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

Goal state before:
pp: case a.refine_1
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S ⊗[R] S
⊢ 0 ∈ Submodule.span S (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)

Tactic:
;

Goal state after:
case a.refine_1
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S ⊗[R] S
⊢ 0 ∈ Submodule.span S (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝¹ : Category.{v, u} C
inst✝ : AddMonoid A
h : ShiftMkCore C A
n : A
X : C
⊢ (h.add n 0).inv.app X = h.zero.hom.app ((h.F n).obj X) ≫ eqToHom ⋯

Tactic:
by
  rw [← cancel_epi ((h.add n 0).hom.app X), Iso.hom_inv_id_app, h.add_zero_hom_app,
    Category.assoc, Iso.inv_hom_id_app_assoc, eqToHom_trans, eqToHom_refl]

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
K : HomologicalComplex V c
p₂ p₃ : ι
⊢ (K.XIsoOfEq ⋯).inv ≫ K.d p₂ p₃ = K.d p₂ p₃

Tactic:
;

Goal state after:
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
K : HomologicalComplex V c
p₂ p₃ : ι
⊢ (K.XIsoOfEq ⋯).inv ≫ K.d p₂ p₃ = K.d p₂ p₃
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
K : HomologicalComplex V c
p₂ p₁ : ι
⊢ K.d p₁ p₂ ≫ (K.XIsoOfEq ⋯).hom = K.d p₁ p₂

Tactic:
;

Goal state after:
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
K : HomologicalComplex V c
p₂ p₁ : ι
⊢ K.d p₁ p₂ ≫ (K.XIsoOfEq ⋯).hom = K.d p₁ p₂
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
K : HomologicalComplex V c
p₃ p₁ : ι
⊢ K.d p₁ p₃ ≫ (K.XIsoOfEq ⋯).inv = K.d p₁ p₃

Tactic:
;

Goal state after:
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
K : HomologicalComplex V c
p₃ p₁ : ι
⊢ K.d p₁ p₃ ≫ (K.XIsoOfEq ⋯).inv = K.d p₁ p₃
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a₁ a₂ a₃ a₁₂ a₂₃ a₁₂₃ : A
h₁₂ : a₁ + a₂ = a₁₂
h₂₃ : a₂ + a₃ = a₂₃
h₁₂₃ : a₁ + a₂ + a₃ = a₁₂₃
⊢ shiftFunctorAdd' C a₁₂ a₃ a₁₂₃ ⋯ ≪≫
      isoWhiskerRight (shiftFunctorAdd' C a₁ a₂ a₁₂ h₁₂) (shiftFunctor C a₃) ≪≫
        (shiftFunctor C a₁).associator (shiftFunctor C a₂) (shiftFunctor C a₃) =
    shiftFunctorAdd' C a₁ a₂₃ a₁₂₃ ⋯ ≪≫ (shiftFunctor C a₁).isoWhiskerLeft (shiftFunctorAdd' C a₂ a₃ a₂₃ h₂₃)

Tactic:
subst h₁₂ h₂₃ h₁₂₃

Goal state after:
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a₁ a₂ a₃ : A
⊢ shiftFunctorAdd' C (a₁ + a₂) a₃ (a₁ + a₂ + a₃) ⋯ ≪≫
      isoWhiskerRight (shiftFunctorAdd' C a₁ a₂ (a₁ + a₂) ⋯) (shiftFunctor C a₃) ≪≫
        (shiftFunctor C a₁).associator (shiftFunctor C a₂) (shiftFunctor C a₃) =
    shiftFunctorAdd' C a₁ (a₂ + a₃) (a₁ + a₂ + a₃) ⋯ ≪≫
      (shiftFunctor C a₁).isoWhiskerLeft (shiftFunctorAdd' C a₂ a₃ (a₂ + a₃) ⋯)
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
K L : HomologicalComplex V c
φ : K ⟶ L
n : ι
⊢ φ.f n ≫ (L.XIsoOfEq ⋯).hom = (K.XIsoOfEq ⋯).hom ≫ φ.f n

Tactic:
;

Goal state after:
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
K L : HomologicalComplex V c
φ : K ⟶ L
n : ι
⊢ φ.f n ≫ (L.XIsoOfEq ⋯).hom = (K.XIsoOfEq ⋯).hom ≫ φ.f n
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
K L : HomologicalComplex V c
φ : K ⟶ L
n : ι
⊢ φ.f n ≫ (L.XIsoOfEq ⋯).inv = (K.XIsoOfEq ⋯).inv ≫ φ.f n

Tactic:
;

Goal state after:
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
K L : HomologicalComplex V c
φ : K ⟶ L
n : ι
⊢ φ.f n ≫ (L.XIsoOfEq ⋯).inv = (K.XIsoOfEq ⋯).inv ≫ φ.f n
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctor C (0 + a)).obj X = (shiftFunctor C a).obj X

