mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.DirectSum.TensorProduct

Goal state before:
pp: case tmul
R : Type u
inst✝⁹ : CommSemiring R
S : Type u_1
inst✝⁸ : Semiring S
inst✝⁷ : Algebra R S
ι₁ : Type v₁
inst✝⁶ : DecidableEq ι₁
M₁ : ι₁ → Type w₁
M₂' : Type w₂'
inst✝⁵ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)
inst✝⁴ : AddCommMonoid M₂'
inst✝³ : (i₁ : ι₁) → Module R (M₁ i₁)
inst✝² : Module R M₂'
inst✝¹ : (i₁ : ι₁) → Module S (M₁ i₁)
inst✝ : ∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)
i : ι₁
x : M₁ i
y : M₂'
⊢ (directSumLeft R S M₁ M₂').symm ((lof S ι₁ (fun i ↦ M₁ i ⊗[R] M₂') i) (x ⊗ₜ[R] y)) =
    (rTensor M₂' (lof R ι₁ M₁ i)) (x ⊗ₜ[R] y)

Tactic:
directSumLeft_symm_lof_tmul,

Goal state after:
case tmul
R : Type u
inst✝⁹ : CommSemiring R
S : Type u_1
inst✝⁸ : Semiring S
inst✝⁷ : Algebra R S
ι₁ : Type v₁
inst✝⁶ : DecidableEq ι₁
M₁ : ι₁ → Type w₁
M₂' : Type w₂'
inst✝⁵ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)
inst✝⁴ : AddCommMonoid M₂'
inst✝³ : (i₁ : ι₁) → Module R (M₁ i₁)
inst✝² : Module R M₂'
inst✝¹ : (i₁ : ι₁) → Module S (M₁ i₁)
inst✝ : ∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)
i : ι₁
x : M₁ i
y : M₂'
⊢ (lof S ι₁ M₁ i) x ⊗ₜ[R] y = (rTensor M₂' (lof R ι₁ M₁ i)) (x ⊗ₜ[R] y)
Module: Mathlib.LinearAlgebra.DirectSum.TensorProduct

Goal state before:
pp: R : Type u
inst✝⁹ : CommSemiring R
S : Type u_1
inst✝⁸ : Semiring S
inst✝⁷ : Algebra R S
ι₂ : Type v₂
inst✝⁶ : DecidableEq ι₂
M₁' : Type w₁'
M₂ : ι₂ → Type w₂
inst✝⁵ : AddCommMonoid M₁'
inst✝⁴ : (i₂ : ι₂) → AddCommMonoid (M₂ i₂)
inst✝³ : Module R M₁'
inst✝² : (i₂ : ι₂) → Module R (M₂ i₂)
inst✝¹ : Module S M₁'
inst✝ : IsScalarTower R S M₁'
m : M₁'
n : ⨁ (i : ι₂), M₂ i
i : ι₂
⊢ ((directSumRight R S M₁' M₂) (m ⊗ₜ[R] n)) i = m ⊗ₜ[R] n i

Tactic:
by
  suffices (DirectSum.component S ι₂ _ i).restrictScalars R ∘ₗ
      (directSumRight R S M₁' M₂).toLinearMap.restrictScalars R ∘ₗ
        (TensorProduct.mk R M₁' (⨁ i, M₂ i) m) =
          (TensorProduct.mk R M₁' (M₂ i) m) ∘ₗ (DirectSum.component R ι₂ M₂ i) by
    simpa using! LinearMap.congr_fun this n
  ext j n
  by_cases hj : j = i
  · subst hj; simp
  · simp [DirectSum.component.of, hj]

Goal state after:
Module: Mathlib.Algebra.Field.Opposite

Goal state before:
pp: α : Type u_1
inst✝ : DivisionSemiring α
q : ℚ≥0
⊢ unop ↑q = unop (↑q.num / ↑q.den)

Tactic:
rw [unop_nnratCast, unop_div, unop_natCast, unop_natCast,
    NNRat.cast_def, div_eq_mul_inv]

