l3lab/ntp-mathlib huggingface dataset:
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · 

Proof state:
case hsingle
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
⊢ (RingEquiv.symm (toFinsuppIso R)) (r • (single R n) a) = r * (RingEquiv.symm (toFinsuppIso R)) ((single R n) a)

Next tactic:
ext i
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        

Proof state:
case hsingle.a
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
⊢ coeff ((RingEquiv.symm (toFinsuppIso R)) (r • (single R n) a)) i =
    coeff (r * (RingEquiv.symm (toFinsuppIso R)) ((single R n) a)) i

Next tactic:
simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        

Proof state:
case hsingle.a
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
⊢ (if n ≤ i then coeff r (i - n) * a else 0) = ∑ x in Finset.antidiagonal i, if n = x.2 then coeff r x.1 * a else 0

Next tactic:
split_ifs with hn
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · 

Proof state:
case pos
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
⊢ coeff r (i - n) * a = ∑ x in Finset.antidiagonal i, if n = x.2 then coeff r x.1 * a else 0

Next tactic:
rw [Finset.sum_eq_single (i - n, n)]
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          

Proof state:
case pos
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
⊢ coeff r (i - n) * a = if n = (i - n, n).2 then coeff r (i - n, n).1 * a else 0
case pos.h₀
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
⊢ ∀ b ∈ Finset.antidiagonal i, b ≠ (i - n, n) → (if n = b.2 then coeff r b.1 * a else 0) = 0
case pos.h₁
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
⊢ (i - n, n) ∉ Finset.antidiagonal i → (if n = (i - n, n).2 then coeff r (i - n, n).1 * a else 0) = 0

Next tactic:
simp only [ite_true]
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · 

Proof state:
case pos.h₀
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
⊢ ∀ b ∈ Finset.antidiagonal i, b ≠ (i - n, n) → (if n = b.2 then coeff r b.1 * a else 0) = 0

Next tactic:
rintro ⟨p, q⟩ hpq1 hpq2
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            

Proof state:
case pos.h₀.mk
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
p q : ℕ
hpq1 : (p, q) ∈ Finset.antidiagonal i
hpq2 : (p, q) ≠ (i - n, n)
⊢ (if n = (p, q).2 then coeff r (p, q).1 * a else 0) = 0

Next tactic:
rw [Finset.mem_antidiagonal] at hpq1
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            rw [Finset.mem_antidiagonal] at hpq1
            

Proof state:
case pos.h₀.mk
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
p q : ℕ
hpq1 : (p, q).1 + (p, q).2 = i
hpq2 : (p, q) ≠ (i - n, n)
⊢ (if n = (p, q).2 then coeff r (p, q).1 * a else 0) = 0

Next tactic:
split_ifs with H
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            rw [Finset.mem_antidiagonal] at hpq1
            split_ifs with H
            · 

Proof state:
case pos
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
p q : ℕ
hpq1 : (p, q).1 + (p, q).2 = i
hpq2 : (p, q) ≠ (i - n, n)
H : n = (p, q).2
⊢ coeff r (p, q).1 * a = 0

Next tactic:
dsimp at H
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            rw [Finset.mem_antidiagonal] at hpq1
            split_ifs with H
            · dsimp at H
              

Proof state:
case pos
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
p q : ℕ
hpq1 : (p, q).1 + (p, q).2 = i
hpq2 : (p, q) ≠ (i - n, n)
H : n = q
⊢ coeff r (p, q).1 * a = 0

Next tactic:
exfalso
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            rw [Finset.mem_antidiagonal] at hpq1
            split_ifs with H
            · dsimp at H
              exfalso
              

Proof state:
case pos.h
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
p q : ℕ
hpq1 : (p, q).1 + (p, q).2 = i
hpq2 : (p, q) ≠ (i - n, n)
H : n = q
⊢ False

Next tactic:
apply hpq2
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            rw [Finset.mem_antidiagonal] at hpq1
            split_ifs with H
            · dsimp at H
              exfalso
              apply hpq2
              

Proof state:
case pos.h
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
p q : ℕ
hpq1 : (p, q).1 + (p, q).2 = i
hpq2 : (p, q) ≠ (i - n, n)
H : n = q
⊢ (p, q) = (i - n, n)

Next tactic:
rw [← hpq1, H]
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            rw [Finset.mem_antidiagonal] at hpq1
            split_ifs with H
            · dsimp at H
              exfalso
              apply hpq2
              rw [← hpq1, H]
              

Proof state:
case pos.h
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
p q : ℕ
hpq1 : (p, q).1 + (p, q).2 = i
hpq2 : (p, q) ≠ (i - n, n)
H : n = q
⊢ (p, q) = ((p, q).1 + (p, q).2 - q, q)

Next tactic:
simp only [add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right]
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            rw [Finset.mem_antidiagonal] at hpq1
            split_ifs with H
            · dsimp at H
              exfalso
              apply hpq2
              rw [← hpq1, H]
              simp only [add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right]
            · 

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
p q : ℕ
hpq1 : (p, q).1 + (p, q).2 = i
hpq2 : (p, q) ≠ (i - n, n)
H : ¬n = (p, q).2
⊢ 0 = 0

Next tactic:
rfl
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            rw [Finset.mem_antidiagonal] at hpq1
            split_ifs with H
            · dsimp at H
              exfalso
              apply hpq2
              rw [← hpq1, H]
              simp only [add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right]
            · rfl
          · 

Proof state:
case pos.h₁
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
⊢ (i - n, n) ∉ Finset.antidiagonal i → (if n = (i - n, n).2 then coeff r (i - n, n).1 * a else 0) = 0

Next tactic:
intro H
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            rw [Finset.mem_antidiagonal] at hpq1
            split_ifs with H
            · dsimp at H
              exfalso
              apply hpq2
              rw [← hpq1, H]
              simp only [add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right]
            · rfl
          · intro H
            

Proof state:
case pos.h₁
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
H : (i - n, n) ∉ Finset.antidiagonal i
⊢ (if n = (i - n, n).2 then coeff r (i - n, n).1 * a else 0) = 0

Next tactic:
exfalso
/-- `PolynomialModule R R` is isomorphic to `R[X]` as an `R[X]` module. -/
noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :=
  { (Polynomial.toFinsuppIso R).symm with
    map_smul' := fun r x => by
      dsimp
      rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
      induction' x using induction_linear with _ _ hp hq n a
      · rw [smul_zero, map_zero, mul_zero]
      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
      · ext i
        simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
        single_apply, ge_iff_le, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
        split_ifs with hn
        · rw [Finset.sum_eq_single (i - n, n)]
          simp only [ite_true]
          · rintro ⟨p, q⟩ hpq1 hpq2
            rw [Finset.mem_antidiagonal] at hpq1
            split_ifs with H
            · dsimp at H
              exfalso
              apply hpq2
              rw [← hpq1, H]
              simp only [add_le_iff_nonpos_left, nonpos_iff_eq_zero, add_tsub_cancel_right]
            · rfl
          · intro H
            exfalso
            

Proof state:
case pos.h₁.h
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
I : Ideal R
S : Type u_3
inst✝³ : CommSemiring S
inst✝² : Algebra S R
inst✝¹ : Module S M
inst✝ : IsScalarTower S R M
src✝ : AddMonoidAlgebra R ℕ ≃+* R[X] := RingEquiv.symm (toFinsuppIso R)
r : R[X]
n : ℕ
a : R
i : ℕ
hn : n ≤ i
H : (i - n, n) ∉ Finset.antidiagonal i
⊢ False

Next tactic:
apply H