l3lab/ntp-mathlib huggingface dataset:
theorem left_unitality (X : Type u) :
    (λ_ ((free R).obj X)).hom =
      (ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro x
  

Proof state:
case H.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x : X
⊢ (LinearMap.compr₂ (TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X)) (λ_ ((free R).obj X)).hom) 1 ∘ₗ
      Finsupp.lsingle x =
    (LinearMap.compr₂ (TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X))
          ((ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).toPrefunctor.obj (λ_ X).hom))
        1 ∘ₗ
      Finsupp.lsingle x

Next tactic:
apply LinearMap.ext_ring
theorem left_unitality (X : Type u) :
    (λ_ ((free R).obj X)).hom =
      (ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  

Proof state:
case H.h.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x : X
⊢ ((LinearMap.compr₂ (TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X)) (λ_ ((free R).obj X)).hom) 1 ∘ₗ
        Finsupp.lsingle x)
      1 =
    ((LinearMap.compr₂ (TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X))
            ((ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).toPrefunctor.obj (λ_ X).hom))
          1 ∘ₗ
        Finsupp.lsingle x)
      1

Next tactic:
apply Finsupp.ext
theorem left_unitality (X : Type u) :
    (λ_ ((free R).obj X)).hom =
      (ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply Finsupp.ext
  

Proof state:
case H.h.h.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x : X
⊢ ∀ (a : X),
    (((LinearMap.compr₂ (TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X)) (λ_ ((free R).obj X)).hom) 1 ∘ₗ
            Finsupp.lsingle x)
          1)
        a =
      (((LinearMap.compr₂ (TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X))
                ((ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).toPrefunctor.obj (λ_ X).hom))
              1 ∘ₗ
            Finsupp.lsingle x)
          1)
        a

Next tactic:
intro x'
theorem left_unitality (X : Type u) :
    (λ_ ((free R).obj X)).hom =
      (ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply Finsupp.ext
  intro x'
  -- Porting note: used to be dsimp [ε, μ]
  

Proof state:
case H.h.h.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x x' : X
⊢ (((LinearMap.compr₂ (TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X)) (λ_ ((free R).obj X)).hom) 1 ∘ₗ
          Finsupp.lsingle x)
        1)
      x' =
    (((LinearMap.compr₂ (TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X))
              ((ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).toPrefunctor.obj (λ_ X).hom))
            1 ∘ₗ
          Finsupp.lsingle x)
        1)
      x'

Next tactic:
let q : X →₀ R := ((λ_ (of R (X →₀ R))).hom) (1 ⊗ₜ[R] Finsupp.single x 1)
theorem left_unitality (X : Type u) :
    (λ_ ((free R).obj X)).hom =
      (ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply Finsupp.ext
  intro x'
  -- Porting note: used to be dsimp [ε, μ]
  let q : X →₀ R := ((λ_ (of R (X →₀ R))).hom) (1 ⊗ₜ[R] Finsupp.single x 1)
  

Proof state:
case H.h.h.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x x' : X
q : X →₀ R := (λ_ (of R (X →₀ R))).hom (1 ⊗ₜ[R] fun₀ | x => 1)
⊢ (((LinearMap.compr₂ (TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X)) (λ_ ((free R).obj X)).hom) 1 ∘ₗ
          Finsupp.lsingle x)
        1)
      x' =
    (((LinearMap.compr₂ (TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X))
              ((ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).toPrefunctor.obj (λ_ X).hom))
            1 ∘ₗ
          Finsupp.lsingle x)
        1)
      x'

Next tactic:
change q x' = Finsupp.mapDomain (λ_ X).hom (finsuppTensorFinsupp' R (𝟙_ (Type u)) X
    (Finsupp.single PUnit.unit 1 ⊗ₜ[R] Finsupp.single x 1)) x'
theorem left_unitality (X : Type u) :
    (λ_ ((free R).obj X)).hom =
      (ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply Finsupp.ext
  intro x'
  -- Porting note: used to be dsimp [ε, μ]
  let q : X →₀ R := ((λ_ (of R (X →₀ R))).hom) (1 ⊗ₜ[R] Finsupp.single x 1)
  change q x' = Finsupp.mapDomain (λ_ X).hom (finsuppTensorFinsupp' R (𝟙_ (Type u)) X
    (Finsupp.single PUnit.unit 1 ⊗ₜ[R] Finsupp.single x 1)) x'
  

Proof state:
case H.h.h.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x x' : X
q : X →₀ R := (λ_ (of R (X →₀ R))).hom (1 ⊗ₜ[R] fun₀ | x => 1)
⊢ q x' =
    (Finsupp.mapDomain (λ_ X).hom
        ((finsuppTensorFinsupp' R (𝟙_ (Type u)) X) ((fun₀ | PUnit.unit => 1) ⊗ₜ[R] fun₀ | x => 1)))
      x'