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

Goal state before:
pp: S : Type u_1
R : Type u_2
inst✝² : InvolutiveStar R
inst✝¹ : SetLike S R
inst✝ : StarMemClass S R
s : S
r : R
hr : star r ∈ s
⊢ star (star r) ∈ s

Tactic:
exact star_mem hr

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

Goal state before:
pp: R : Type u_1
inst✝¹ : AddGroup R
inst✝ : StarAddMonoid R
x : R
⊢ x ∈ selfAdjoint R ↔ star x = x

Tactic:
rw [← AddSubgroup.mem_carrier]
  exact Iff.rfl

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

Goal state before:
pp: R : Type u_1
inst✝¹ : AddCommGroup R
inst✝ : StarAddMonoid R
x : R
⊢ x ∈ skewAdjoint R ↔ star x = -x

Tactic:
rw [← AddSubgroup.mem_carrier]
  exact Iff.rfl

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

Goal state before:
pp: ι : Type u_1
R₂ : Type u_3
M : Type u_4
inst✝² : AddCommMonoid M
inst✝¹ : CommSemiring R₂
inst✝ : Module R₂ M
e : Basis ι R₂ M
w : ι → R₂ˣ
i : ι
⊢ (e.unitsSMul w).coord i = (w i)⁻¹ • e.coord i

Tactic:
by
  classical
    apply e.ext
    intro j
    trans ((unitsSMul e w).coord i) ((w j)⁻¹ • (unitsSMul e w) j)
    · simp [Basis.unitsSMul, ← mul_smul]
    simp only [Basis.coord_apply, LinearMap.smul_apply, Basis.repr_self, Units.smul_def,
      map_smul, Finsupp.single_apply]
    split_ifs with h <;> simp [h]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.RowCol

Goal state before:
pp: n : Type u_3
α : Type v
inst✝¹ : DecidableEq n
inst✝ : Zero α
v : n → α
i : n
x : α
⊢ (diagonal v).updateRow i (Pi.single i x) = diagonal (Function.update v i x)

Tactic:
by
  rw [← diagonal_transpose, updateRow_transpose, diagonal_updateCol_single, diagonal_transpose]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.RowCol

Goal state before:
pp: l : Type u_1
m : Type u_2
n : Type u_3
α : Type v
inst✝² : DecidableEq l
inst✝¹ : Fintype m
inst✝ : NonUnitalNonAssocSemiring α
A : Matrix l m α
i : l
r : m → α
B : Matrix m n α
⊢ A.updateRow i r * B = (A * B).updateRow i (r ᵥ* B)

Tactic:
by
  ext i' j'
  obtain rfl | hi := eq_or_ne i' i
  · simp [mul_apply, vecMul, dotProduct]
  · simp [mul_apply, hi]

Goal state after:
Module: Mathlib.Algebra.Field.Subfield.Basic

Goal state before:
pp: K : Type u
L : Type v
inst✝¹ : DivisionRing K
inst✝ : DivisionRing L
f : K →+* L
x : ↥f.fieldRange
⊢ f (f.rangeRestrictFieldEquiv.symm x) = ↑x

Tactic:
← rangeRestrictFieldEquiv_apply_coe,

Goal state after:
K : Type u
L : Type v
inst✝¹ : DivisionRing K
inst✝ : DivisionRing L
f : K →+* L
x : ↥f.fieldRange
⊢ ↑(f.rangeRestrictFieldEquiv (f.rangeRestrictFieldEquiv.symm x)) = ↑x
Module: Mathlib.RingTheory.OreLocalization.Ring

Goal state before:
pp: case c.c.c
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * (↑sa * r₂) = rb * ↑s₁
hs₃rasb : ↑sb * ra * ↑s₃ ∈ S
ha' : ↑(sb * sa * s₂) = ↑sb * ra * ↑s₃
rc : R
sc : ↥S
hc : ↑sc * (↑sb * ra * r₃) = rc * ↑s₁
hc' : ((↑sb * ra) • r₃ /ₒ (sb * sa * s₂)) • (r₁ /ₒ s₁) = rc • r₁ /ₒ (sc * (sb * sa * s₂))
⊢ ((↑sc * rb * ↑s₁ + rc * ↑s₁) /ₒ (sc * sb * (sa * s₂))) • (r₁ /ₒ s₁) =
    (↑sc • rb • r₁ + rc • r₁) /ₒ (sc * (sb * (sa * s₂)))

Tactic:
← add_mul

Goal state after:
case c.c.c
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * (↑sa * r₂) = rb * ↑s₁
hs₃rasb : ↑sb * ra * ↑s₃ ∈ S
ha' : ↑(sb * sa * s₂) = ↑sb * ra * ↑s₃
rc : R
sc : ↥S
hc : ↑sc * (↑sb * ra * r₃) = rc * ↑s₁
hc' : ((↑sb * ra) • r₃ /ₒ (sb * sa * s₂)) • (r₁ /ₒ s₁) = rc • r₁ /ₒ (sc * (sb * sa * s₂))
⊢ ((↑sc * rb + rc) * ↑s₁ /ₒ (sc * sb * (sa * s₂))) • (r₁ /ₒ s₁) = (↑sc • rb • r₁ + rc • r₁) /ₒ (sc * (sb * (sa * s₂)))
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
ι₁ : Type u_6
ι₂ : Type u_7
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Fintype ι₁
inst✝¹ : Fintype ι₂
inst✝ : DecidableEq ι₁
b₁ : Basis ι₁ R M₁
b₂ : Basis ι₂ R M₂
ij : ι₂ × ι₁
k : ι₁
this : DecidableEq ι₂
⊢ ((b₁.linearMap b₂) ij) (b₁ k) = if ij.2 = k then b₂ ij.1 else 0

Tactic:
linearMap_apply,

Goal state after:
R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
ι₁ : Type u_6
ι₂ : Type u_7
inst✝⁷ : CommSemiring R
inst✝⁶ : AddCommMonoid M₁
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Fintype ι₁
inst✝¹ : Fintype ι₂
inst✝ : DecidableEq ι₁
b₁ : Basis ι₁ R M₁
b₂ : Basis ι₂ R M₂
ij : ι₂ × ι₁
k : ι₁
this : DecidableEq ι₂
⊢ ((toLin b₁ b₂) ((stdBasis R ι₂ ι₁) ij)) (b₁ k) = if ij.2 = k then b₂ ij.1 else 0
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝⁴ : CommRing R
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R M
n : ℕ
N : Submodule R M
snf : SmithNormalForm N ι n
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : M →ₗ[R] M
hf : ∀ (x : M), f x ∈ N
hf' : ∀ x ∈ N, f x ∈ N
i : Fin n
⊢ (LinearMap.toMatrix snf.bN snf.bN) (f.restrict hf') i i = (LinearMap.toMatrix snf.bM snf.bM) f (snf.f i) (snf.f i)

Tactic:
by
  rw [LinearMap.toMatrix_apply, LinearMap.toMatrix_apply,
    snf.repr_apply_embedding_eq_repr_smul ⟨_, (hf _)⟩]
  congr
  ext
  simp [snf.snf]

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: case intro
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
N : Submodule R M
inst✝ : Finite ι
b : Basis ι R M
h : finrank R ↥N = finrank R M
val✝ : Fintype ι
⊢ ∃ b' a ab', ∀ (i : ι), ↑(ab' i) = a i • b' i

Tactic:
let ⟨bM, bN, f, a, snf⟩ := N.smithNormalFormOfRankEq b h

Goal state after:
case intro
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
N : Submodule R M
inst✝ : Finite ι
b : Basis ι R M
h : finrank R ↥N = finrank R M
val✝ : Fintype ι
bM : Basis ι R M
bN : Basis (Fin (Fintype.card ι)) R ↥N
f : Fin (Fintype.card ι) ↪ ι
a : Fin (Fintype.card ι) → R
snf : ∀ (i : Fin (Fintype.card ι)), ↑(bN i) = a i • bM (f i)
⊢ ∃ b' a ab', ∀ (i : ι), ↑(ab' i) = a i • b' i
Module: Mathlib.RingTheory.Localization.FractionRing

Goal state before:
pp: A : Type u_4
inst✝⁴ : CommRing A
K : Type u_5
inst✝³ : Field K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
L : Type u_8
inst✝ : Field L
g : A →+* L
f : K →+* L
h : f.comp (algebraMap A K) = g
s : Set L
hs : g.range = Subring.closure s
⊢ f.fieldRange = Subfield.closure s

Tactic:
rw [ringHom_fieldRange_eq_of_comp_eq h, hs]
  ext
  simp_rw [Subfield.mem_closure_iff, Subring.closure_eq]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
M : Submonoid R
x : R
y : ↥M
⊢ mk x y = mk' (Localization M) x y

Tactic:
mk_eq_monoidOf_mk'_apply,

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
M : Submonoid R
x : R
y : ↥M
⊢ (monoidOf M).mk' x y = mk' (Localization M) x y
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: R : Type u_1
inst✝²² : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝²¹ : CommSemiring S
inst✝²⁰ : Algebra R S
Rₘ : Type u_4
Sₙ : Type u_5
Rₘ' : Type u_6
Sₙ' : Type u_7
inst✝¹⁹ : CommSemiring Rₘ
inst✝¹⁸ : CommSemiring Sₙ
inst✝¹⁷ : CommSemiring Rₘ'
inst✝¹⁶ : CommSemiring Sₙ'
inst✝¹⁵ : Algebra R Rₘ
inst✝¹⁴ : Algebra S Sₙ
inst✝¹³ : Algebra R Rₘ'
inst✝¹² : Algebra S Sₙ'
inst✝¹¹ : Algebra R Sₙ
inst✝¹⁰ : Algebra Rₘ Sₙ
inst✝⁹ : Algebra Rₘ' Sₙ'
inst✝⁸ : Algebra R Sₙ'
N : Submonoid S
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization N Sₙ
inst✝⁵ : IsLocalization M Rₘ'
inst✝⁴ : IsLocalization N Sₙ'
inst✝³ : IsScalarTower R Rₘ Sₙ
inst✝² : IsScalarTower R S Sₙ
inst✝¹ : IsScalarTower R Rₘ' Sₙ'
inst✝ : IsScalarTower R S Sₙ'
x : R
⊢ (((↑(algEquiv N Sₙ Sₙ')).comp (algebraMap Rₘ Sₙ)).comp (algebraMap R Rₘ)) x =
    (((algebraMap Rₘ' Sₙ').comp ↑(algEquiv M Rₘ Rₘ')).comp (algebraMap R Rₘ)) x

Tactic:
simp only [RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, AlgEquiv.commutes]

Goal state after:
R : Type u_1
inst✝²² : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝²¹ : CommSemiring S
inst✝²⁰ : Algebra R S
Rₘ : Type u_4
Sₙ : Type u_5
Rₘ' : Type u_6
Sₙ' : Type u_7
inst✝¹⁹ : CommSemiring Rₘ
inst✝¹⁸ : CommSemiring Sₙ
inst✝¹⁷ : CommSemiring Rₘ'
inst✝¹⁶ : CommSemiring Sₙ'
inst✝¹⁵ : Algebra R Rₘ
inst✝¹⁴ : Algebra S Sₙ
inst✝¹³ : Algebra R Rₘ'
inst✝¹² : Algebra S Sₙ'
inst✝¹¹ : Algebra R Sₙ
inst✝¹⁰ : Algebra Rₘ Sₙ
inst✝⁹ : Algebra Rₘ' Sₙ'
inst✝⁸ : Algebra R Sₙ'
N : Submonoid S
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization N Sₙ
inst✝⁵ : IsLocalization M Rₘ'
inst✝⁴ : IsLocalization N Sₙ'
inst✝³ : IsScalarTower R Rₘ Sₙ
inst✝² : IsScalarTower R S Sₙ
inst✝¹ : IsScalarTower R Rₘ' Sₙ'
inst✝ : IsScalarTower R S Sₙ'
x : R
⊢ (algEquiv N Sₙ Sₙ') ((algebraMap Rₘ Sₙ) ((algebraMap R Rₘ) x)) = (algebraMap Rₘ' Sₙ') ((algebraMap R Rₘ') x)
Module: Mathlib.Algebra.Algebra.Epi

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Field A
inst✝¹ : Algebra R A
inst✝ : IsFractionRing R A
a b : R
f : R →+* A := algebraMap R A
hf : f = algebraMap R A
hb : f b ≠ 0
⊢ f a ⊗ₜ[R] (1 / f b) = (b • (f a / f b)) ⊗ₜ[R] (1 / f b)

Tactic:
rw [smul_def, mul_div_cancel₀ _ hb]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Field A
inst✝¹ : Algebra R A
inst✝ : IsFractionRing R A
a b : R
f : R →+* A := algebraMap R A
hf : f = algebraMap R A
hb : f b ≠ 0
⊢ (f a / f b) ⊗ₜ[R] (b • (1 / f b)) = (f a / f b) ⊗ₜ[R] 1

Tactic:
rw [smul_def, mul_div_cancel₀ _ hb]

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

Goal state before:
pp: α : Type u_1
s✝ t : Set α
a : α
inst✝¹ : Star α
inst✝ : TrivialStar α
s : Set α
⊢ s⋆ = s

Tactic:
← star_preimage

Goal state after:
α : Type u_1
s✝ t : Set α
a : α
inst✝¹ : Star α
inst✝ : TrivialStar α
s : Set α
⊢ star ⁻¹' s = s
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : NonUnitalRing β
⊢ NonUnitalRing α

Tactic:
let zero := e.zero
  let add := e.add
  let mul := e.mul
  let neg := e.Neg
  let sub := e.sub
  let nsmul := e.smul ℕ
  let zsmul := e.smul ℤ
  apply e.injective.nonUnitalRing _ <;> intros <;> exact e.apply_symm_apply _

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

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
A : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module A N
inst✝ : IsScalarTower R A N
l : M →ₗ[R] N
⊢ (liftBaseChange A l).range = Submodule.span A ↑l.range

Tactic:
apply le_antisymm
  · rintro _ ⟨x, rfl⟩
    induction x using TensorProduct.induction_on
    · simp
    · rw [LinearMap.liftBaseChange_tmul]
      exact Submodule.smul_mem _ _ (Submodule.subset_span ⟨_, rfl⟩)
    · rw [map_add]
      exact add_mem ‹_› ‹_›
  · rw [Submodule.span_le]
    rintro _ ⟨x, rfl⟩
    exact ⟨1 ⊗ₜ x, by simp⟩

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

Goal state before:
pp: R : Type u_3
A : Type u_4
B : Type u_5
C : Type u_6
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : Semiring B
inst✝² : Algebra R B
inst✝¹ : Semiring C
inst✝ : Algebra R C
x : A
y : C
⊢ x ⊗ₜ[R] 1 ⊗ₜ[A] (1 ⊗ₜ[R] y) = 1 ⊗ₜ[A] (x ⊗ₜ[R] y)

Tactic:
by
  trans x • 1 ⊗ₜ[A] (1 ⊗ₜ[R] y)
  · simp [Algebra.smul_def]
  · simp [← tmul_smul, smul_tmul' (M := A)]

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

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : MulZeroClass R
inst✝² : AddZeroClass A
inst✝¹ : Mul A
inst✝ : SMulWithZero R A
a : A
⊢ IsIdempotentElem ↑a ↔ IsIdempotentElem a

Tactic:
simp only [IsIdempotentElem, ← inr_mul, inr_injective.eq_iff]

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 : R
⊢ 0 ≤ star x ↔ 0 ≤ x

Tactic:
by
  simpa using star_le_star_iff (x := 0) (y := x)

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Nontrivial R
s : Submodule R M
h : 0 < 0
eq : s = ⊥
⊢ False

Tactic:
exact lt_irrefl _ h

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✝¹² : CommSemiring R
inst✝¹¹ : NonUnitalNonAssocSemiring A
inst✝¹⁰ : Module R A
inst✝⁹ : Star A
inst✝⁸ : NonUnitalNonAssocSemiring B
inst✝⁷ : Module R B
inst✝⁶ : Star B
inst✝⁵ : NonUnitalNonAssocSemiring C
inst✝⁴ : Module R C
inst✝³ : Star C
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : StarHomClass F A B
φ : F
⊢ ∀ {a : B},
    a ∈ (NonUnitalAlgHom.range (NonUnitalAlgHomClass.toNonUnitalAlgHom φ)).carrier →
      star a ∈ (NonUnitalAlgHom.range (NonUnitalAlgHomClass.toNonUnitalAlgHom φ)).carrier

Tactic:
rintro _ ⟨a, rfl⟩

Goal state after:
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹² : CommSemiring R
inst✝¹¹ : NonUnitalNonAssocSemiring A
inst✝¹⁰ : Module R A
inst✝⁹ : Star A
inst✝⁸ : NonUnitalNonAssocSemiring B
inst✝⁷ : Module R B
inst✝⁶ : Star B
inst✝⁵ : NonUnitalNonAssocSemiring C
inst✝⁴ : Module R C
inst✝³ : Star C
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : StarHomClass F A B
φ : F
a : A
⊢ star (↑(NonUnitalAlgHomClass.toNonUnitalAlgHom φ) a) ∈
    (NonUnitalAlgHom.range (NonUnitalAlgHomClass.toNonUnitalAlgHom φ)).carrier
Module: Mathlib.LinearAlgebra.Basis.VectorSpace

Goal state before:
pp: case neg
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 := Basis.ofVectorSpace K ↥f.ker
s : Set V := ⇑f.ker.subtype '' Set.range ⇑b₁
hs : span K s = f.ker
n : Set V := insert v s
H₁ : LinearIndepOn K _root_.id n
H₂ : ⊤ ≤ span K n
b : Basis (↑n) K V := Basis.mk H₁ ⋯
i : ↑n := ⟨v, ⋯⟩
hi : b i = v
j : ↑n
h : ¬i = j
⊢ f (b j) = (f (b i) • b.coord i) (b j)

Tactic:
suffices f (b j) = 0 by
      simp [Finsupp.single_eq_of_ne h, this]
    rw [← mem_ker, ← hs, Basis.coe_mk]
    apply subset_span
    apply Or.resolve_left (Set.mem_insert_iff.mpr j.prop)
    simp [← hi, b, Subtype.coe_inj, Ne.symm h]

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)
r : R
⊢ ∀ (x : ↥(iSup K)),
    Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ (r • x) =
      (MonoidHom.id R) r • Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ x

Tactic:
dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply,
          inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast]

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)
r : R
⊢ ∀ (x : ↥(iSup K)),
    Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ (r • x) =
      (MonoidHom.id R) r • Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ x
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)
⊢ Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ 0 = 0

Tactic:
dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply,
          inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast]

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)
⊢ Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ 0 = 0
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
⊢ finrank R M = 0

Tactic:
by
  delta finrank
  rw [h, zero_toNat]

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)
⊢ ∀ (x y : ↥(iSup K)),
    Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ (x + y) =
      Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ x +
        Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ y

Tactic:
dsimp only [SetLike.coe_sort_coe, NonUnitalAlgHom.coe_comp, Function.comp_apply,
          inclusion_mk, Eq.ndrec, id_eq, eq_mpr_eq_cast]

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)
⊢ ∀ (x y : ↥(iSup K)),
    Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ (x + y) =
      Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ x +
        Set.iUnionLift (fun i ↦ ↑(K i)) (fun i x ↦ (f i) x) ⋯ ↑(iSup K) ⋯ y
Module: Mathlib.LinearAlgebra.Dimension.Finite

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : IsDomain R
inst✝² : IsTorsionFree R M
inst✝¹ : StrongRankCondition R
S : Submodule R M
inst✝ : Module.Finite R ↥S
⊢ 1 ≤ finrank R ↥S ↔ S ≠ ⊥

Tactic:
contrapose!; rw [Nat.lt_one_iff, finrank_eq_zero]

Goal state after: