rw [← mul_assoc, ← mul_assoc, Algebra.commutes]
      _ = algebraMap R A (counit y) * ∑ i ∈ ℛx.index, ℛx.left i * antipode R (ℛx.right i) := by
        rw [← Finset.mul_sum]
      _ = algebraMap R A (counit y) * (counit (R := R) x • (1 : A)) := by
        rw [sum_mul_antipode_eq_smul ℛx]
      _ = (counit (R := R) x * counit y) • (1 : A) := by
        simp only [Algebra.smul_def, mul_one, ← map_mul, mul_comm (counit x)]
      _ = (counit (R := R) y • counit x) • (1 : A) := by
        simp only [smul_eq_mul, mul_comm (counit y)]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
A : Type v
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b x y : A
ℛx : Coalgebra.Repr R x := ℛ R x
ℛy : Coalgebra.Repr R y := ℛ R y
calc :
  Eq (α := A)
    (∑ i ∈ ℛx.index,
      ∑ j ∈ ℛy.index,
        ℛx.left i * ℛy.left j *
          HMul.hMul (α := A) (β := A) ((antipode R : (a : A) → A) (ℛy.right j) : A)
            ((antipode R : (a : A) → A) (ℛx.right i) : A))
    (∑ i ∈ ℛx.index,
      ∑ j ∈ ℛy.index,
        ℛx.left i *
          HMul.hMul (β := A) (HMul.hMul (β := A) (ℛy.left j) ((antipode R : (a : A) → A) (ℛy.right j) : A))
            ((antipode R : (a : A) → A) (ℛx.right i) : A))
⊢ ∑ y_1 ∈ ℛx.index,
      ∑ x_1 ∈ ℛy.index,
        HMul.hMul (β := A) (ℛx.left y_1 * ℛy.left x_1)
          ((LinearMap.mul' R A : (a : A ⊗[R] A) → A)
              ((TensorProduct.map (antipode R) (antipode R) : (a : A ⊗[R] A) → A ⊗[R] A)
                  (((↑(TensorProduct.comm R A A) : A ⊗[R] A →ₗ[R] A ⊗[R] A) : (a : A ⊗[R] A) → A ⊗[R] A)
                    (ℛx.right y_1 ⊗ₜ[R] ℛy.right x_1)) :
                A ⊗[R] A) :
            A) =
    HSMul.hSMul (β := A) (HSMul.hSMul (α := R) (β := R) ((counit : (a : A) → R) y : R) ((counit : (a : A) → R) x : R)) 1
have calc₁ : Eq (α := A)
  (∑ i ∈ ℛx.index,
    ∑ j ∈ ℛy.index,
      ℛx.left i *
        HMul.hMul (β := A) (HMul.hMul (β := A) (ℛy.left j) ((antipode R : (a : A) → A) (ℛy.right j) : A))
          ((antipode R : (a : A) → A) (ℛx.right i) : A))
  (∑ i ∈ ℛx.index,
    ℛx.left i *
      HMul.hMul (α := A) (β := A)
        (∑ j ∈ ℛy.index, HMul.hMul (β := A) (ℛy.left j) ((antipode R : (a : A) → A) (ℛy.right j) : A) : A)
        ((antipode R : (a : A) → A) (ℛx.right i) : A))
Tactic: calc ∑ i ∈ ℛx.index, ∑ j ∈ ℛy.index,
        (ℛx.left i * ℛy.left j) * (antipode R (ℛy.right j) * antipode R (ℛx.right i))
      _ = ∑ i ∈ ℛx.index, ∑ j ∈ ℛy.index,
          ℛx.left i * (ℛy.left j * antipode R (ℛy.right j) * antipode R (ℛx.right i)) := by
        simp [mul_assoc]
      _ = ∑ i ∈ ℛx.index, ℛx.left i *
          ((∑ j ∈ ℛy.index, ℛy.left j * antipode R (ℛy.right j)) * antipode R (ℛx.right i)) := by
        simp [Finset.sum_mul, Finset.mul_sum]
      _ = ∑ i ∈ ℛx.index, ℛx.left i *
          (counit y • 1 * antipode R (ℛx.right i)) := by
        rw [sum_mul_antipode_eq_smul ℛy]
      _ = ∑ i ∈ ℛx.index, ℛx.left i *
          (algebraMap R A (counit y) * antipode R (ℛx.right i)) := by
        simp [Algebra.smul_def]
      _ = ∑ i ∈ ℛx.index, algebraMap R A (counit y) * (ℛx.left i * antipode R (ℛx.right i)) := by
        congr 1; ext i; rw [← mul_assoc, ← mul_assoc, Algebra.commutes]
      _ = algebraMap R A (counit y) * ∑ i ∈ ℛx.index, ℛx.left i * antipode R (ℛx.right i) := by
        rw [← Finset.mul_sum]
      _ = algebraMap R A (counit y) * (counit (R := R) x • (1 : A)) := by
        rw [sum_mul_antipode_eq_smul ℛx]
      _ = (counit (R := R) x * counit y) • (1 : A) := by
        simp only [Algebra.smul_def, mul_one, ← map_mul, mul_comm (counit x)]
