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

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
inst✝¹³ : CommSemiring R
inst✝¹² : CommSemiring A
inst✝¹¹ : CommSemiring B
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Algebra R A
inst✝⁸ : Algebra R B
inst✝⁷ : Module A M
inst✝⁶ : Module B M
inst✝⁵ : Module R M
D✝ D1 D2 : Derivation R A M
r : R
a b : A
N : Type u_5
inst✝⁴ : AddCommMonoid N
inst✝³ : Module A N
inst✝² : Module R N
inst✝¹ : IsScalarTower R A M
inst✝ : IsScalarTower R A N
f : M →ₗ[A] N
e : M ≃ₗ[A] N
D : Derivation R A N
⊢ (↑e).compDer.toFun ((↑e.symm).compDer D) = D

Tactic:
ext a

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
inst✝¹³ : CommSemiring R
inst✝¹² : CommSemiring A
inst✝¹¹ : CommSemiring B
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Algebra R A
inst✝⁸ : Algebra R B
inst✝⁷ : Module A M
inst✝⁶ : Module B M
inst✝⁵ : Module R M
D✝ D1 D2 : Derivation R A M
r : R
a✝ b : A
N : Type u_5
inst✝⁴ : AddCommMonoid N
inst✝³ : Module A N
inst✝² : Module R N
inst✝¹ : IsScalarTower R A M
inst✝ : IsScalarTower R A N
f : M →ₗ[A] N
e : M ≃ₗ[A] N
D : Derivation R A N
a : A
⊢ ((↑e).compDer.toFun ((↑e.symm).compDer D)) a = D a
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case h_nhds
R : Type u
inst✝ : CommSemiring R
p : PrimeSpectrum R
U : Set (PrimeSpectrum R)
s : Set R
hs : zeroLocus s = Uᶜ
f : R
hfs : f ∈ s
hfp : f ∉ ↑p.asIdeal
⊢ ↑(basicOpen f) ⊆ U

Tactic:
rw [← Set.compl_subset_compl, ← hs, basicOpen_eq_zeroLocus_compl, compl_compl]

Goal state after:
case h_nhds
R : Type u
inst✝ : CommSemiring R
p : PrimeSpectrum R
U : Set (PrimeSpectrum R)
s : Set R
hs : zeroLocus s = Uᶜ
f : R
hfs : f ∈ s
hfp : f ∉ ↑p.asIdeal
⊢ zeroLocus s ⊆ zeroLocus {f}
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : Monoid G
⊢ exponent G = 0 ↔ ∀ n > 0, ∃ g, g ^ n ≠ 1

Tactic:
rw [exponent_eq_zero_iff, ExponentExists]
  push Not
  rfl

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic

Goal state before:
pp: G : Type u_2
inst✝ : Group G
H : Subgroup G
hp : Fact (Nat.Prime (Nat.card G))
this✝ : Finite G
this : Nat.card ↥H ∣ Nat.card G
⊢ H = ⊥ ∨ H = ⊤

Tactic:
Nat.dvd_prime hp.1,

Goal state after:
G : Type u_2
inst✝ : Group G
H : Subgroup G
hp : Fact (Nat.Prime (Nat.card G))
this✝ : Finite G
this : Nat.card ↥H = 1 ∨ Nat.card ↥H = Nat.card G
⊢ H = ⊥ ∨ H = ⊤
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case refine_2
ι : Type u_1
R : ι → Type u_2
inst✝¹ : (i : ι) → CommRing (R i)
inst✝ : DecidableEq ι
i : ι
f : R i
p : PrimeSpectrum ((i : ι) → R i)
hp : p ∈ ↑(basicOpen (Pi.single i f))
this : p ∈ Set.range (comap (Pi.evalRingHom R i))
⊢ p ∈ comap (Pi.evalRingHom R i) '' ↑(basicOpen f)

Tactic:
obtain ⟨q, rfl⟩ := this

Goal state after:
case refine_2
ι : Type u_1
R : ι → Type u_2
inst✝¹ : (i : ι) → CommRing (R i)
inst✝ : DecidableEq ι
i : ι
f : R i
q : PrimeSpectrum (R i)
hp : comap (Pi.evalRingHom R i) q ∈ ↑(basicOpen (Pi.single i f))
⊢ comap (Pi.evalRingHom R i) q ∈ comap (Pi.evalRingHom R i) '' ↑(basicOpen f)
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
x : R
⊢ x ∈ vanishingIdeal (Set.range (comap f)) ↔ x ∈ (RingHom.ker f).radical

Tactic:
rw [RingHom.ker_eq_comap_bot, ← Ideal.comap_radical, Ideal.radical_eq_sInf]

Goal state after:
R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
x : R
⊢ x ∈ vanishingIdeal (Set.range (comap f)) ↔ x ∈ Ideal.comap f (sInf {J | ⊥ ≤ J ∧ J.IsPrime})
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
s✝ : Set (PrimeSpectrum R)
x✝ : ∃ s, s.Finite ∧ s✝ = (zeroLocus s)ᶜ
s : Set R
hs : s.Finite
e : s✝ = (zeroLocus s)ᶜ
⊢ (zeroLocus ↑hs.toFinset)ᶜ = s✝

Tactic:
by simpa using e.symm

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
s✝ : Set (PrimeSpectrum R)
x✝ : ∃ t, (zeroLocus ↑t)ᶜ = s✝
s : Finset R
e : (zeroLocus ↑s)ᶜ = s✝
⊢ s✝ = (zeroLocus ↑s)ᶜ

Tactic:
by simpa using e.symm

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
⊢ {x | Minimal (fun x ↦ ∃ x₀, (IsClosed x₀ ∧ IsIrreducible x₀) ∧ vanishingIdeal x₀ = x) x} =
    {x | Minimal Ideal.IsPrime x}

Tactic:
← funext (@Set.mem_ofPred_eq _ · Ideal.IsPrime),

Goal state after:
R : Type u
inst✝ : CommSemiring R
⊢ {x | Minimal (fun x ↦ ∃ x₀, (IsClosed x₀ ∧ IsIrreducible x₀) ∧ vanishingIdeal x₀ = x) x} =
    {x | Minimal (Membership.mem {y | y.IsPrime}) x}
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
this : (fixedPoints ↥↑Q ↑(orbit G P)).Nonempty
R : ↑(orbit G P)
hR : R ∈ fixedPoints ↥↑Q ↑(orbit G P)
⊢ ∃ g, g • P = Q

Tactic:
by
        rw [← Sylow.ext (H.mp hR)]
        exact R.2

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: α : Type u_1
inst✝¹ : CommGroup α
inst✝ : IsSimpleGroup α
g : α
hg : ∀ (x : α), x ∈ Subgroup.zpowers g
hα : orderOf g ≠ 1
h : orderOf g = 0
⊢ ¬orderOf g ∣ 37 ∧ Nat.gcd 37 (orderOf g) ≠ 1

Tactic:
by simp [h]

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

Goal state before:
pp: M : Type u_4
inst✝³ : AddCommGroup M
R : Type u_6
inst✝² : CommRing R
inst✝¹ : Module R M
inst✝ : IsSemisimpleModule R M
⊢ (⨅ i, (↑i).annihilator).IsRadical

Tactic:
exact Ideal.isRadical_iInf _ fun i ↦ (i.2.annihilator_isMaximal).isPrime.isRadical

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : M
h : m = 0
⊢ torsionOf R M m = ⊤

Tactic:
by simp [h]

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

Goal state before:
pp: R : Type u_3
A : Type u_1
M : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
a : A
inst✝⁵ : Algebra R A
inst✝⁴ : AddCommMonoid M
inst✝³ : Module A M
inst✝² : Module R M
inst✝¹ : IsScalarTower R A M
inst✝ : FaithfulSMul A M
p : R[X]
h : ∀ (m : M), (aeval a) p • m = 0
⊢ ∀ (a_1 : M), (aeval a) p • a_1 = 0 • a_1

Tactic:
by simp [h]

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

Goal state before:
pp: R : Type u_3
A : Type u_1
M : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
a : A
inst✝⁵ : Algebra R A
inst✝⁴ : AddCommMonoid M
inst✝³ : Module A M
inst✝² : Module R M
inst✝¹ : IsScalarTower R A M
inst✝ : FaithfulSMul A M
p : R[X]
h : (aeval a) p = 0
⊢ ∀ (m : M), (aeval a) p • m = 0

Tactic:
by simp [h]

