l3lab/ntp-mathlib huggingface dataset:
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
  apply LinearMap.ext_ring
  

Proof state:
case H.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)
      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 ∘ₗ
        Finsupp.lsingle y)
      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'
  intro y
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  

Proof state:
case H.H.h.h.h.h.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
y : Y
⊢ ∀ (a : Z),
    ((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)
          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 y)
          1 ∘ₗ
        Finsupp.lsingle a

Next tactic:
intro z
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
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro z
  

Proof state:
case H.H.h.h.h.h.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
y : Y
z : Z
⊢ ((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)
        1 ∘ₗ
      Finsupp.lsingle z =
    ((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)
        1 ∘ₗ
      Finsupp.lsingle z

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
  apply Finsupp.lhom_ext'
  intro y
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro z
  apply LinearMap.ext_ring
  

Proof state:
case H.H.h.h.h.h.h.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
y : Y
z : Z
⊢ (((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)
          1 ∘ₗ
        Finsupp.lsingle z)
      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 ∘ₗ
            Finsupp.lsingle y)
          1 ∘ₗ
        Finsupp.lsingle z)
      1

Next tactic:
apply Finsupp.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'
  intro y
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro z
  apply LinearMap.ext_ring
  apply Finsupp.ext
  

Proof state:
case H.H.h.h.h.h.h.h.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
y : Y
z : Z
⊢ ∀ (a : X ⊗ Y ⊗ Z),
    ((((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)
              1 ∘ₗ
            Finsupp.lsingle z)
          1)
        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 y)
              1 ∘ₗ
            Finsupp.lsingle z)
          1)
        a

Next tactic:
intro a
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
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro z
  apply LinearMap.ext_ring
  apply Finsupp.ext
  intro a
  -- Porting note: used to be dsimp [μ]
  

Proof state:
case H.H.h.h.h.h.h.h.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
y : Y
z : Z
a : X ⊗ Y ⊗ Z
⊢ ((((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)
            1 ∘ₗ
          Finsupp.lsingle z)
        1)
      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 y)
            1 ∘ₗ
          Finsupp.lsingle z)
        1)
      a

Next tactic:
change Finsupp.mapDomain (α_ X Y Z).hom (finsuppTensorFinsupp' R (X ⊗ Y) Z
    (finsuppTensorFinsupp' R X Y
    (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single y 1) ⊗ₜ[R] Finsupp.single z 1)) a =
    finsuppTensorFinsupp' R X (Y ⊗ Z)
    (Finsupp.single x 1 ⊗ₜ[R]
      finsuppTensorFinsupp' R Y Z (Finsupp.single y 1 ⊗ₜ[R] Finsupp.single z 1)) a
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
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro z
  apply LinearMap.ext_ring
  apply Finsupp.ext
  intro a
  -- Porting note: used to be dsimp [μ]
  change Finsupp.mapDomain (α_ X Y Z).hom (finsuppTensorFinsupp' R (X ⊗ Y) Z
    (finsuppTensorFinsupp' R X Y
    (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single y 1) ⊗ₜ[R] Finsupp.single z 1)) a =
    finsuppTensorFinsupp' R X (Y ⊗ Z)
    (Finsupp.single x 1 ⊗ₜ[R]
      finsuppTensorFinsupp' R Y Z (Finsupp.single y 1 ⊗ₜ[R] Finsupp.single z 1)) a
  -- extra `rfl` after leanprover/lean4#2466
  

Proof state:
case H.H.h.h.h.h.h.h.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
y : Y
z : Z
a : X ⊗ Y ⊗ Z
⊢ (Finsupp.mapDomain (α_ X Y Z).hom
        ((finsuppTensorFinsupp' R (X ⊗ Y) Z)
          ((finsuppTensorFinsupp' R X Y) ((fun₀ | x => 1) ⊗ₜ[R] fun₀ | y => 1) ⊗ₜ[R] fun₀ | z => 1)))
      a =
    ((finsuppTensorFinsupp' R X (Y ⊗ Z))
        ((fun₀ | x => 1) ⊗ₜ[R] (finsuppTensorFinsupp' R Y Z) ((fun₀ | y => 1) ⊗ₜ[R] fun₀ | z => 1)))
      a