      _ = (counit (R := R) y • counit x) • (1 : A) := by
        simp only [smul_eq_mul, mul_comm (counit y)]
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
inst : CommSemiring R
⊢ Eq (α := R →ₗ[R] R) (LinearMap.mul' R R ∘ₗ LinearMap.rTensor R LinearMap.id ∘ₗ CoalgebraStruct.comul)
    (Algebra.linearMap R R ∘ₗ CoalgebraStruct.counit)
have h : Eq (α := R) ((LinearMap.mul' R R ∘ₗ LinearMap.rTensor R LinearMap.id ∘ₗ CoalgebraStruct.comul : (a : R) → R) 1 : R)
  ((Algebra.linearMap R R ∘ₗ CoalgebraStruct.counit : (a : R) → R) 1 : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.Basic (goal)
Goal:
R : Type u
inst : CommSemiring R
⊢ Eq (α := R →ₗ[R] R) (LinearMap.mul' R R ∘ₗ LinearMap.lTensor R LinearMap.id ∘ₗ CoalgebraStruct.comul)
    (Algebra.linearMap R R ∘ₗ CoalgebraStruct.counit)
have h : Eq (α := R) ((LinearMap.mul' R R ∘ₗ LinearMap.lTensor R LinearMap.id ∘ₗ CoalgebraStruct.comul : (a : R) → R) 1 : R)
  ((Algebra.linearMap R R ∘ₗ CoalgebraStruct.counit : (a : R) → R) 1 : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.GroupLike (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a b : A
⊢ Eq.{u_2 + 1} (α := Aˣ)
    ({ val := (↑1 : A), inv := (antipode R : (a : A) → A) (↑1 : A), val_inv := ⋯, inv_val := ⋯ } : Aˣ) (1 : Aˣ)
have huv : Eq.{u_2 + 1} (α := A)
  ((↑{ val := (↑1 : A), inv := (antipode R : (a : A) → A) (↑1 : A), val_inv := sorry, inv_val := sorry } : A) : A)
  ((↑1 : A) : A)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.GroupLike (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a_1 b_1 : A
a b : GroupLike R A
⊢ { val := (↑(a * b) : A), inv := (antipode R : (a : A) → A) (↑(a * b) : A), val_inv := ⋯, inv_val := ⋯ : Aˣ } =
    { val := (↑a : A), inv := (antipode R : (a : A) → A) (↑a : A), val_inv := ⋯, inv_val := ⋯ : Aˣ } *
      { val := (↑b : A), inv := (antipode R : (a : A) → A) (↑b : A), val_inv := ⋯, inv_val := ⋯ : Aˣ }
have huv : (↑{ val := (↑(a * b) : A), inv := (antipode R : (a : A) → A) (↑(a * b) : A), val_inv := sorry, inv_val := sorry : Aˣ } : A) =
  (↑({ val := (↑a : A), inv := (antipode R : (a : A) → A) (↑a : A), val_inv := sorry, inv_val := sorry : Aˣ } *
        { val := (↑b : A), inv := (antipode R : (a : A) → A) (↑b : A), val_inv := sorry, inv_val := sorry : Aˣ }) :
    A)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.GroupLike (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
a_1 b : A
a : GroupLike R A
⊢ a⁻¹ * a = 1
have val : (↑(a⁻¹ * a) : A) = (↑1 : A)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
G : Type u_3
inst_3 : Group G
⊢ Eq (α := A[G] →ₗ[R] A[G]) (LinearMap.mul' R A[G] ∘ₗ LinearMap.rTensor A[G] (antipode R) ∘ₗ comul)
    (Algebra.linearMap R A[G] ∘ₗ counit)
have H : ∀ (a : G) (b : A),
  Eq (α := A[G])
    (((LinearMap.mul' R A[G] ∘ₗ LinearMap.rTensor A[G] (antipode R) ∘ₗ comul) ∘ₗ lsingle a : (a : A) → A[G]) b : A[G])
    (((Algebra.linearMap R A[G] ∘ₗ counit) ∘ₗ lsingle a : (a : A) → A[G]) b : A[G])
Tactic: ext a b : 2
-- Lean 4 proof step from Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommSemiring R
inst_1 : Semiring A
inst_2 : HopfAlgebra R A
G : Type u_3
inst_3 : Group G
⊢ Eq (α := A[G] →ₗ[R] A[G]) (LinearMap.mul' R A[G] ∘ₗ LinearMap.lTensor A[G] (antipode R) ∘ₗ comul)
    (Algebra.linearMap R A[G] ∘ₗ counit)
have H : ∀ (a : G) (b : A),
  Eq (α := A[G])
    (((LinearMap.mul' R A[G] ∘ₗ LinearMap.lTensor A[G] (antipode R) ∘ₗ comul) ∘ₗ lsingle a : (a : A) → A[G]) b : A[G])
    (((Algebra.linearMap R A[G] ∘ₗ counit) ∘ₗ lsingle a : (a : A) → A[G]) b : A[G])
Tactic: ext a b : 2