Tactic:
;

Goal state after:
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctor C (0 + a)).obj X = (shiftFunctor C a).obj X
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctor C a).obj X = (shiftFunctor C (0 + a)).obj X

Tactic:
;

Goal state after:
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctor C a).obj X = (shiftFunctor C (0 + a)).obj X
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctor C (a + 0)).obj X = (shiftFunctor C a).obj X

Tactic:
;

Goal state after:
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctor C (a + 0)).obj X = (shiftFunctor C a).obj X
Module: Mathlib.Algebra.Homology.HomologicalComplex

Goal state before:
pp: case hnc
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
⊢ ∀ (j_1 : ι), ¬c.Rel j_1 j

Tactic:
;

Goal state after:
case hnc
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
⊢ ∀ (j_1 : ι), ¬c.Rel j_1 j
Module: Mathlib.Algebra.Homology.HomologicalComplex

Goal state before:
pp: case hnc
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
j i : ι
h : ¬c.Rel i j
hi : c.Rel i j
this : c.prev j = i
⊢ False

Tactic:
;

Goal state after:
case hnc
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
j i : ι
h : ¬c.Rel i j
hi : c.Rel i j
this : c.prev j = i
⊢ False
Module: Mathlib.CategoryTheory.Shift.Basic

Goal state before:
pp: C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctor C a).obj X = (shiftFunctor C (a + 0)).obj X

Tactic:
;

Goal state after:
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctor C a).obj X = (shiftFunctor C (a + 0)).obj X
Module: Mathlib.Algebra.Homology.HomologicalComplex

Goal state before:
pp: case hnc
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)
⊢ ¬∃ j, c.Rel i j

Tactic:
;

Goal state after:
case hnc
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)
⊢ ¬∃ j, c.Rel i j
Module: Mathlib.Algebra.Homology.HomologicalComplex

Goal state before:
pp: case hnc
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i j : ι
h : ¬c.Rel i j
hj : c.Rel i j
this : c.next i = j
⊢ False

Tactic:
;

Goal state after:
case hnc
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i j : ι
h : ¬c.Rel i j
hj : c.Rel i j
this : c.next i = j
⊢ False
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
i j : A
h : i + j = 0
X : C
⊢ j + i = 0

Tactic:
;

Goal state after:
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
i j : A
h : i + j = 0
X : C
⊢ j + i = 0
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_3
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x✝ : S ⊗[R] S
hx✝ : x✝ ∈ ideal R S
hx : x✝ ∈ Submodule.span S (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)
x y : S ⊗[R] S
hx₁ : x ∈ ideal R S
hx₂ : (fromIdeal R S) ⟨x, hx₁⟩ ∈ Submodule.span S (Set.range ⇑(D R S))
hy₁ : y ∈ ideal R S
hy₂ : (fromIdeal R S) ⟨y, hy₁⟩ ∈ Submodule.span S (Set.range ⇑(D R S))
⊢ ∃ (hx : x + y ∈ ideal R S), (fromIdeal R S) ⟨x + y, hx⟩ ∈ Submodule.span S (Set.range ⇑(D R S))

Tactic:
;

Goal state after:
case refine_3
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x✝ : S ⊗[R] S
hx✝ : x✝ ∈ ideal R S
hx : x✝ ∈ Submodule.span S (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)
x y : S ⊗[R] S
hx₁ : x ∈ ideal R S
hx₂ : (fromIdeal R S) ⟨x, hx₁⟩ ∈ Submodule.span S (Set.range ⇑(D R S))
hy₁ : y ∈ ideal R S
hy₂ : (fromIdeal R S) ⟨y, hy₁⟩ ∈ Submodule.span S (Set.range ⇑(D R S))
⊢ ∃ (hx : x + y ∈ ideal R S), (fromIdeal R S) ⟨x + y, hx⟩ ∈ Submodule.span S (Set.range ⇑(D 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
h : Function.Surjective ⇑(algebraMap R S)
x : S
⊢ (D R S) x ∈ ↑⊥

Tactic:
;

Goal state after:
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
h : Function.Surjective ⇑(algebraMap R S)
x : S
⊢ (D R S) x ∈ ↑⊥
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
h : Function.Surjective ⇑(algebraMap R S)
x : R
⊢ (D R S) ((algebraMap R S) x) ∈ ↑⊥

Tactic:
;

Goal state after:
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
h : Function.Surjective ⇑(algebraMap R S)
x : R
⊢ (D R S) ((algebraMap R S) x) ∈ ↑⊥
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
X₀ X₁ X₂ : V
d₀ : X₁ ⟶ X₀
d₁ : X₂ ⟶ X₁
s : d₁ ≫ d₀ = 0
succ : (S : ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0
S : ShortComplex V
⊢ (succ S).fst = (succ S).fst

Tactic:
;

Goal state after:
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
X₀ X₁ X₂ : V
d₀ : X₁ ⟶ X₀
d₁ : X₂ ⟶ X₁
s : d₁ ≫ d₀ = 0
succ : (S : ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0
S : ShortComplex V
⊢ (succ S).fst = (succ S).fst
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
X₀ X₁ X₂ : V
d₀ : X₁ ⟶ X₀
d₁ : X₂ ⟶ X₁
s : d₁ ≫ d₀ = 0
succ : (S : ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0
S : ShortComplex V
⊢ S.X₁ = S.X₁

Tactic:
;

Goal state after:
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
X₀ X₁ X₂ : V
d₀ : X₁ ⟶ X₀
d₁ : X₂ ⟶ X₁
s : d₁ ≫ d₀ = 0
succ : (S : ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0
S : ShortComplex V
⊢ S.X₁ = S.X₁
Module: Mathlib.Algebra.Homology.Additive

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹³ : Category.{v, u} V
inst✝¹² : Preadditive V
W✝ : Type u_2
inst✝¹¹ : Category.{v_1, u_2} W✝
inst✝¹⁰ : Preadditive W✝
W₁ : Type u_3
W₂ : Type u_4
inst✝⁹ : Category.{v_2, u_3} W₁
inst✝⁸ : Category.{v_3, u_4} W₂
inst✝⁷ : HasZeroMorphisms W₁
inst✝⁶ : HasZeroMorphisms W₂
c : ComplexShape ι
C D : HomologicalComplex V c
f✝ : C ⟶ D
i : ι
inst✝⁵ : HasZeroObject W₁
inst✝⁴ : HasZeroObject W₂
W : Type u_5
inst✝³ : Category.{v_4, u_5} W
inst✝² : Preadditive W
inst✝¹ : HasZeroObject W
inst✝ : DecidableEq ι
j : ι
x✝¹ x✝ : W
f g : x✝¹ ⟶ x✝
⊢ ((single W c j).map (f + g)).f j = ((single W c j).map f + (single W c j).map g).f j

Tactic:
;

Goal state after:
ι : Type u_1
V : Type u
inst✝¹³ : Category.{v, u} V
inst✝¹² : Preadditive V
W✝ : Type u_2
inst✝¹¹ : Category.{v_1, u_2} W✝
inst✝¹⁰ : Preadditive W✝
W₁ : Type u_3
W₂ : Type u_4
inst✝⁹ : Category.{v_2, u_3} W₁
inst✝⁸ : Category.{v_3, u_4} W₂
inst✝⁷ : HasZeroMorphisms W₁
inst✝⁶ : HasZeroMorphisms W₂
c : ComplexShape ι
C D : HomologicalComplex V c
f✝ : C ⟶ D
i : ι
inst✝⁵ : HasZeroObject W₁
inst✝⁴ : HasZeroObject W₂
W : Type u_5
inst✝³ : Category.{v_4, u_5} W
inst✝² : Preadditive W
inst✝¹ : HasZeroObject W
inst✝ : DecidableEq ι
j : ι
x✝¹ x✝ : W
f g : x✝¹ ⟶ x✝
⊢ ((single W c j).map (f + g)).f j = ((single W c j).map f + (single W c j).map g).f j
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_2.refine_2
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
x✝ : ↥(KaehlerDifferential.ideal R S)
hx : x✝ ∈ Submodule.restrictScalars S (KaehlerDifferential.ideal R S • ⊤)
x y : ↥(KaehlerDifferential.ideal R S)
ex : (D.tensorProductTo ∘ₗ ↑S (Submodule.subtype (KaehlerDifferential.ideal R S))) x = 0
ey : (D.tensorProductTo ∘ₗ ↑S (Submodule.subtype (KaehlerDifferential.ideal R S))) y = 0
⊢ (D.tensorProductTo ∘ₗ ↑S (Submodule.subtype (KaehlerDifferential.ideal R S))) (x + y) = 0

Tactic:
;

Goal state after:
case refine_2.refine_2
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
x✝ : ↥(KaehlerDifferential.ideal R S)
hx : x✝ ∈ Submodule.restrictScalars S (KaehlerDifferential.ideal R S • ⊤)
x y : ↥(KaehlerDifferential.ideal R S)
ex : (D.tensorProductTo ∘ₗ ↑S (Submodule.subtype (KaehlerDifferential.ideal R S))) x = 0
ey : (D.tensorProductTo ∘ₗ ↑S (Submodule.subtype (KaehlerDifferential.ideal R S))) y = 0
⊢ (D.tensorProductTo ∘ₗ ↑S (Submodule.subtype (KaehlerDifferential.ideal R S))) (x + y) = 0
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
f f' : Ω[S⁄R] →ₗ[S] M
hf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S)
x : Ω[S⁄R]
⊢ x ∈ ⊤

Tactic:
;

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
f f' : Ω[S⁄R] →ₗ[S] M
hf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S)
x : