l3lab/ntp-mathlib huggingface dataset:
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((uncurry R N P ((M ⊗[R] N) ⊗[R] P)) ((curry (mk R (M ⊗[R] N) P)) m)) (n ⊗ₜ[R] p) = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [compr₂_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((uncurry R N P ((M ⊗[R] N) ⊗[R] P)) ((curry (mk R (M ⊗[R] N) P)) m)) (n ⊗ₜ[R] p) = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [comp_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((uncurry R N P ((M ⊗[R] N) ⊗[R] P)) ((curry (mk R (M ⊗[R] N) P)) m)) (n ⊗ₜ[R] p) = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [mk_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|rw [mk_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((uncurry R N P ((M ⊗[R] N) ⊗[R] P)) ((curry (mk R (M ⊗[R] N) P)) m)) (n ⊗ₜ[R] p) = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [flip_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|rw [mk_apply]|rw [flip_apply]
        |

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((uncurry R N P ((M ⊗[R] N) ⊗[R] P)) ((curry (mk R (M ⊗[R] N) P)) m)) (n ⊗ₜ[R] p) = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [lcurry_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|rw [mk_apply]|rw [flip_apply]
        |rw [lcurry_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((uncurry R N P ((M ⊗[R] N) ⊗[R] P)) ((curry (mk R (M ⊗[R] N) P)) m)) (n ⊗ₜ[R] p) = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [uncurry_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ (((curry (mk R (M ⊗[R] N) P)) m) n) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|rw [mk_apply]|rw [flip_apply]
        |rw [lcurry_apply]|rw [uncurry_apply]|rw [curry_apply]|rw [id_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ (((curry (mk R (M ⊗[R] N) P)) m) n) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [lift.tmul]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ (((curry (mk R (M ⊗[R] N) P)) m) n) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [compr₂_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ (((curry (mk R (M ⊗[R] N) P)) m) n) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [comp_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ (((curry (mk R (M ⊗[R] N) P)) m) n) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [mk_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|rw [mk_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ (((curry (mk R (M ⊗[R] N) P)) m) n) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [flip_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|rw [mk_apply]|rw [flip_apply]
        |

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ (((curry (mk R (M ⊗[R] N) P)) m) n) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [lcurry_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|rw [mk_apply]|rw [flip_apply]
        |rw [lcurry_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ (((curry (mk R (M ⊗[R] N) P)) m) n) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [uncurry_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|rw [mk_apply]|rw [flip_apply]
        |rw [lcurry_apply]|rw [uncurry_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ (((curry (mk R (M ⊗[R] N) P)) m) n) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [curry_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((mk R (M ⊗[R] N) P) (m ⊗ₜ[R] n)) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|rw [mk_apply]|rw [flip_apply]
        |rw [lcurry_apply]|rw [uncurry_apply]|rw [curry_apply]|rw [id_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((mk R (M ⊗[R] N) P) (m ⊗ₜ[R] n)) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [lift.tmul]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((mk R (M ⊗[R] N) P) (m ⊗ₜ[R] n)) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [compr₂_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((mk R (M ⊗[R] N) P) (m ⊗ₜ[R] n)) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [comp_apply]
/-- The associator for tensor product of R-modules, as a linear equivalence. -/
protected def assoc : (M ⊗[R] N) ⊗[R] P ≃ₗ[R] M ⊗[R] N ⊗[R] P := by
  refine
      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)
        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)
        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)
        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>
    repeat'
      first
        |rw [lift.tmul]|rw [compr₂_apply]|rw [comp_apply]|

Proof state:
case refine_2
R : Type u_1
inst✝¹⁶ : CommSemiring R
R' : Type u_2
inst✝¹⁵ : Monoid R'
R'' : Type u_3
inst✝¹⁴ : Semiring R''
M : Type u_4
N : Type u_5
P : Type u_6
Q : Type u_7
S : Type u_8
T : Type u_9
inst✝¹³ : AddCommMonoid M
inst✝¹² : AddCommMonoid N
inst✝¹¹ : AddCommMonoid P
inst✝¹⁰ : AddCommMonoid Q
inst✝⁹ : AddCommMonoid S
inst✝⁸ : AddCommMonoid T
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Module R S
inst✝² : Module R T
inst✝¹ : DistribMulAction R' M
inst✝ : Module R'' M
p : P
m : M
n : N
⊢ ((mk R (M ⊗[R] N) P) (m ⊗ₜ[R] n)) p = LinearMap.id ((m ⊗ₜ[R] n) ⊗ₜ[R] p)

Next tactic:
rw [mk_apply]