Next tactic:
simp_rw [finsuppTensorFinsupp'_single_tmul_single,
    ModuleCat.MonoidalCategory.leftUnitor_hom_apply, mul_one,
    Finsupp.mapDomain_single, CategoryTheory.leftUnitor_hom_apply, one_smul]
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  

Proof state:
R : Type u
inst✝ : CommRing R
X : Type u
⊢ (ρ_ ((free R).obj X)).hom =
    (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).toPrefunctor.obj (ρ_ X).hom

Next tactic:
intros
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  intros
  -- Porting note: broken ext
  

Proof state:
R : Type u
inst✝ : CommRing R
X : Type u
⊢ (ρ_ ((free R).obj X)).hom =
    (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).toPrefunctor.obj (ρ_ X).hom

Next tactic:
apply TensorProduct.ext
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  

Proof state:
case H
R : Type u
inst✝ : CommRing R
X : Type u
⊢ LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))) (ρ_ ((free R).obj X)).hom =
    LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R)))
      ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).toPrefunctor.obj (ρ_ X).hom)

Next tactic:
apply Finsupp.lhom_ext'
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  

Proof state:
case H.h
R : Type u
inst✝ : CommRing R
X : Type u
⊢ ∀ (a : X),
    LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))) (ρ_ ((free R).obj X)).hom ∘ₗ
        Finsupp.lsingle a =
      LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R)))
          ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).toPrefunctor.obj (ρ_ X).hom) ∘ₗ
        Finsupp.lsingle a

Next tactic:
intro x
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  

Proof state:
case H.h
R : Type u
inst✝ : CommRing R
X : Type u
x : X
⊢ LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))) (ρ_ ((free R).obj X)).hom ∘ₗ
      Finsupp.lsingle x =
    LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R)))
        ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).toPrefunctor.obj (ρ_ X).hom) ∘ₗ
      Finsupp.lsingle x

Next tactic:
apply LinearMap.ext_ring
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  

Proof state:
case H.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x : X
⊢ (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))) (ρ_ ((free R).obj X)).hom ∘ₗ
        Finsupp.lsingle x)
      1 =
    (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R)))
          ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).toPrefunctor.obj (ρ_ X).hom) ∘ₗ
        Finsupp.lsingle x)
      1

Next tactic:
apply LinearMap.ext_ring
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply LinearMap.ext_ring
  

Proof state:
case H.h.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x : X
⊢ ((LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))) (ρ_ ((free R).obj X)).hom ∘ₗ
          Finsupp.lsingle x)
        1)
      1 =
    ((LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R)))
            ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).toPrefunctor.obj (ρ_ X).hom) ∘ₗ
          Finsupp.lsingle x)
        1)
      1

Next tactic:
apply Finsupp.ext
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply LinearMap.ext_ring
  apply Finsupp.ext
  

Proof state:
case H.h.h.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x : X
⊢ ∀ (a : X),
    (((LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))) (ρ_ ((free R).obj X)).hom ∘ₗ
              Finsupp.lsingle x)
            1)
          1)
        a =
      (((LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R)))
                ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).toPrefunctor.obj (ρ_ X).hom) ∘ₗ
              Finsupp.lsingle x)
            1)
          1)
        a

Next tactic:
intro x'
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply LinearMap.ext_ring
  apply Finsupp.ext
  intro x'
  -- Porting note: used to be dsimp [ε, μ]
  

Proof state:
case H.h.h.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x x' : X
⊢ (((LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))) (ρ_ ((free R).obj X)).hom ∘ₗ
            Finsupp.lsingle x)
          1)
        1)
      x' =
    (((LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R)))
              ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).toPrefunctor.obj (ρ_ X).hom) ∘ₗ
            Finsupp.lsingle x)
          1)
        1)
      x'

Next tactic:
let q : X →₀ R := ((ρ_ (of R (X →₀ R))).hom) (Finsupp.single x 1 ⊗ₜ[R] 1)
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply LinearMap.ext_ring
  apply Finsupp.ext
  intro x'
  -- Porting note: used to be dsimp [ε, μ]
  let q : X →₀ R := ((ρ_ (of R (X →₀ R))).hom) (Finsupp.single x 1 ⊗ₜ[R] 1)
  

Proof state:
case H.h.h.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x x' : X
q : X →₀ R := (ρ_ (of R (X →₀ R))).hom ((fun₀ | x => 1) ⊗ₜ[R] 1)
⊢ (((LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))) (ρ_ ((free R).obj X)).hom ∘ₗ
            Finsupp.lsingle x)
          1)
        1)
      x' =
    (((LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R)))
              ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).toPrefunctor.obj (ρ_ X).hom) ∘ₗ
            Finsupp.lsingle x)
          1)
        1)
      x'