Next tactic:
simp_rw [finsuppTensorFinsupp'_single_tmul_single, Finsupp.mapDomain_single, mul_one,
    CategoryTheory.associator_hom_apply]
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
  apply LinearMap.ext_ring
  apply Finsupp.lhom_ext'
  intro z
  apply LinearMap.ext_ring
  apply Finsupp.ext
  intro a
  -- Porting note: used to be dsimp [μ]
  change Finsupp.mapDomain (α_ X Y Z).hom (finsuppTensorFinsupp' R (X ⊗ Y) Z
    (finsuppTensorFinsupp' R X Y
    (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single y 1) ⊗ₜ[R] Finsupp.single z 1)) a =
    finsuppTensorFinsupp' R X (Y ⊗ Z)
    (Finsupp.single x 1 ⊗ₜ[R]
      finsuppTensorFinsupp' R Y Z (Finsupp.single y 1 ⊗ₜ[R] Finsupp.single z 1)) a
  -- extra `rfl` after leanprover/lean4#2466
  simp_rw [finsuppTensorFinsupp'_single_tmul_single, Finsupp.mapDomain_single, mul_one,
    CategoryTheory.associator_hom_apply]; 

Proof state:
case H.H.h.h.h.h.h.h.h
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
y : Y
z : Z
a : X ⊗ Y ⊗ Z
⊢ (fun₀ | (x, y, z) => 1) a = (fun₀ | (x, y, z) => 1) a

Next tactic:
rfl
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  

Proof state:
R : Type u
inst✝ : CommRing R
⊢ IsIso LaxMonoidal.ε

Next tactic:
refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
  · -- Porting note: broken ext
    

Proof state:
case refine'_1
R : Type u
inst✝ : CommRing R
⊢ LaxMonoidal.ε ≫ Finsupp.lapply PUnit.unit = 𝟙 (𝟙_ (ModuleCat R))

Next tactic:
apply LinearMap.ext_ring
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
  · -- Porting note: broken ext
    apply LinearMap.ext_ring
    -- Porting note: simp used to be able to close this goal
    

Proof state:
case refine'_1.h
R : Type u
inst✝ : CommRing R
⊢ (LaxMonoidal.ε ≫ Finsupp.lapply PUnit.unit) 1 = (𝟙 (𝟙_ (ModuleCat R))) 1

Next tactic:
dsimp
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
  · -- Porting note: broken ext
    apply LinearMap.ext_ring
    -- Porting note: simp used to be able to close this goal
    dsimp
    

Proof state:
case refine'_1.h
R : Type u
inst✝ : CommRing R
⊢ (ε R ≫ Finsupp.lapply PUnit.unit) 1 = (𝟙 (𝟙_ (ModuleCat R))) 1

Next tactic:
erw [ModuleCat.comp_def, LinearMap.comp_apply, ε_apply, Finsupp.lapply_apply,
      Finsupp.single_eq_same, id_apply]
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
  · -- Porting note: broken ext
    apply LinearMap.ext_ring
    -- Porting note: simp used to be able to close this goal
    dsimp
    erw [ModuleCat.comp_def, LinearMap.comp_apply, ε_apply, Finsupp.lapply_apply,
      Finsupp.single_eq_same, id_apply]
  · -- Porting note: broken ext
    

Proof state:
case refine'_2
R : Type u
inst✝ : CommRing R
⊢ Finsupp.lapply PUnit.unit ≫ LaxMonoidal.ε = 𝟙 ((free R).obj (𝟙_ (Type u)))

Next tactic:
apply Finsupp.lhom_ext'
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
  · -- Porting note: broken ext
    apply LinearMap.ext_ring
    -- Porting note: simp used to be able to close this goal
    dsimp
    erw [ModuleCat.comp_def, LinearMap.comp_apply, ε_apply, Finsupp.lapply_apply,
      Finsupp.single_eq_same, id_apply]
  · -- Porting note: broken ext
    apply Finsupp.lhom_ext'
    

Proof state:
case refine'_2.h
R : Type u
inst✝ : CommRing R
⊢ ∀ (a : 𝟙_ (Type u)),
    (Finsupp.lapply PUnit.unit ≫ LaxMonoidal.ε) ∘ₗ Finsupp.lsingle a =
      𝟙 ((free R).obj (𝟙_ (Type u))) ∘ₗ Finsupp.lsingle a

Next tactic:
intro ⟨⟩
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
  · -- Porting note: broken ext
    apply LinearMap.ext_ring
    -- Porting note: simp used to be able to close this goal
    dsimp
    erw [ModuleCat.comp_def, LinearMap.comp_apply, ε_apply, Finsupp.lapply_apply,
      Finsupp.single_eq_same, id_apply]
  · -- Porting note: broken ext
    apply Finsupp.lhom_ext'
    intro ⟨⟩
    

Proof state:
case refine'_2.h
R : Type u
inst✝ : CommRing R
⊢ (Finsupp.lapply PUnit.unit ≫ LaxMonoidal.ε) ∘ₗ Finsupp.lsingle PUnit.unit =
    𝟙 ((free R).obj (𝟙_ (Type u))) ∘ₗ Finsupp.lsingle PUnit.unit

Next tactic:
apply LinearMap.ext_ring
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
  · -- Porting note: broken ext
    apply LinearMap.ext_ring
    -- Porting note: simp used to be able to close this goal
    dsimp
    erw [ModuleCat.comp_def, LinearMap.comp_apply, ε_apply, Finsupp.lapply_apply,
      Finsupp.single_eq_same, id_apply]
  · -- Porting note: broken ext
    apply Finsupp.lhom_ext'
    intro ⟨⟩
    apply LinearMap.ext_ring
    

Proof state:
case refine'_2.h.h
R : Type u
inst✝ : CommRing R
⊢ ((Finsupp.lapply PUnit.unit ≫ LaxMonoidal.ε) ∘ₗ Finsupp.lsingle PUnit.unit) 1 =
    (𝟙 ((free R).obj (𝟙_ (Type u))) ∘ₗ Finsupp.lsingle PUnit.unit) 1

Next tactic:
apply Finsupp.ext
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
  · -- Porting note: broken ext
    apply LinearMap.ext_ring
    -- Porting note: simp used to be able to close this goal
    dsimp
    erw [ModuleCat.comp_def, LinearMap.comp_apply, ε_apply, Finsupp.lapply_apply,
      Finsupp.single_eq_same, id_apply]
  · -- Porting note: broken ext
    apply Finsupp.lhom_ext'
    intro ⟨⟩
    apply LinearMap.ext_ring
    apply Finsupp.ext
    

Proof state:
case refine'_2.h.h.h
R : Type u
inst✝ : CommRing R
⊢ ∀ (a : 𝟙_ (Type u)),
    (((Finsupp.lapply PUnit.unit ≫ LaxMonoidal.ε) ∘ₗ Finsupp.lsingle PUnit.unit) 1) a =
      ((𝟙 ((free R).obj (𝟙_ (Type u))) ∘ₗ Finsupp.lsingle PUnit.unit) 1) a

Next tactic:
intro ⟨⟩
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
  · -- Porting note: broken ext
    apply LinearMap.ext_ring
    -- Porting note: simp used to be able to close this goal
    dsimp
    erw [ModuleCat.comp_def, LinearMap.comp_apply, ε_apply, Finsupp.lapply_apply,
      Finsupp.single_eq_same, id_apply]
  · -- Porting note: broken ext
    apply Finsupp.lhom_ext'
    intro ⟨⟩
    apply LinearMap.ext_ring
    apply Finsupp.ext
    intro ⟨⟩
    -- Porting note: simp used to be able to close this goal
    

Proof state:
case refine'_2.h.h.h
R : Type u
inst✝ : CommRing R
⊢ (((Finsupp.lapply PUnit.unit ≫ LaxMonoidal.ε) ∘ₗ Finsupp.lsingle PUnit.unit) 1) PUnit.unit =
    ((𝟙 ((free R).obj (𝟙_ (Type u))) ∘ₗ Finsupp.lsingle PUnit.unit) 1) PUnit.unit

Next tactic:
dsimp
instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _) := by
  refine' ⟨⟨Finsupp.lapply PUnit.unit, ⟨_, _⟩⟩⟩
  · -- Porting note: broken ext
    apply LinearMap.ext_ring
    -- Porting note: simp used to be able to close this goal
    dsimp
    erw [ModuleCat.comp_def, LinearMap.comp_apply, ε_apply, Finsupp.lapply_apply,
      Finsupp.single_eq_same, id_apply]
  · -- Porting note: broken ext
    apply Finsupp.lhom_ext'
    intro ⟨⟩
    apply LinearMap.ext_ring
    apply Finsupp.ext
    intro ⟨⟩
    -- Porting note: simp used to be able to close this goal
    dsimp
    

Proof state:
case refine'_2.h.h.h
R : Type u
inst✝ : CommRing R
⊢ ((Finsupp.lapply PUnit.unit ≫ ε R) fun₀ | PUnit.unit => 1) PUnit.unit =
    ((𝟙 (of R (𝟙_ (Type u) →₀ R))) fun₀ | PUnit.unit => 1) PUnit.unit

Next tactic:
erw [ModuleCat.comp_def, LinearMap.comp_apply, ε_apply, Finsupp.lapply_apply,
      Finsupp.single_eq_same]
/-- The free functor `Type u ⥤ ModuleCat R`, as a monoidal functor. -/
def monoidalFree : MonoidalFunctor (Type u) (ModuleCat.{u} R) :=
  { LaxMonoidalFunctor.of (free R).obj with
    -- Porting note: used to be dsimp
    ε_isIso := (by 

Proof state:
R : Type u
inst✝ : CommRing R
src✝ : LaxMonoidalFunctor (Type u) (ModuleCat R) := LaxMonoidalFunctor.of (free R).toPrefunctor.obj
⊢ IsIso LaxMonoidal.ε

Next tactic:
infer_instance
/-- The free functor `Type u ⥤ ModuleCat R`, as a monoidal functor. -/
def monoidalFree : MonoidalFunctor (Type u) (ModuleCat.{u} R) :=
  { LaxMonoidalFunctor.of (free R).obj with
    -- Porting note: used to be dsimp
    ε_isIso := (by infer_instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _))
    μ_isIso := fun X Y => by 

Proof state:
R : Type u
inst✝ : CommRing R
src✝ : LaxMonoidalFunctor (Type u) (ModuleCat R) := LaxMonoidalFunctor.of (free R).toPrefunctor.obj
X Y : Type u
⊢ IsIso (LaxMonoidalFunctor.μ (LaxMonoidalFunctor.mk src✝.toFunctor src✝.ε src✝.μ) X Y)

Next tactic:
dsimp
/-- The free functor `Type u ⥤ ModuleCat R`, as a monoidal functor. -/
def monoidalFree : MonoidalFunctor (Type u) (ModuleCat.{u} R) :=
  { LaxMonoidalFunctor.of (free R).obj with
    -- Porting note: used to be dsimp
    ε_isIso := (by infer_instance : IsIso (@LaxMonoidal.ε _ _ _ _ _ _ (free R).obj _ _))
    μ_isIso := fun X Y => by dsimp; 

Proof state:
R : Type u
inst✝ : CommRing R
src✝ : LaxMonoidalFunctor (Type u) (ModuleCat R) := LaxMonoidalFunctor.of (free R).toPrefunctor.obj
X Y : Type u
⊢ IsIso (Free.μ R X Y).hom

Next tactic:
infer_instance
instance categoryFree : Category (Free R C) where
  Hom := fun X Y : C => (X ⟶ Y) →₀ R
  id := fun X : C => Finsupp.single (𝟙 X) 1
  comp {X Y Z : C} f g :=
    (f.sum (fun f' s => g.sum (fun g' t => Finsupp.single (f' ≫ g') (s * t))) : (X ⟶ Z) →₀ R)
  assoc {W X Y Z} f g h := by
    

Proof state:
R : Type u_1
inst✝¹ : CommRing R
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : Free R C
f : W ⟶ X
g : X ⟶ Y
h : Y ⟶ Z
⊢ (f ≫ g) ≫ h = f ≫ g ≫ h

Next tactic:
dsimp
instance categoryFree : Category (Free R C) where
  Hom := fun X Y : C => (X ⟶ Y) →₀ R
  id := fun X : C => Finsupp.single (𝟙 X) 1
  comp {X Y Z : C} f g :=
    (f.sum (fun f' s => g.sum (fun g' t => Finsupp.single (f' ≫ g') (s * t))) : (X ⟶ Z) →₀ R)
  assoc {W X Y Z} f g h := by
    dsimp
    -- This imitates the proof of associativity for `MonoidAlgebra`.
    

Proof state:
R : Type u_1
inst✝¹ : CommRing R
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : Free R C
f : W ⟶ X
g : X ⟶ Y
h : Y ⟶ Z
⊢ (sum (sum f fun f' s => sum g fun g' t => fun₀ | f' ≫ g' => s * t) fun f' s =>
      sum h fun g' t => fun₀ | f' ≫ g' => s * t) =
    sum f fun f' s =>
      sum (sum g fun f' s => sum h fun g' t => fun₀ | f' ≫ g' => s * t) fun g' t => fun₀ | f' ≫ g' => s * t

Next tactic:
simp only [sum_sum_index, sum_single_index, single_zero, single_add, eq_self_iff_true,
      forall_true_iff, forall₃_true_iff, add_mul, mul_add, Category.assoc, mul_assoc,
      zero_mul, mul_zero, sum_zero, sum_add]
instance : Preadditive (Free R C) where
  homGroup X Y := Finsupp.addCommGroup
  add_comp X Y Z f f' g := by
    

Proof state:
R : Type u_1
inst✝¹ : CommRing R
C : Type u
inst✝ : Category.{v, u} C
X Y Z : Free R C
f f' : X ⟶ Y
g : Y ⟶ Z
⊢ (f + f') ≫ g = f ≫ g + f' ≫ g

Next tactic:
dsimp [CategoryTheory.categoryFree]
instance : Preadditive (Free R C) where
  homGroup X Y := Finsupp.addCommGroup
  add_comp X Y Z f f' g := by
    dsimp [CategoryTheory.categoryFree]
    

Proof state:
R : Type u_1
inst✝¹ : CommRing R
C : Type u
inst✝ : Category.{v, u} C
X Y Z : Free R C
f f' : X ⟶ Y
g : Y ⟶ Z
⊢ (sum (f + f') fun f' s => sum g fun g' t => fun₀ | f' ≫ g' => s * t) =
    (sum f fun f' s => sum g fun g' t => fun₀ | f' ≫ g' => s * t) +
      sum f' fun f' s => sum g fun g' t => fun₀ | f' ≫ g' => s * t

Next tactic:
rw [Finsupp.sum_add_index']
instance : Preadditive (Free R C) where
  homGroup X Y := Finsupp.addCommGroup
  add_comp X Y Z f f' g := by
    dsimp [CategoryTheory.categoryFree]
    rw [Finsupp.sum_add_index'] <;> · 

Proof state:
case h_zero
R : Type u_1
inst✝¹ : CommRing R
C : Type u
inst✝ : Category.{v, u} C
X Y Z : Free R C
f f' : X ⟶ Y
g : Y ⟶ Z
⊢ ∀ (a : X ⟶ Y), (sum g fun g' t => fun₀ | a ≫ g' => 0 * t) = 0

Next tactic:
simp [add_mul]