mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Localization.Defs

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
x : R
y : ↥M
z₁ z₂ : S
⊢ (mk' S x y + z₁) * (algebraMap R S) ↑y = z₂ * (algebraMap R S) ↑y ↔
    mk' S x y * (algebraMap R S) ↑y + z₁ * (algebraMap R S) ↑y = z₂ * (algebraMap R S) ↑y

Tactic:
right_distrib

Goal state after:
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
x : R
y : ↥M
z₁ z₂ : S
⊢ mk' S x y * (algebraMap R S) ↑y + z₁ * (algebraMap R S) ↑y = z₂ * (algebraMap R S) ↑y ↔
    mk' S x y * (algebraMap R S) ↑y + z₁ * (algebraMap R S) ↑y = z₂ * (algebraMap R S) ↑y
Module: Mathlib.RingTheory.Localization.Defs

Goal state before:
pp: R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
hM : ∀ m ∈ M, IsRegular m
r : R
m : ↥M
⊢ IsRegular (mk' S r m) ↔ IsRegular r

Tactic:
by
  have (n : M) (x y : R) : n * x = n * y ↔ x = y := (hM _ n.2).1.eq_iff
  simp +contextual only [← isLeftRegular_iff_isRegular, IsLeftRegular, Function.Injective,
    (mk'_surjective M).forall, ← mk'_mul, Prod.forall, Subtype.forall, IsLocalization.eq,
    Submonoid.coe_mul, this, exists_const, mul_assoc]
  simp_rw [← mul_left_comm r]
  exact ⟨fun h a b ↦ by simpa using h a 1 M.one_mem b 1 M.one_mem, fun h ha s hs b t ht ↦ @h _ _⟩

Goal state after:
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝¹⁹ : CommSemiring R✝
M✝ N : Submonoid R✝
S✝ : Type u_2
inst✝¹⁸ : CommSemiring S✝
inst✝¹⁷ : Algebra R✝ S✝
P✝ : Type u_3
inst✝¹⁶ : CommSemiring P✝
inst✝¹⁵ : IsLocalization M✝ S✝
g : R✝ →+* P✝
hg : ∀ (y : ↥M✝), IsUnit (g ↑y)
A : Type u_4
inst✝¹⁴ : CommSemiring A
R : Type u_5
inst✝¹³ : CommSemiring R
inst✝¹² : Algebra A R
M : Submonoid R
S : Type u_6
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra A S
inst✝⁹ : Algebra R S
inst✝⁸ : IsScalarTower A R S
inst✝⁷ : IsLocalization M S
P : Type u_7
inst✝⁶ : CommSemiring P
inst✝⁵ : Algebra A P
T : Submonoid P
Q : Type u_8
inst✝⁴ : CommSemiring Q
inst✝³ : Algebra A Q
inst✝² : Algebra P Q
inst✝¹ : IsScalarTower A P Q
inst✝ : IsLocalization T Q
h : R ≃ₐ[A] P
H : Submonoid.map h M = T
x✝ : A
⊢ (map Q ↑h ⋯) ((algebraMap A S) x✝) = (algebraMap A Q) x✝

Tactic:
rw [IsScalarTower.algebraMap_apply A R S, map_eq,
    RingHom.coe_coe, AlgEquiv.commutes, IsScalarTower.algebraMap_apply A P Q]

Goal state after:
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
M : Submonoid R
hM : Algebra.algebraMapSubmonoid S M ≤ IsUnit.submonoid S
h : Bijective ⇑(algebraMap R S)
x : R
⊢ ∃ x_1, (algebraMap R S) x * (algebraMap R S) ↑x_1.2 = (algebraMap R S) x_1.1

Tactic:
use ⟨x, 1⟩

Goal state after:
case h
R : Type u_1
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
M : Submonoid R
hM : Algebra.algebraMapSubmonoid S M ≤ IsUnit.submonoid S
h : Bijective ⇑(algebraMap R S)
x : R
⊢ (algebraMap R S) x * (algebraMap R S) ↑(x, 1).2 = (algebraMap R S) (x, 1).1
Module: Mathlib.RingTheory.Localization.Defs

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
inst✝¹ : IsLocalization M S
T : Type u_4
inst✝ : CommSemiring T
f : S →+* T
⊢ Injective ⇑f ↔ ∀ (x y : R), (algebraMap R S) x = (algebraMap R S) y ↔ f ((algebraMap R S) x) = f ((algebraMap R S) y)

Tactic:
← IsLocalization.lift_of_comp (M := M) f,

Goal state after:
R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
inst✝¹ : IsLocalization M S
T : Type u_4
inst✝ : CommSemiring T
f : S →+* T
⊢ Injective ⇑(lift ⋯) ↔
    ∀ (x y : R), (algebraMap R S) x = (algebraMap R S) y ↔ (lift ⋯) ((algebraMap R S) x) = (lift ⋯) ((algebraMap R S) y)
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case exists_of_eq
R : Type u_1
inst✝⁵ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁴ : CommSemiring S
inst✝³ : Algebra R S
P : Type u_3
inst✝² : CommSemiring P
inst✝¹ : Algebra R P
inst✝ : IsLocalization M S
h : S ≃ₐ[R] P
⊢ ∀ {x y : R}, (algebraMap R P) x = (algebraMap R P) y → ∃ c, ↑c * x = ↑c * y

Tactic:
intro x y
    rw [← h.symm.toEquiv.injective.eq_iff, ← IsLocalization.eq_iff_exists M S, ← h.symm.commutes, ←
      h.symm.commutes]
    exact id

Goal state after:
Module: Mathlib.Algebra.Algebra.Epi

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommSemiring A
inst✝¹ : Algebra R A
inst✝ : Algebra.IsEpi R A
a b : A
this : ∀ (a b : A), a ⊗ₜ[R] b = a • b • 1 ⊗ₜ[R] 1
⊢ a ⊗ₜ[R] b = b ⊗ₜ[R] a

Tactic:
rw [this a b, this b a, smul_comm]

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

Goal state before:
pp: α : Type u_1
s✝ t : Set α
a : α
inst✝ : InvolutiveStar α
s : Set α
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
simp only [← star_preimage, preimage_preimage, star_star, preimage_id']

Goal state after:
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
S : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Semiring S
inst✝ : Algebra R S
h : Function.Surjective ⇑(algebraMap R S)
⊢ ⊤ ≤ ((mk R S M) 1).range

Tactic:
← span_tmul_eq_top,

Goal state after:
R : Type u_1
M : Type u_2
S : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Semiring S
inst✝ : Algebra R S
h : Function.Surjective ⇑(algebraMap R S)
⊢ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} ≤ ((mk R S M) 1).range
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type u_1
S : Type u_3
T : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
inst✝¹ : Ring T
inst✝ : Algebra R T
h : Function.Surjective ⇑(algebraMap R T)
⊢ ⊤ ≤ ((mk R S T).flip 1).range

Tactic:
← span_tmul_eq_top,

Goal state after:
R : Type u_1
S : Type u_3
T : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
inst✝¹ : Ring T
inst✝ : Algebra R T
h : Function.Surjective ⇑(algebraMap R T)
⊢ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} ≤ ((mk R S T).flip 1).range
Module: Mathlib.RingTheory.TensorProduct.Basic

Goal state before:
pp: R : Type u_1
S : Type u_3
T : Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
inst✝¹ : Ring T
inst✝ : Algebra R T
h : Function.Surjective ⇑(algebraMap R T)
s : S
r : R
⊢ (r • s) ⊗ₜ[R] 1 ∈ ↑((mk R S T).flip 1).range

Tactic:
exact ⟨r • s, rfl⟩

Goal state after:
Module: Mathlib.Algebra.Algebra.Spectrum.Quasispectrum

Goal state before:
pp: R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
r : R
hr : IsUnit r
⊢ IsQuasiregular (-(hr.unit⁻¹ • a)) ↔ IsUnit ((algebraMap R A) r - a)

Tactic:
isQuasiregular_iff_isUnit,

Goal state after:
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
r : R
hr : IsUnit r
⊢ IsUnit (1 + -(hr.unit⁻¹ • a)) ↔ IsUnit ((algebraMap R A) r - a)
Module: Mathlib.Algebra.Algebra.Subalgebra.Centralizer

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommSemiring R
A : Type u_2
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u_3
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Set A
inst✝ : Module.Free R B
⊢ centralizer R (⇑includeLeft '' S) = (Algebra.TensorProduct.map (centralizer R S).val (AlgHom.id R B)).range

Tactic:
by
  ext w
  constructor
  · intro hw
    rw [mem_centralizer_iff] at hw
    let ℬ := Module.Free.chooseBasis R B
    obtain ⟨b, rfl⟩ := TensorProduct.eq_repr_basis_right ℬ w
    refine Subalgebra.sum_mem _ fun j hj => ⟨⟨b j, ?_⟩ ⊗ₜ[R] ℬ j, by simp⟩
    rw [Subalgebra.mem_centralizer_iff]
    intro x hx
    suffices x • b = b.mapRange (· * x) (by simp) from Finsupp.ext_iff.1 this j
    specialize hw (x ⊗ₜ[R] 1) ⟨x, hx, rfl⟩
    simp only [Finsupp.sum, Finset.mul_sum, Algebra.TensorProduct.tmul_mul_tmul, one_mul,
      Finset.sum_mul, mul_one] at hw
    refine TensorProduct.sum_tmul_basis_right_injective ℬ ?_
    simp only [Finsupp.coe_lsum]
    rw [sum_of_support_subset (s := b.support) (hs := Finsupp.support_smul) (h := by simp),
      sum_of_support_subset (s := b.support) (hs := support_mapRange) (h := by simp)]
    simpa only [Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul, LinearMap.flip_apply,
      TensorProduct.mk_apply, Finsupp.mapRange_apply] using hw
  · rintro ⟨w, rfl⟩
    rw [Subalgebra.mem_centralizer_iff]
    rintro _ ⟨x, hx, rfl⟩
    induction w using TensorProduct.induction_on with
    | zero => simp
    | tmul b c =>
      simp [Subalgebra.mem_centralizer_iff _ |>.1 b.2 x hx]
    | add y z hy hz => rw [map_add, mul_add, hy, hz, add_mul]

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

Goal state before:
pp: R : Type u
S : Type u'
M : Type v
M' : Type v'
inst✝¹² : Semiring R
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : AddCommMonoid M'
inst✝⁸ : Module R M
inst✝⁷ : StrongRankCondition R
inst✝⁶ : StrongRankCondition S
inst✝⁵ : Module S M
inst✝⁴ : Module S M'
inst✝³ : Free S M'
inst✝² : Algebra S R
inst✝¹ : IsScalarTower S R M
inst✝ : Free R M
fst✝¹ : Type v
bM : Basis fst✝¹ R M
fst✝ : Type v'
bN : Basis fst✝ S M'
⊢ Module.rank R (M ⊗[S] M') = Cardinal.lift.{v', v} #fst✝¹ * Cardinal.lift.{v, v'} #fst✝

Tactic:
← (bM.tensorProduct bN).mk_eq_rank'',

Goal state after:
R : Type u
S : Type u'
M : Type v
M' : Type v'
inst✝¹² : Semiring R
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : AddCommMonoid M'
inst✝⁸ : Module R M
inst✝⁷ : StrongRankCondition R
inst✝⁶ : StrongRankCondition S
inst✝⁵ : Module S M
inst✝⁴ : Module S M'
inst✝³ : Free S M'
inst✝² : Algebra S R
inst✝¹ : IsScalarTower S R M
inst✝ : Free R M
fst✝¹ : Type v
bM : Basis fst✝¹ R M
fst✝ : Type v'
bN : Basis fst✝ S M'
⊢ #(fst✝¹ × fst✝) = Cardinal.lift.{v', v} #fst✝¹ * Cardinal.lift.{v, v'} #fst✝
Module: Mathlib.LinearAlgebra.Dimension.Constructions

Goal state before:
pp: R : Type u
ι : Type w
inst✝³ : Semiring R
inst✝² : StrongRankCondition R
inst✝¹ : Finite ι
inst✝ : Nontrivial R
s : Set ι
this : Fintype ι
⊢ finrank R ↥(spanSubset R s) = s.ncard

Tactic:
rw [Pi.spanSubset, finrank_span_set_eq_card <| (Pi.basisFun R ι).linearIndepOn _ |>.id_image,
    Set.toFinset_card, Fintype.card_eq_nat_card, Nat.card_coe_set_eq]

Goal state after:
R : Type u
ι : Type w
inst✝³ : Semiring R
inst✝² : StrongRankCondition R
inst✝¹ : Finite ι
inst✝ : Nontrivial R
s : Set ι
this : Fintype ι
⊢ (⇑(basisFun R ι) '' s).ncard = s.ncard
Module: Mathlib.LinearAlgebra.Basis.VectorSpace

Goal state before:
pp: case right.inr
K : Type u_3
V : Type u_4
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
v : V
hv : v ≠ 0
T : Submodule K V
hT : T < K ∙ v
a : K
hs : a • v ∈ T
hz : a • v ≠ 0
h : a ≠ 0
⊢ v ∈ T

Tactic:
rwa [T.smul_mem_iff h] at hs

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

Goal state before:
pp: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁷ : CommSemiring R
inst✝¹⁶ : NonUnitalSemiring A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : Module R A
inst✝¹³ : NonUnitalSemiring B
inst✝¹² : StarRing B
inst✝¹¹ : Module R B
inst✝¹⁰ : FunLike F A B
inst✝⁹ : NonUnitalAlgHomClass F R A B
inst✝⁸ : StarHomClass F A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝⁷ : StarRing R
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : StarModule R A
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
⊢ ↥T →⋆ₙₐ[R] B

Tactic:
subst hT

Goal state after:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁷ : CommSemiring R
inst✝¹⁶ : NonUnitalSemiring A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : Module R A
inst✝¹³ : NonUnitalSemiring B
inst✝¹² : StarRing B
inst✝¹¹ : Module R B
inst✝¹⁰ : FunLike F A B
inst✝⁹ : NonUnitalAlgHomClass F R A B
inst✝⁸ : StarHomClass F A B
S : NonUnitalStarSubalgebra R A
ι : Type u_1
inst✝⁷ : StarRing R
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : SMulCommClass R A A
inst✝⁴ : StarModule R A
inst✝³ : IsScalarTower R B B
inst✝² : SMulCommClass R B B
inst✝¹ : StarModule R B
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
⊢ ↥(iSup K) →⋆ₙₐ[R] B
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
B : Type w
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : NonUnitalSemiring B
inst✝⁶ : StarRing B
inst✝⁵ : Module R B
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
x : ↥(K i)
hx : ↑x ∈ T
⊢ (iSupLift K dir f hf T hT) ⟨↑x, hx⟩ = (f i) x

Tactic:
subst hT

Goal state after:
R : Type u
A : Type v
B : Type w
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : NonUnitalSemiring B
inst✝⁶ : StarRing B
inst✝⁵ : Module R B
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
i : ι
x : ↥(K i)
hx : ↑x ∈ iSup K
⊢ (iSupLift K dir f hf (iSup K) ⋯) ⟨↑x, hx⟩ = (f i) x
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
B : Type w
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : NonUnitalSemiring B
inst✝⁶ : StarRing B
inst✝⁵ : Module R B
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
T : NonUnitalStarSubalgebra R A
hT : T = iSup K
i : ι
x : ↥T
hx : ↑x ∈ K i
⊢ (iSupLift K dir f hf T hT) x = (f i) ⟨↑x, hx⟩

Tactic:
subst hT

Goal state after:
R : Type u
A : Type v
B : Type w
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : NonUnitalSemiring B
inst✝⁶ : StarRing B
inst✝⁵ : Module R B
ι : Type u_1
inst✝⁴ : StarRing R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : StarModule R A
inst✝ : Nonempty ι
K : ι → NonUnitalStarSubalgebra R A
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) K
f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B
hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)
i : ι
x : ↥(iSup K)
hx : ↑x ∈ K i
⊢ (iSupLift K dir f hf (iSup K) ⋯) x = (f i) ⟨↑x, hx⟩
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
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
this : LinearIndepOn K id (Subtype.val '' Basis.ofVectorSpaceIndex K ↥S)
b : Basis (↑(this.extend ⋯)) K V := Basis.extend this
b_eq : b = Basis.extend this
i2 : Fintype ↑(Subtype.val '' Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype
⊢ Fintype.card ↑(this.extend ⋯) ≤ Fintype.card ↑(Subtype.val '' Basis.ofVectorSpaceIndex K ↥S)

Tactic:
rw [Set.card_image_of_injective _ Subtype.coe_injective, ← finrank_eq_card_basis bS, ←
            finrank_eq_card_basis b, h]

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
inst✝ : FiniteDimensional K V
f : V →ₗ[K] V
hinj : Injective ⇑f
h : finrank K ↥f.range = finrank K V
⊢ Surjective ⇑f

Tactic:
exact range_eq_top.1 (eq_top_of_finrank_eq h)

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
⊢ (∃ v₀, ∀ (v : ↥W), ∃ r, r • v₀ = v) → ∃ a, (∀ v ∈ W, ∃ r, r • a = v) ∧ a ∈ W

pp: case mpr
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
⊢ (∃ a, (∀ v ∈ W, ∃ r, r • a = v) ∧ a ∈ W) → ∃ v₀, ∀ (v : ↥W), ∃ r, r • v₀ = v

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

Goal state after:
case mpr
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
⊢ (∃ a, (∀ v ∈ W, ∃ r, r • a = v) ∧ a ∈ W) → ∃ v₀, ∀ (v : ↥W), ∃ r, r • v₀ = v
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
S : Subalgebra F E
inst✝¹ : Nontrivial E
inst✝ : Free F ↥S
⊢ Module.rank F ↥S = 1 ↔ S = ⊥

Tactic:
by
  refine ⟨fun h ↦ Subalgebra.eq_bot_of_rank_le_one h.le, ?_⟩
  rintro rfl
  obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := F) (M := (⊥ : Subalgebra F E))
  refine le_antisymm ?_ ?_
  · have := lift_rank_range_le (Algebra.linearMap F E)
    rwa [← one_eq_range, rank_self, lift_one, lift_le_one_iff,
      ← Algebra.toSubmodule_bot, rank_toSubmodule] at this
  · by_contra H
    rw [not_le, Cardinal.lt_one_iff] at H
    have := mk_eq_zero_iff.1 (H ▸ b.mk_eq_rank'')
    have := b.repr.toEquiv.subsingleton
    exact one_ne_zero congr((⊥ : Subalgebra F E).val $(Subsingleton.elim 1 0))

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: case right_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 c✝ + liftFun R X f b✝ * liftFun R X f c✝

Tactic:
right_distrib

Goal state after:
case right_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 c✝ + liftFun R X f b✝ * liftFun R X f c✝ =
    liftFun R X f a✝ * liftFun R X f c✝ + liftFun R X f b✝ * liftFun R X f c✝
Module: Mathlib.LinearAlgebra.LinearPMap

Goal state before:
pp: case h
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
f g : E →ₛₗ.[σ] F
h' : f + g = 0
x : E
this : f.domain = ⊤ ∧ g.domain = ⊤
⊢ x ∈ ⊤ ↔ x ∈ g.domain

Tactic:
this.2

Goal state after:
case h
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring S
inst✝⁶ : Ring T
σ : R →+* S
τ : S →+* T
E : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_5
inst✝³ : AddCommGroup F
inst✝² : Module S F
G : Type u_6
inst✝¹ : AddCommGroup G
inst✝ : Module T G
f g : E →ₛₗ.[σ] F
h' : f + g = 0
x : E
this : f.domain = ⊤ ∧ g.domain = ⊤
⊢ x ∈ ⊤ ↔ x ∈ ⊤
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: 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 g : FreeAlgebra R X →ₐ[R] A
w : ⇑f ∘ ι R = ⇑g ∘ ι R
⊢ f = g

Tactic:
rw [← lift_symm_apply, ← lift_symm_apply] at w
  exact (lift R).symm.injective w

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

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
x : R
n : ℕ
⊢ eval x (↑n * p) = ↑n * eval x p

Tactic:
rw [← C_eq_natCast, eval_C_mul]

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: R : Type u_1
X : Type u_2
inst✝ : CommSemiring R
S : Subalgebra R (FreeAlgebra R X) := Algebra.adjoin R (Set.range (ι R))
⊢ S = ⊤

Tactic:
refine top_unique fun x hx => ?_

Goal state after:
R : Type u_1
X : Type u_2
inst✝ : CommSemiring R
S : Subalgebra R (FreeAlgebra R X) := Algebra.adjoin R (Set.range (ι R))
x : FreeAlgebra R X
hx : x ∈ ⊤
⊢ x ∈ S
Module: Mathlib.Data.Nat.Prime.Defs

Goal state before:
pp: case mp
n : ℕ
h : n.minFac = 1
hn : ¬n = 1
⊢ False

Tactic:
have := minFac_prime hn

Goal state after:
case mp
n : ℕ
h : n.minFac = 1
hn : ¬n = 1
this : Prime n.minFac
⊢ False
Module: Mathlib.Algebra.CharP.Defs

Goal state before:
pp: case zero
R : Type u_1
inst✝¹ : AddMonoidWithOne R
inst✝ : CharZero R
⊢ Nat.Prime 1 ∨ 1 = 1

Tactic:
exact .inr rfl

Goal state after:
Module: Mathlib.Algebra.CharP.Defs

Goal state before:
pp: case prime
R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : Nontrivial R
p : ℕ
hp : CharP R p
hprime✝ : Nat.Prime 1
hchar✝ : CharP R 1
⊢ p = 0

Tactic:
exact False.elim (CharP.char_ne_one R 1 rfl)

Goal state after:
Module: Mathlib.Algebra.CharP.Defs

Goal state before:
pp: case prime
R : Type u_1
inst✝¹ : NonAssocSemiring R
inst✝ : Nontrivial R
hprime✝ : Nat.Prime 1
hchar✝ : CharP R 1
⊢ CharZero R

Tactic:
exact False.elim (CharP.char_ne_one R 1 rfl)

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

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
T : Type u_3
A : Type u_4
B : Type u_5
C : Type u_6
M : Type u_7
N : Type u_8
O : Type u_9
inst✝⁹ : Monoid M
inst✝⁸ : CommSemiring R
V : Type u_10
W : Type u_11
inst✝⁷ : AddCommMonoid V
inst✝⁶ : Module R V
inst✝⁵ : Module R[M] V
inst✝⁴ : IsScalarTower R R[M] V
inst✝³ : AddCommMonoid W
inst✝² : Module R W
inst✝¹ : Module R[M] W
inst✝ : IsScalarTower R R[M] W
f : V →ₗ[R] W
h : ∀ (g : M) (v : V), f (single g 1 • v) = single g 1 • f v
c : R[M]
v : V
g : M
r : R
⊢ ∀ (x : R[M]),
    g ∉ x.coeff.support →
      r ≠ 0 →
        f (x • v) = (RingHom.id R[M]) x • f v → f ((single g r + x) • v) = (RingHom.id R[M]) (single g r + x) • f v

Tactic:
c'

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
T : Type u_3
A : Type u_4
B : Type u_5
C : Type u_6
M : Type u_7
N : Type u_8
O : Type u_9
inst✝⁹ : Monoid M
inst✝⁸ : CommSemiring R
V : Type u_10
W : Type u_11
inst✝⁷ : AddCommMonoid V
inst✝⁶ : Module R V
inst✝⁵ : Module R[M] V
inst✝⁴ : IsScalarTower R R[M] V
inst✝³ : AddCommMonoid W
inst✝² : Module R W
inst✝¹ : Module R[M] W
inst✝ : IsScalarTower R R[M] W
f : V →ₗ[R] W
h : ∀ (g : M) (v : V), f (single g 1 • v) = single g 1 • f v
c : R[M]
v : V
g : M
r : R
c' : R[M]
⊢ g ∉ c'.coeff.support →
    r ≠ 0 →
      f (c' • v) = (RingHom.id R[M]) c' • f v → f ((single g r + c') • v) = (RingHom.id R[M]) (single g r + c') • f v
Module: Mathlib.Algebra.Polynomial.Coeff

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
n d : ℕ
⊢ (p * X ^ n).coeff (d + n) = p.coeff d

Tactic:
coeff_mul_X_pow

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
n d : ℕ
⊢ p.coeff d = p.coeff d
Module: Mathlib.Algebra.Polynomial.Coeff

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
n d : ℕ
r : R
⊢ (p * X ^ n).coeff (d + n) * r = p.coeff d * r

Tactic:
coeff_mul_X_pow

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
n d : ℕ
r : R
⊢ p.coeff d * r = p.coeff d * r
Module: Mathlib.Algebra.Polynomial.Coeff

Goal state before:
pp: R : Type u
inst✝ : Semiring R
p : R[X]
n d : ℕ
r : R
⊢ r * (X ^ n * p).coeff (d + n) = r * p.coeff d

Tactic:
X_pow_mul,

Goal state after:
R : Type u
inst✝ : Semiring R
p : R[X]
n d : ℕ
r : R
⊢ r * (p * X ^ n).coeff (d + n) = r * p.coeff d