Next tactic:
change q x' = Finsupp.mapDomain (ρ_ X).hom (finsuppTensorFinsupp' R X (𝟙_ (Type u))
    (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single PUnit.unit 1)) x'
theorem right_unitality (X : Type u) :
    (ρ_ ((free R).obj X)).hom =
      (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply LinearMap.ext_ring
  apply Finsupp.ext
  intro x'
  -- Porting note: used to be dsimp [ε, μ]
  let q : X →₀ R := ((ρ_ (of R (X →₀ R))).hom) (Finsupp.single x 1 ⊗ₜ[R] 1)
  change q x' = Finsupp.mapDomain (ρ_ X).hom (finsuppTensorFinsupp' R X (𝟙_ (Type u))
    (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single PUnit.unit 1)) x'
  

Proof state:
case H.h.h.h.h
R : Type u
inst✝ : CommRing R
X : Type u
x x' : X
q : X →₀ R := (ρ_ (of R (X →₀ R))).hom ((fun₀ | x => 1) ⊗ₜ[R] 1)
⊢ q x' =
    (Finsupp.mapDomain (ρ_ X).hom
        ((finsuppTensorFinsupp' R X (𝟙_ (Type u))) ((fun₀ | x => 1) ⊗ₜ[R] fun₀ | PUnit.unit => 1)))
      x'

Next tactic:
simp_rw [finsuppTensorFinsupp'_single_tmul_single,
    ModuleCat.MonoidalCategory.rightUnitor_hom_apply, mul_one,
    Finsupp.mapDomain_single, CategoryTheory.rightUnitor_hom_apply, one_smul]
theorem associativity (X Y Z : Type u) :
    ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
      (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom := by
  

Proof state:
R : Type u
inst✝ : CommRing R
X Y Z : Type u
⊢ ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom =
    (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
      (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom

Next tactic:
intros
theorem associativity (X Y Z : Type u) :
    ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
      (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom := by
  intros
  -- Porting note: broken ext
  

Proof state:
R : Type u
inst✝ : CommRing R
X Y Z : Type u
⊢ ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom =
    (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
      (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom

Next tactic:
apply TensorProduct.ext
theorem associativity (X Y Z : Type u) :
    ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
      (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  

Proof state:
case H
R : Type u
inst✝ : CommRing R
X Y Z : Type u
⊢ LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
      (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom) =
    LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
      ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)

Next tactic:
apply TensorProduct.ext
theorem associativity (X Y Z : Type u) :
    ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
      (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply TensorProduct.ext
  

Proof state:
case H.H
R : Type u
inst✝ : CommRing R
X Y Z : Type u
⊢ LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
      (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
        (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)) =
    LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
      (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
        ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
          (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom))

Next tactic:
apply Finsupp.lhom_ext'
theorem associativity (X Y Z : Type u) :
    ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
      (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  

Proof state:
case H.H.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
⊢ ∀ (a : X),
    LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
          (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
            (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫
              (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)) ∘ₗ
        Finsupp.lsingle a =
      LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
          (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
            ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
              (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)) ∘ₗ
        Finsupp.lsingle a

Next tactic:
intro x
theorem associativity (X Y Z : Type u) :
    ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
      (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  

Proof state:
case H.H.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
⊢ LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
        (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
          (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫
            (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)) ∘ₗ
      Finsupp.lsingle x =
    LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
        (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
          ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
            (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)) ∘ₗ
      Finsupp.lsingle x

Next tactic:
apply LinearMap.ext_ring
theorem associativity (X Y Z : Type u) :
    ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
      (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  

Proof state:
case H.H.h.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
⊢ (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
          (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
            (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫
              (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)) ∘ₗ
        Finsupp.lsingle x)
      1 =
    (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
          (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
            ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
              (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)) ∘ₗ
        Finsupp.lsingle x)
      1

Next tactic:
apply Finsupp.lhom_ext'
theorem associativity (X Y Z : Type u) :
    ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
      (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  

Proof state:
case H.H.h.h.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
⊢ ∀ (a : Y),
    (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
              (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
                (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫
                  (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)) ∘ₗ
            Finsupp.lsingle x)
          1 ∘ₗ
        Finsupp.lsingle a =
      (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
              (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
                ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
                  (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)) ∘ₗ
            Finsupp.lsingle x)
          1 ∘ₗ
        Finsupp.lsingle a

Next tactic:
intro y
theorem associativity (X Y Z : Type u) :
    ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom =
      (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
        (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom := by
  intros
  -- Porting note: broken ext
  apply TensorProduct.ext
  apply TensorProduct.ext
  apply Finsupp.lhom_ext'
  intro x
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro y
  

Proof state:
case H.H.h.h.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
y : Y
⊢ (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
            (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
              (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫
                (μ R (X ⊗ Y) Z).hom ≫ map (free R).toPrefunctor.obj (α_ X Y Z).hom)) ∘ₗ
          Finsupp.lsingle x)
        1 ∘ₗ
      Finsupp.lsingle y =
    (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y))
            (LinearMap.compr₂ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z))
              ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫
                (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)) ∘ₗ
          Finsupp.lsingle x)
        1 ∘ₗ
      Finsupp.lsingle y

Next tactic:
apply LinearMap.ext_ring