mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Module A M
inst✝⁵ : Module B M
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
inst✝² : IsScalarTower R A M
inst✝¹ : IsScalarTower R B M
inst✝ : SMulCommClass A B M
m : M
⊢ 0 • m = 0

Tactic:
by simp only [(· • ·), map_zero, LinearMap.zero_apply]

Goal state after:
Module: Mathlib.Algebra.Order.Star.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁷ : Semiring R
inst✝⁶ : StarRing R
inst✝⁵ : NonUnitalSemiring A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : StarModule R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
r✝ : R
a✝ : A
r : R
a : A
⊢ (fun s ↦ star s * s) (r • a) = (fun s ↦ star s * s) r • (fun s ↦ star s * s) a

Tactic:
by simp [star_smul, smul_mul_smul_comm]

Goal state after:
Module: Mathlib.Algebra.Order.Star.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : NonUnitalSemiring R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
inst✝ : StarOrderedRing R
x y : R
⊢ star x < y ↔ x < star y

Tactic:
← star_lt_star_iff,

Goal state after:
R : Type u_1
inst✝³ : NonUnitalSemiring R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
inst✝ : StarOrderedRing R
x y : R
⊢ star (star x) < star y ↔ x < star y
Module: Mathlib.LinearAlgebra.Dimension.Constructions

Goal state before:
pp: R : Type u
M : Type v
M' : Type v'
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid M'
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Nontrivial R
⊢ Module.rank R (ULift.{v', v} M) + Module.rank R (ULift.{v, v'} M') ≤ Module.rank R (M × M')

Tactic:
exact (rank_add_rank_le_rank_prod R _).trans_eq
    (ULift.moduleEquiv.prodCongr ULift.moduleEquiv).rank_eq

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.Constructions

Goal state before:
pp: R : Type u
inst✝⁵ : Semiring R
inst✝⁴ : StrongRankCondition R
M η : Type u
inst✝³ : Fintype η
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Free R M
⊢ #η * Module.rank R M = ↑(Fintype.card η) * Module.rank R M

Tactic:
Cardinal.mk_fintype

Goal state after:
R : Type u
inst✝⁵ : Semiring R
inst✝⁴ : StrongRankCondition R
M η : Type u
inst✝³ : Fintype η
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Free R M
⊢ ↑(Fintype.card η) * Module.rank R M = ↑(Fintype.card η) * Module.rank R M
Module: Mathlib.LinearAlgebra.Basis.VectorSpace

Goal state before:
pp: K : Type u_3
V : Type u_4
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
p : Submodule K V
v : V
hv : v ∉ p
⊢ ∃ f, f v ≠ 0 ∧ p ≤ f.ker

Tactic:
rcases LinearMap.exists_extend_of_notMem (0 : p →ₗ[K] K) hv 1 with ⟨f, hpf, hfv⟩
  refine ⟨f, by simp [hfv], fun x hx ↦ ?_⟩
  simpa using congr($hpf ⟨x, hx⟩)

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.VectorSpace

Goal state before:
pp: case hs.hs
K : Type u_6
V : Type u_7
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : V →ₗ[K] K
v : V
hfv : f v ≠ 0
b₁ : Basis (↑(Basis.ofVectorSpaceIndex K ↥f.ker)) K ↥f.ker := ⋯
s : Set V := ⋯
hs : span K s = f.ker
n : Set V := ⋯
⊢ LinearIndepOn K _root_.id (Set.range ⇑b₁)

Tactic:
exact b₁.linearIndependent.linearIndepOn_id

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommSemiring R
inst✝⁶ : StarRing R
inst✝⁵ : NonUnitalSemiring A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : StarModule R A
⊢ ↑⊥ = {0}

Tactic:
by
  simp only [Set.ext_iff, NonUnitalStarAlgebra.mem_bot, SetLike.mem_coe, Set.mem_singleton_iff,
    forall_const]

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.Finite

Goal state before:
pp: R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
h : Module.rank R M = 0
⊢ toNat (Module.rank R M) = 0

Tactic:
h,

Goal state after:
R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
h : Module.rank R M = 0
⊢ toNat 0 = 0
Module: Mathlib.LinearAlgebra.Dimension.Finite

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
S : Submodule R M
h : S = ⊥
⊢ Module.rank R ↥S = 0

Tactic:
h,

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
S : Submodule R M
h : S = ⊥
⊢ Module.rank R ↥⊥ = 0
Module: Mathlib.LinearAlgebra.Basis.VectorSpace

Goal state before:
pp: case hs.hs
K : Type u_6
V : Type u_7
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : V →ₗ[K] K
hf : f ≠ 0
v : ↥f.ker
hv : ↑v ≠ 0
this : LinearIndepOn K _root_.id {v}
b₁ : Basis (↑(this.extend ⋯)) K ↥f.ker := ⋯
w : V
hw : f w = 1
s : Set V := ⋯
hs : span K s = f.ker
hvs : ↑v ∈ s
n : Set V := ⋯
⊢ LinearIndepOn K _root_.id (Set.range ⇑b₁)

Tactic:
exact b₁.linearIndependent.linearIndepOn_id

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Defs

Goal state before:
pp: K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
⊢ FiniteDimensional K ↥S

Tactic:
by
  infer_instance

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommSemiring R
inst✝⁶ : StarRing R
inst✝⁵ : NonUnitalSemiring A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : StarModule R A
s : Set A
⊢ adjoin R s ≤ centralizer R ↑(centralizer R s)

Tactic:
by
  rw [← toNonUnitalSubalgebra_le_iff, centralizer_toNonUnitalSubalgebra,
    adjoin_toNonUnitalSubalgebra]
  convert! NonUnitalAlgebra.adjoin_le_centralizer_centralizer R (s ∪ star s)
  rw [StarMemClass.star_coe_eq]
  simp

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Basic

Goal state before:
pp: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W W' : Submodule K V
inst✝ : FiniteDimensional K ↥W
f : ↥W ≃ₗ[K] ↥W'
Q : Submodule K V
hQ : IsCompl W Q
eQ : (↥W × ↥Q) ≃ₗ[K] V := W.prodEquivOfIsCompl Q hQ
Q' : Submodule K V
hQ' : IsCompl W' Q'
eQ' : (↥W' × ↥Q') ≃ₗ[K] V := W'.prodEquivOfIsCompl Q' hQ'
⊢ Module.rank K ↥W < ℵ₀

Tactic:
exact Module.rank_lt_aleph0 K ↥W

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition

Goal state before:
pp: case mp
K : Type u
V : Type v
inst✝⁴ : Ring K
inst✝³ : StrongRankCondition K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
W : Submodule K V
inst✝ : Free K ↥W
m : V
hm : m ∈ W
f : ↥W → K
hf : ∀ (v : ↥W), f v • ⟨m, hm⟩ = v
⊢ ∃ a, (∀ v ∈ W, ∃ r, r • a = v) ∧ a ∈ W

Tactic:
exact ⟨m, ⟨fun v hv => ⟨f ⟨v, hv⟩, congr_arg ((↑) : W → V) (hf ⟨v, hv⟩)⟩, hm⟩⟩

Goal state after:
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: case w.refine_1
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Ring S
σ : R →+* S
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module S F
f g : E →ₛₗ.[σ] F
h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y
x : ↥(f.domain ⊔ g.domain) → E
hx : ∀ (z : ↥(f.domain ⊔ g.domain)), x z ∈ f.domain
y : ↥(f.domain ⊔ g.domain) → E
hy : ∀ (z : ↥(f.domain ⊔ g.domain)), y z ∈ g.domain
hxy : ∀ (z : ↥(f.domain ⊔ g.domain)), x z + y z = ↑z
fg : ↥(f.domain ⊔ g.domain) → F := fun z ↦ ↑f ⟨x z, ⋯⟩ + ↑g ⟨y z, ⋯⟩
fg_eq : ∀ (x' : ↥f.domain) (y' : ↥g.domain) (z' : ↥(f.domain ⊔ g.domain)), ↑x' + ↑y' = ↑z' → fg z' = ↑f x' + ↑g y'
z₁ : E
hz₁ : z₁ ∈ f.domain ⊔ g.domain
z₂ : E
hz₂ : z₂ ∈ f.domain ⊔ g.domain
⊢ z₁ + ↑⟨z₂, hz₂⟩ = z₁ + z₂

Tactic:
coe_mk

Goal state after:
case w.refine_1
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : Ring S
σ : R →+* S
E : Type u_4
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_5
inst✝¹ : AddCommGroup F
inst✝ : Module S F
f g : E →ₛₗ.[σ] F
h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y
x : ↥(f.domain ⊔ g.domain) → E
hx : ∀ (z : ↥(f.domain ⊔ g.domain)), x z ∈ f.domain
y : ↥(f.domain ⊔ g.domain) → E
hy : ∀ (z : ↥(f.domain ⊔ g.domain)), y z ∈ g.domain
hxy : ∀ (z : ↥(f.domain ⊔ g.domain)), x z + y z = ↑z
fg : ↥(f.domain ⊔ g.domain) → F := fun z ↦ ↑f ⟨x z, ⋯⟩ + ↑g ⟨y z, ⋯⟩
fg_eq : ∀ (x' : ↥f.domain) (y' : ↥g.domain) (z' : ↥(f.domain ⊔ g.domain)), ↑x' + ↑y' = ↑z' → fg z' = ↑f x' + ↑g y'
z₁ : E
hz₁ : z₁ ∈ f.domain ⊔ g.domain
z₂ : E
hz₂ : z₂ ∈ f.domain ⊔ g.domain
⊢ z₁ + z₂ = z₁ + z₂
Module: Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition

Goal state before:
pp: K V : Type u
inst✝⁵ : Ring K
inst✝⁴ : StrongRankCondition K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : Free K V
inst✝ : Module.Finite K V
⊢ #V = #K ^ Module.rank K V

Tactic:
by
  simpa using lift_cardinalMk_eq_lift_cardinalMk_field_pow_lift_rank K V

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition

Goal state before:
pp: F : Type u_1
E : Type u_2
inst✝⁵ : CommRing F
inst✝⁴ : StrongRankCondition F
inst✝³ : Ring E
inst✝² : Algebra F E
inst✝¹ : Nontrivial E
inst✝ : Free F E
⊢ ⊥ = ⊤ ↔ Module.rank F E = 1

Tactic:
have := Module.Free.of_equiv (Subalgebra.topEquiv (R := F) (A := E)).toLinearEquiv.symm
  rw [← rank_top, Subalgebra.rank_eq_one_iff, eq_comm]

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: R : Type u_1
X : Type u_2
inst✝ : CommSemiring R
⊢ ∀ (a : FreeAlgebra R X), 0 + a = a

Tactic:
rintro ⟨⟩
    exact Quot.sound Rel.zero_add

Goal state after:
Module: Mathlib.LinearAlgebra.FiniteDimensional.Basic

Goal state before:
pp: case a
K : Type u
V : Type v
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
k : ℕ
h : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)
m : ℕ
⊢ comap (f ^ m) (LinearMap.ker (f ^ (k + 1))) ≤ comap (f ^ m) (LinearMap.ker (f ^ k))

Tactic:
h,

Goal state after:
case a
K : Type u
V : Type v
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
k : ℕ
h : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)
m : ℕ
⊢ comap (f ^ m) (LinearMap.ker (f ^ (k + 1))) ≤ comap (f ^ m) (LinearMap.ker (f ^ k.succ))
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: case left_distrib
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝¹ : FreeAlgebra R X
a b a✝ b✝ c✝ : Pre R X
⊢ liftFun R X f a✝ * (liftFun R X f b✝ + liftFun R X f c✝) =
    liftFun R X f a✝ * liftFun R X f b✝ + liftFun R X f a✝ * liftFun R X f c✝

Tactic:
rw [left_distrib]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Eval.Defs

Goal state before:
pp: case hf
R : Type u
S : Type v
a : R
inst✝¹ : Semiring R
p : R[X]
inst✝ : Semiring S
f : R →+* S
x : S
h : Commute (f a) x
⊢ ∀ (k : ℕ), Commute (f ((C a).coeff k)) x

Tactic:
intro k

Goal state after:
case hf
R : Type u
S : Type v
a : R
inst✝¹ : Semiring R
p : R[X]
inst✝ : Semiring S
f : R →+* S
x : S
h : Commute (f a) x
k : ℕ
⊢ Commute (f ((C a).coeff k)) x
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: R : Type u_1
X : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
x : X
r : R
h : ι R x = (algebraMap R (FreeAlgebra R X)) r
f0 : FreeAlgebra R X →ₐ[R] R := (lift R) 0
f1 : FreeAlgebra R X →ₐ[R] R := (lift R) 1
hf0 : f0 (ι R x) = 0
hf1 : f1 (ι R x) = 1
⊢ False

Tactic:
h,

Goal state after:
R : Type u_1
X : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
x : X
r : R
h : ι R x = (algebraMap R (FreeAlgebra R X)) r
f0 : FreeAlgebra R X →ₐ[R] R := (lift R) 0
f1 : FreeAlgebra R X →ₐ[R] R := (lift R) 1
hf0 : f0 ((algebraMap R (FreeAlgebra R X)) r) = 0
hf1 : f1 (ι R x) = 1
⊢ False
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: R : Type u_1
X : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
x : X
r : R
h : ι R x = (algebraMap R (FreeAlgebra R X)) r
f0 : FreeAlgebra R X →ₐ[R] R := (lift R) 0
f1 : FreeAlgebra R X →ₐ[R] R := (lift R) 1
hf0 : r = 0
hf1 : f1 (ι R x) = 1
⊢ False

Tactic:
h,

Goal state after:
R : Type u_1
X : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
x : X
r : R
h : ι R x = (algebraMap R (FreeAlgebra R X)) r
f0 : FreeAlgebra R X →ₐ[R] R := (lift R) 0
f1 : FreeAlgebra R X →ₐ[R] R := (lift R) 1
hf0 : r = 0
hf1 : f1 ((algebraMap R (FreeAlgebra R X)) r) = 1
⊢ False
Module: Mathlib.Algebra.Polynomial.Eval.Defs

Goal state before:
pp: R : Type u
inst✝ : Ring R
p q : R[X]
n : ℕ
⊢ (p * X - p * ↑n).comp q = p.comp q * (q - ↑n)

Tactic:
sub_comp,

Goal state after:
R : Type u
inst✝ : Ring R
p q : R[X]
n : ℕ
⊢ (p * X).comp q - (p * ↑n).comp q = p.comp q * (q - ↑n)
Module: Mathlib.Algebra.CharP.Defs

Goal state before:
pp: R : Type u_1
inst✝¹ : AddMonoidWithOne R
p : ℕ
inst✝ : CharP R p
k a : ℕ
⊢ ↑(p * a) = 0

Tactic:
rw [CharP.cast_eq_zero_iff R p]

Goal state after:
R : Type u_1
inst✝¹ : AddMonoidWithOne R
p : ℕ
inst✝ : CharP R p
k a : ℕ
⊢ p ∣ p * a
Module: Mathlib.Algebra.MonoidAlgebra.Degree

Goal state before:
pp: R : Type u_1
A : Type u_3
B : Type u_5
inst✝⁶ : SemilatticeSup B
inst✝⁵ : OrderBot B
inst✝⁴ : Semiring R
inst✝³ : AddMonoid A
inst✝² : AddMonoid B
inst✝¹ : AddLeftMono B
inst✝ : AddRightMono B
degb : A → B
degb0 : degb 0 ≤ 0
degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b
n : ℕ
f : R[A]
⊢ (f ^ n).coeff.support.sup degb ≤ n • f.coeff.support.sup degb

Tactic:
rw [← List.prod_replicate, ← List.sum_replicate]
  refine (sup_support_list_prod_le degb0 degbm _).trans_eq ?_
  rw [List.map_replicate]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Degree.Defs

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
h : p.degree = ↑n
⊢ WithBot.unbotD 0 p.degree = n

Tactic:
h,

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
h : p.degree = ↑n
⊢ WithBot.unbotD 0 ↑n = n
Module: Mathlib.Algebra.Polynomial.Degree.Defs

Goal state before:
pp: case pos
R : Type u
a : R
inst✝ : Semiring R
h : a = 0
⊢ (C a).degree ≤ 0

Tactic:
h,

Goal state after:
case pos
R : Type u
a : R
inst✝ : Semiring R
h : a = 0
⊢ (C 0).degree ≤ 0
Module: Mathlib.Algebra.Polynomial.Degree.Defs

Goal state before:
pp: R : Type u
a : R
inst✝ : Semiring R
n : ℕ
ha : a ≠ 0
⊢ ((monomial n) a).degree = ↑n

Tactic:
rw [degree, support_monomial n ha, max_singleton, Nat.cast_withBot]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Degree.Defs

Goal state before:
pp: R : Type u
inst✝ : Semiring R
n : ℕ
a : R
this : DecidableEq R := Classical.decEq R
h : a = 0
⊢ ((monomial n) a).degree ≤ ↑n

Tactic:
h,

Goal state after:
R : Type u
inst✝ : Semiring R
n : ℕ
a : R
this : DecidableEq R := Classical.decEq R
h : a = 0
⊢ ((monomial n) 0).degree ≤ ↑n
Module: Mathlib.Algebra.Polynomial.Degree.Defs

Goal state before:
pp: R : Type u
inst✝ : Ring R
p : R[X]
⊢ (-p).leadingCoeff = -p.leadingCoeff

Tactic:
rw [leadingCoeff, leadingCoeff, natDegree_neg, coeff_neg]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.degree = q.degree
hlc : p.leadingCoeff + q.leadingCoeff ≠ 0
⊢ max p.degree q.degree = p.degree

Tactic:
h,

Goal state after:
R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.degree = q.degree
hlc : p.leadingCoeff + q.leadingCoeff ≠ 0
⊢ max q.degree q.degree = q.degree
Module: Mathlib.Algebra.Polynomial.Degree.Defs

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
inst✝ : Nontrivial R
c : R
hc : (C c).Monic
⊢ c ≠ 0

Tactic:
by
  rintro rfl
  simp [Monic] at hc

Goal state after:
Module: Mathlib.Algebra.Polynomial.Degree.Defs

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p q : R[X]
⊢ p.degree + q.degree ≤ ↑p.natDegree + ↑q.natDegree

Tactic:
apply add_le_add <;> apply degree_le_natDegree

Goal state after:
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.leadingCoeff * q.leadingCoeff ≠ 0
⊢ p ≠ 0

Tactic:
refine mt ?_ h

Goal state after:
R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.leadingCoeff * q.leadingCoeff ≠ 0
⊢ p = 0 → p.leadingCoeff * q.leadingCoeff = 0
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.leadingCoeff * q.leadingCoeff ≠ 0
hp : p ≠ 0
⊢ q ≠ 0

Tactic:
refine mt ?_ h

Goal state after:
R : Type u
inst✝ : Semiring R
p q : R[X]
h : p.leadingCoeff * q.leadingCoeff ≠ 0
hp : p ≠ 0
⊢ q = 0 → p.leadingCoeff * q.leadingCoeff = 0
Module: Mathlib.Algebra.MonoidAlgebra.Degree

Goal state before:
pp: R : Type u_1
A : Type u_3
B : Type u_5
inst✝⁴ : Semiring R
inst✝³ : LinearOrder B
inst✝² : OrderBot B
p : R[A]
D : A → B
inst✝¹ : Nonempty A
inst✝ : Nontrivial R
hp : leadingCoeff D p = 1
h : p = 0
⊢ False

Tactic:
h,

Goal state after:
R : Type u_1
A : Type u_3
B : Type u_5
inst✝⁴ : Semiring R
inst✝³ : LinearOrder B
inst✝² : OrderBot B
p : R[A]
D : A → B
inst✝¹ : Nonempty A
inst✝ : Nontrivial R
h : p = 0
hp : leadingCoeff D 0 = 1
⊢ False
Module: Mathlib.Data.Nat.WithBot

Goal state before:
pp: n m : WithBot ℕ
⊢ n + m = 3 ↔ n = 0 ∧ m = 3 ∨ n = 1 ∧ m = 2 ∨ n = 2 ∧ m = 1 ∨ n = 3 ∧ m = 0

Tactic:
by
  cases n
  · simp [WithBot.bot_add]
  cases m
  · simp [WithBot.add_bot]
  simp [← WithBot.coe_add, Nat.add_eq_three_iff]

Goal state after:
Module: Mathlib.Algebra.MonoidAlgebra.Degree

Goal state before:
pp: case refine_2
A : Type u_3
B : Type u_5
inst✝³ : LinearOrder B
inst✝² : OrderBot B
D : A → B
inst✝¹ : AddZeroClass A
hD : Function.Injective D
R : Type u_8
inst✝ : Ring R
p q : R[A]
hd : supDegree D p = supDegree D q
hc : leadingCoeff D p = leadingCoeff D q
he : ¬(p - q).coeff (Function.invFun D (supDegree D (p - q))) = 0
h : supDegree D (p - q) = supDegree D p
⊢ (p - q).coeff (Function.invFun D (supDegree D (p - q))) = 0

Tactic:
h,

Goal state after:
case refine_2
A : Type u_3
B : Type u_5
inst✝³ : LinearOrder B
inst✝² : OrderBot B
D : A → B
inst✝¹ : AddZeroClass A
hD : Function.Injective D
R : Type u_8
inst✝ : Ring R
p q : R[A]
hd : supDegree D p = supDegree D q
hc : leadingCoeff D p = leadingCoeff D q
he : ¬(p - q).coeff (Function.invFun D (supDegree D (p - q))) = 0
h : supDegree D (p - q) = supDegree D p
⊢ (p - q).coeff (Function.invFun D (supDegree D p)) = 0
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
h : p = C (p.coeff 0)
⊢ p.natDegree = 0

Tactic:
h,

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
h : p = C (p.coeff 0)
⊢ (C (p.coeff 0)).natDegree = 0
Module: Mathlib.Algebra.Polynomial.Degree.Operations

Goal state before:
pp: R : Type u
inst✝ : Ring R
p q : R[X]
h : q.degree < p.degree
⊢ (p - q).leadingCoeff = p.leadingCoeff

Tactic:
by
  rw [← q.degree_neg] at h
  rw [sub_eq_add_neg, leadingCoeff_add_of_degree_lt' h]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Eval.Coeff

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+* S
hf : Function.Surjective ⇑f
p : S[X]
⊢ ∀ (p q : S[X]), (∃ a, map f a = p) → (∃ a, map f a = q) → ∃ a, map f a = p + q

Tactic:
by rintro _ _ ⟨p, rfl⟩ ⟨q, rfl⟩; exact ⟨p + q, Polynomial.map_add f⟩

Goal state after:
Module: Mathlib.Algebra.Polynomial.Eval.Coeff

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+* S
h✝ : Function.Surjective (map f)
s : S
p : R[X]
h : map f p = C s
⊢ f (p.coeff 0) = s

Tactic:
by simpa using congr(coeff $h 0)

Goal state after:
Module: Mathlib.Algebra.Polynomial.Eval.Degree

Goal state before:
pp: R : Type u
S : Type v
inst✝² : Semiring R
inst✝¹ : CommRing S
inst✝ : IsDomain S
φ : R →+* S
f : R[X]
hf : IsUnit f.leadingCoeff
H : IsUnit (map φ f)
dz : f.degree = 0
⊢ IsUnit f

pp: case hf
R : Type u
S : Type v
inst✝² : Semiring R
inst✝¹ : CommRing S
inst✝ : IsDomain S
φ : R →+* S
f : R[X]
hf : IsUnit f.leadingCoeff
H : IsUnit (map φ f)
dz : (map φ f).degree = 0
⊢ φ f.leadingCoeff ≠ 0

Tactic:
· rw [eq_C_of_degree_eq_zero dz]
    refine IsUnit.map C ?_
    convert! hf
    change coeff f 0 = coeff f (natDegree f)
    rw [(degree_eq_iff_natDegree_eq _).1 dz]
    · rfl
    rintro rfl
    simp at H

Goal state after:
case hf
R : Type u
S : Type v
inst✝² : Semiring R
inst✝¹ : CommRing S
inst✝ : IsDomain S
φ : R →+* S
f : R[X]
hf : IsUnit f.leadingCoeff
H : IsUnit (map φ f)
dz : (map φ f).degree = 0
⊢ φ f.leadingCoeff ≠ 0
Module: Mathlib.Data.Finsupp.Antidiagonal

Goal state before:
pp: case refine_2
α : Type u
inst✝¹ : DecidableEq α
β : Type u_1
inst✝ : DecidableEq β
f : α ↪ β
y : α →₀ ℕ
u v : β →₀ ℕ
h : u + v = embDomain f y
⊢ comapDomain (⇑f) (u + v) ⋯ = y

Tactic:
h,

Goal state after:
case refine_2
α : Type u
inst✝¹ : DecidableEq α
β : Type u_1
inst✝ : DecidableEq β
f : α ↪ β
y : α →₀ ℕ
u v : β →₀ ℕ
h : u + v = embDomain f y
⊢ comapDomain (⇑f) (embDomain f y) ⋯ = y
Module: Mathlib.Data.Finsupp.Antidiagonal

Goal state before:
pp: case refine_2
α : Type u
inst✝¹ : DecidableEq α
β : Type u_1
inst✝ : DecidableEq β
x : α →₀ ℕ
y : β →₀ ℕ
u v : α ⊕ β →₀ ℕ
h : u + v = x.sumElim y
⊢ comapDomain Sum.inl (u + v) ⋯ = x

Tactic:
h,

Goal state after:
case refine_2
α : Type u
inst✝¹ : DecidableEq α
β : Type u_1
inst✝ : DecidableEq β
x : α →₀ ℕ
y : β →₀ ℕ
u v : α ⊕ β →₀ ℕ
h : u + v = x.sumElim y
⊢ comapDomain Sum.inl (x.sumElim y) ⋯ = x
Module: Mathlib.Data.Finsupp.Antidiagonal

Goal state before:
pp: case refine_3
α : Type u
inst✝¹ : DecidableEq α
β : Type u_1
inst✝ : DecidableEq β
x : α →₀ ℕ
y : β →₀ ℕ
u v : α ⊕ β →₀ ℕ
h : u + v = x.sumElim y
⊢ comapDomain Sum.inr (u + v) ⋯ = y

Tactic:
h,

Goal state after:
case refine_3
α : Type u
inst✝¹ : DecidableEq α
β : Type u_1
inst✝ : DecidableEq β
x : α →₀ ℕ
y : β →₀ ℕ
u v : α ⊕ β →₀ ℕ
h : u + v = x.sumElim y
⊢ comapDomain Sum.inr (x.sumElim y) ⋯ = y
Module: Mathlib.Algebra.Polynomial.AlgebraMap

Goal state before:
pp: R : Type u
S : Type v
T : Type w
A : Type z
A' : Type u_1
B : Type u_2
a b : R
n : ℕ
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
p✝ q r✝ : R[X]
r : R
p : A[X]
⊢ (r • p).toFinsupp = (C ((algebraMap R A) r) * p).toFinsupp

Tactic:
rw [toFinsupp_smul, toFinsupp_mul, toFinsupp_C]

Goal state after:
R : Type u
S : Type v
T : Type w
A : Type z
A' : Type u_1
B : Type u_2
a b : R
n : ℕ
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
p✝ q r✝ : R[X]
r : R
p : A[X]
⊢ r • p.toFinsupp = AddMonoidAlgebra.single 0 ((algebraMap R A) r) * p.toFinsupp
Module: Mathlib.Algebra.Polynomial.AlgebraMap

Goal state before:
pp: R : Type u
inst✝⁵ : CommSemiring R
S : Type u_3
T : Type u_4
U : Type u_5
inst✝⁴ : Semiring S
inst✝³ : CommSemiring T
inst✝² : Semiring U
inst✝¹ : Algebra R S
inst✝ : Algebra T U
φ : R →+* T
ψ : S →+* U
h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)
p : R[X]
a : S
⊢ ψ ((aeval a) p) = eval (ψ a) (map ((algebraMap T U).comp φ) p)

Tactic:
h,

Goal state after:
R : Type u
inst✝⁵ : CommSemiring R
S : Type u_3
T : Type u_4
U : Type u_5
inst✝⁴ : Semiring S
inst✝³ : CommSemiring T
inst✝² : Semiring U
inst✝¹ : Algebra R S
inst✝ : Algebra T U
φ : R →+* T
ψ : S →+* U
h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)
p : R[X]
a : S
⊢ ψ ((aeval a) p) = eval (ψ a) (map (ψ.comp (algebraMap R S)) p)
Module: Mathlib.Order.PiLex

Goal state before:
pp: case refine_2
ι : Type u_1
β : ι → Type u_2
r : ι → ι → Prop
s : {i : ι} → β i → β i → Prop
inst✝² : Preorder ι
inst✝¹ : (i : ι) → LT (β i)
inst✝ : ∀ (i : ι), DenselyOrdered (β i)
a₁✝ a₂ : Lex ((i : ι) → β i)
i : ι
h : ∀ (j : ι), (fun x1 x2 ↦ x1 < x2) j i → a₁✝ j = a₂ j
hi : a₁✝ i < a₂ i
a : β i
ha₁ : a₁✝ i < a
ha₂ : a < a₂ i
⊢ (fun {i} x1 x2 ↦ x1 < x2) (a₁✝ i) (Function.update a₂ i a i)

Tactic:
rwa [Function.update_self i a]

Goal state after:
Module: Mathlib.Algebra.Polynomial.AlgebraMap

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : CommSemiring R
p : R[X]
inst✝ : Semiring S
f : R →+* S
hf : Function.Injective ⇑f
r : R
h : eval₂ f (f r) p = 0
⊢ eval₂ f (f r) p = f 0

Tactic:
h,

Goal state after:
R : Type u
S : Type v
inst✝¹ : CommSemiring R
p : R[X]
inst✝ : Semiring S
f : R →+* S
hf : Function.Injective ⇑f
r : R
h : eval₂ f (f r) p = 0
⊢ 0 = f 0