Goal state after:
Module: Mathlib.Order.JordanHolder

Goal state before:
pp: case succ
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih : ∀ (s₁ s₂ : CompositionSeries X), head s₁ = head s₂ → last s₁ = last s₂ → s₁.length = n → s₁.Equivalent s₂
s₁ s₂ : CompositionSeries X
hb : head s₁ = head s₂
ht : last s₁ = last s₂
hle : s₁.length = n + 1
h0s₂ : 0 < s₂.length
t : CompositionSeries X
htb : head t = head s₁
htl : t.length + 1 = s₁.length
htt : last t = (eraseLast s₂).last
hteq : s₁.Equivalent (snoc t (last s₁) ⋯)
this : t.Equivalent (eraseLast s₂)
⊢ Equivalent (snoc t (last s₂) ⋯) ((eraseLast s₂).snoc (last s₂) ⋯)

Tactic:
exact Equivalent.snoc this (by simpa [htt] using iso_refl)

Goal state after:
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: case h
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
⊢ ∀ n ≥ 0, I • (I.stableFiltration N).N n = (I.stableFiltration N).N (n + 1)

Tactic:
intro n _

Goal state after:
case h
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
n : ℕ
a✝ : n ≥ 0
⊢ I • (I.stableFiltration N).N n = (I.stableFiltration N).N (n + 1)
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: A : Type u_3
inst✝ : AddCommGroup A
n : ℤ
⊢ A[-n] = A[n]

Tactic:
ext a

Goal state after:
A : Type u_3
inst✝ : AddCommGroup A
n : ℤ
a : A
⊢ a ∈ A[-n] ↔ a ∈ A[n]
Module: Mathlib.Algebra.Module.Torsion.Basic

Goal state before:
pp: M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : IsAddTorsionFree M
x y : M
⊢ (Set.range fun n ↦ x + n • y).Infinite ↔ y ≠ 0

Tactic:
refine ⟨fun h hy ↦ by simp [hy] at h, fun h ↦ Set.infinite_range_of_injective fun r s hrs ↦ ?_⟩
  rw [add_right_inj, ← natCast_zsmul, ← natCast_zsmul] at hrs
  simpa using smul_left_injective _ h hrs

Goal state after:
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: case mp
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
H :
  ∀ (i : ℕ), single R i '' ↑(F.N i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i)))
⊢ ∀ n ≥ n₀, I • F.N n = F.N (n + 1)

Tactic:
n

Goal state after:
case mp
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
H :
  ∀ (i : ℕ), single R i '' ↑(F.N i) ⊆ ↑(Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i)))
n : ℕ
⊢ n ≥ n₀ → I • F.N n = F.N (n + 1)
Module: Mathlib.RingTheory.Filtration

Goal state before:
pp: case mp
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
hF' : ∀ (i : ℕ), (F.N i).FG
⊢ F.submodule.FG → ∃ n₀, F.submodule = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i))

Tactic:
rintro H

Goal state after:
case mp
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
hF' : ∀ (i : ℕ), (F.N i).FG
H : F.submodule.FG
⊢ ∃ n₀, F.submodule = Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), single R i '' ↑(F.N i))
Module: Mathlib.RingTheory.Ideal.Cotangent

Goal state before:
pp: R : Type u
S : Type v
S' : Type w
inst✝¹⁰ : CommRing R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra S R
inst✝⁷ : CommSemiring S'
inst✝⁶ : Algebra S' R
inst✝⁵ : Algebra S S'
inst✝⁴ : IsScalarTower S S' R
I : Ideal R
A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
⊢ R ⧸ I ^ 2 ⊔ I ≃+* R ⧸ I

Tactic:
exact Ideal.quotEquivOfEq (sup_eq_right.mpr <| Ideal.pow_le_self two_ne_zero)

Goal state after:
Module: Mathlib.RingTheory.Ideal.Cotangent

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
fg : (maximalIdeal R).FG
s : Set ↥(maximalIdeal R)
hs_card : Cardinal.mk ↑s = ⊤.spanRank
hs_span : Submodule.span R s = ⊤
⊢ Submodule.span (ResidueField R) (⇑(maximalIdeal R).toCotangent '' s) = ⊤

Tactic:
rw [← Submodule.restrictScalars_eq_top_iff R,
        Submodule.restrictScalars_span R (ResidueField R) Ideal.Quotient.mk_surjective,
        ← Submodule.map_span, hs_span, Submodule.map_top, Ideal.toCotangent_range]

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Presheaf

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
M : PresheafOfModules R
U V W : Cᵒᵖ
i : U ⟶ V
j : V ⟶ W
x : ↑(M.obj U)
⊢ (ConcreteCategory.hom (M.map (i ≫ j))) x = (ConcreteCategory.hom (M.map j)) ((ConcreteCategory.hom (M.map i)) x)

Tactic:
by
  rw [M.map_comp]; rfl

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

Goal state before:
pp: case a
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S ⊗[R] S
hx : (TensorProduct.lmul' R) x = 0
this : x - (TensorProduct.lmul' R) x ⊗ₜ[R] 1 = x
⊢ x - (TensorProduct.lmul' R) x ⊗ₜ[R] 1 ∈ Submodule.span S (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)

Tactic:
clear this hx

Goal state after:
case a
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S ⊗[R] S
⊢ x - (TensorProduct.lmul' R) x ⊗ₜ[R] 1 ∈ Submodule.span S (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)
Module: Mathlib.Algebra.Homology.HomologicalComplex

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
j : ι
h : ¬c.Rel (c.prev j) j
⊢ (C.xPrevIsoSelf h).inv ≫ C.dTo j = 0

Tactic:
by simp [h]

Goal state after:
Module: Mathlib.Algebra.Homology.HomologicalComplex

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i : ι
h : ¬c.Rel i (c.next i)
⊢ C.dFrom i ≫ (C.xNextIsoSelf h).hom = 0

Tactic:
by simp [h]

Goal state after:
Module: Mathlib.Algebra.Homology.HomologicalComplex

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C C₁ C₂ C₃ : HomologicalComplex V c
f : (i : ι) → C₁.X i ≅ C₂.X i
hf : ∀ (i j : ι), c.Rel i j → (f i).hom ≫ C₂.d i j = C₁.d i j ≫ (f j).hom
i j : ι
hij : c.Rel i j
⊢ (f i).inv ≫ (C₁.d i j ≫ (f j).hom) ≫ (f j).inv = (f i).inv ≫ ((f i).hom ≫ C₂.d i j) ≫ (f j).inv

Tactic:
rw [hf i j hij]

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

Goal state before:
pp: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
a b : S
⊢ (DLinearMap R S) (a * b) = a • (DLinearMap R S) b + b • (DLinearMap R S) a

Tactic:
have : LinearMap.CompatibleSMul { x // x ∈ ideal R S } Ω[S⁄R] S (S ⊗[R] S) := inferInstance
      dsimp [KaehlerDifferential.DLinearMap_apply]
      rw [← LinearMap.map_smul_of_tower (ideal R S).toCotangent,
        ← LinearMap.map_smul_of_tower (ideal R S).toCotangent,
        ← map_add (ideal R S).toCotangent, Ideal.toCotangent_eq, pow_two]
      convert!
        Submodule.mul_mem_mul (KaehlerDifferential.one_smul_sub_smul_one_mem_ideal R a :)
          (KaehlerDifferential.one_smul_sub_smul_one_mem_ideal R b :) using 1
      simp only [Submodule.coe_add,
        TensorProduct.tmul_mul_tmul, mul_sub, sub_mul, mul_comm b, Submodule.coe_smul_of_tower,
        smul_sub, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
      ring_nf

Goal state after:
Module: Mathlib.Algebra.Homology.Additive

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹¹ : Category.{v, u} V
inst✝¹⁰ : Preadditive V
W : Type u_2
inst✝⁹ : Category.{v_1, u_2} W
inst✝⁸ : Preadditive W
W₁ : Type u_3
W₂ : Type u_4
inst✝⁷ : Category.{v_2, u_3} W₁
inst✝⁶ : Category.{v_3, u_4} W₂
inst✝⁵ : HasZeroMorphisms W₁
inst✝⁴ : HasZeroMorphisms W₂
c✝ : ComplexShape ι
C D : HomologicalComplex V c✝
f : C ⟶ D
i✝ : ι
inst✝³ : HasZeroObject W₁
inst✝² : HasZeroObject W₂
F : W₁ ⥤ W₂
inst✝¹ : F.PreservesZeroMorphisms
c : ComplexShape ι
inst✝ : DecidableEq ι
j : ι
X : W₁
i : ι
h : i = j
⊢ ((single W₁ c j ⋙ F.mapHomologicalComplex c).obj X).X i = ((F ⋙ single W₂ c j).obj X).X i

Tactic:
by simp [h]

Goal state after:
Module: Mathlib.Algebra.Homology.Additive

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹¹ : Category.{v, u} V
inst✝¹⁰ : Preadditive V
W : Type u_2
inst✝⁹ : Category.{v_1, u_2} W
inst✝⁸ : Preadditive W
W₁ : Type u_3
W₂ : Type u_4
inst✝⁷ : Category.{v_2, u_3} W₁
inst✝⁶ : Category.{v_3, u_4} W₂
inst✝⁵ : HasZeroMorphisms W₁
inst✝⁴ : HasZeroMorphisms W₂
c✝ : ComplexShape ι
C D : HomologicalComplex V c✝
f : C ⟶ D
i✝ : ι
inst✝³ : HasZeroObject W₁
inst✝² : HasZeroObject W₂
F : W₁ ⥤ W₂
inst✝¹ : F.PreservesZeroMorphisms
c : ComplexShape ι
inst✝ : DecidableEq ι
j : ι
X : W₁
i : ι
h : i = j
⊢ ((F ⋙ single W₂ c j).obj X).X i = ((single W₁ c j ⋙ F.mapHomologicalComplex c).obj X).X i

Tactic:
by simp [h]

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

Goal state before:
pp: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
⊢ D.liftKaehlerDifferential.compDer (KaehlerDifferential.D R S) = D

Tactic:
ext a

Goal state after:
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
D : Derivation R S M
a : S
⊢ (D.liftKaehlerDifferential.compDer (KaehlerDifferential.D R S)) a = D a
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
X : C
m n p m' n' p' : A
hm : m' + m = 0
hn : n' + n = 0
hp : p' + p = 0
h : m + n = p
⊢ n' + m' + m = n'

Tactic:
by rw [add_assoc, hm, add_zero]

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

Goal state before:
pp: case a
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : EssFiniteType R S
⊢ ideal R S ≤ Ideal.span ↑(Finset.image (fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) (EssFiniteType.finset R S))

Tactic:
← KaehlerDifferential.span_range_eq_ideal,

Goal state after:
case a
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : EssFiniteType R S
⊢ Ideal.span (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) ≤
    Ideal.span ↑(Finset.image (fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) (EssFiniteType.finset R S))
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
a b : S
⊢ ((derivationQuotKerTotal R S).liftKaehlerDifferential ∘ₗ linearCombination S ⇑(D R S)) (single a b) = b𝖣a

Tactic:
conv_rhs => rw [← Finsupp.smul_single_one a b, map_smul]

Goal state after:
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
a b : S
⊢ ((derivationQuotKerTotal R S).liftKaehlerDifferential ∘ₗ linearCombination S ⇑(D R S)) (single a b) = b • 1𝖣a
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
A : Type u_2
B : Type u_3
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : Algebra R B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Function.Surjective ⇑(algebraMap A B)
⊢ Submodule.map (mapRange.linearMap (Algebra.linearMap A B) ∘ₗ lmapDomain A A ⇑(algebraMap A B)) (kerTotal R A) ⊔
      Submodule.span A (Set.range fun x ↦ single ((algebraMap S B) x) 1) =
    Submodule.restrictScalars A (kerTotal S B)

Tactic:
rw [KaehlerDifferential.kerTotal, Submodule.map_span, KaehlerDifferential.kerTotal,
    Submodule.restrictScalars_span _ _ h]

Goal state after:
R : Type u
S : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
A : Type u_2
B : Type u_3
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : Algebra R B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Function.Surjective ⇑(algebraMap A B)
⊢ Submodule.span A
        (⇑(mapRange.linearMap (Algebra.linearMap A B) ∘ₗ lmapDomain A A ⇑(algebraMap A B)) ''
          (((Set.range fun x ↦ single x.1 1 + single x.2 1 - single (x.1 + x.2) 1) ∪
              Set.range fun x ↦ single x.2 x.1 + single x.1 x.2 - single (x.1 * x.2) 1) ∪
            Set.range fun x ↦ single ((algebraMap R A) x) 1)) ⊔
      Submodule.span A (Set.range fun x ↦ single ((algebraMap S B) x) 1) =
    Submodule.span A
      (((Set.range fun x ↦ single x.1 1 + single x.2 1 - single (x.1 + x.2) 1) ∪
          Set.range fun x ↦ single x.2 x.1 + single x.1 x.2 - single (x.1 * x.2) 1) ∪
        Set.range fun x ↦ single ((algebraMap S B) x) 1)
Module: Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
E₁ : Type u_3
E₂ : Type u_4
inst✝¹ : Category.{v_3, u_3} E₁
inst✝ : Category.{v_4, u_4} E₂
F₁ : C₁ ⥤ E₁
F₂ : C₂ ⥤ E₂
h₁ : W₁.IsInvertedBy F₁
h₂ : W₂.IsInvertedBy F₂
x✝¹ x✝ : C₁ × C₂
f : x✝¹ ⟶ x✝
hf : W₁.prod W₂ f
⊢ IsIso ((F₁.prod F₂).map f)

Tactic:
by
  rw [isIso_prod_iff]
  exact ⟨h₁ _ hf.1, h₂ _ hf.2⟩

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} C₃
W : MorphismProperty C₁
F : C₁ ⥤ C₂
G : C₂ ⥤ C₃
inst✝ : G.ReflectsIsomorphisms
⊢ W.IsInvertedBy (F ⋙ G) ↔ W.IsInvertedBy F

Tactic:
constructor
  · intro h X Y f hf
    have : IsIso (G.map (F.map f)) := h _ hf
    exact isIso_of_reflects_iso (F.map f) G
  · intro hF
    exact IsInvertedBy.of_comp W F hF G

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

Goal state before:
pp: case h
R : Type u
inst✝⁶ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
h : Function.Surjective ⇑(algebraMap A B)
x : A →₀ A
hx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A B) ∘ₗ lmapDomain A A ⇑(algebraMap A B)).ker
c : B
a✝ : c ∈ Finset.image (⇑(algebraMap A B)) x.support
this : ∑ i ∈ x.support with (algebraMap A B) i = c, x i ∈ RingHom.ker (algebraMap A B)
a : A
ha : (algebraMap A B) a = c
⊢ ∑ x_1 ∈ {i ∈ x.support | (algebraMap A B) i = c}.attach, x ↑x_1 • 1 ⊗ₜ[A] ((D R A) ↑x_1 - (D R A) a) =
    1 ⊗ₜ[A] ∑ x_1 ∈ x.support with (algebraMap A B) x_1 = c, x x_1 • (D R A) x_1

Tactic:
← TensorProduct.tmul_smul

Goal state after:
case h
R : Type u
inst✝⁶ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
h : Function.Surjective ⇑(algebraMap A B)
x : A →₀ A
hx : x ∈ ↑(mapRange.linearMap (Algebra.linearMap A B) ∘ₗ lmapDomain A A ⇑(algebraMap A B)).ker
c : B
a✝ : c ∈ Finset.image (⇑(algebraMap A B)) x.support
this : ∑ i ∈ x.support with (algebraMap A B) i = c, x i ∈ RingHom.ker (algebraMap A B)
a : A
ha : (algebraMap A B) a = c
⊢ ∑ x_1 ∈ {i ∈ x.support | (algebraMap A B) i = c}.attach, 1 ⊗ₜ[A] (x ↑x_1 • ((D R A) ↑x_1 - (D R A) a)) =
    1 ⊗ₜ[A] ∑ x_1 ∈ x.support with (algebraMap A B) x_1 = c, x x_1 • (D R A) x_1
Module: Mathlib.Algebra.Homology.HomotopyCofiber

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
i j : ι
hij : c.Rel j i
⊢ inlX φ i j hij ≫ (XIsoBiprod φ j i hij).hom = biprod.inl

Tactic:
have := HasHomotopyCofiber.hasBinaryBiproduct φ _ _ hij

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
i j : ι
hij : c.Rel j i
this : HasBinaryBiproduct (F.X i) (G.X j)
⊢ inlX φ i j hij ≫ (XIsoBiprod φ j i hij).hom = biprod.inl
Module: Mathlib.Algebra.Homology.HomotopyCofiber

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
j : ι
hij : c.Rel j (c.next j)
⊢ inrX φ j ≫ (XIsoBiprod φ j (c.next j) hij).hom = biprod.inr

Tactic:
have := HasHomotopyCofiber.hasBinaryBiproduct φ _ _ hij

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
j : ι
hij : c.Rel j (c.next j)
this : HasBinaryBiproduct (F.X (c.next j)) (G.X j)
⊢ inrX φ j ≫ (XIsoBiprod φ j (c.next j) hij).hom = biprod.inr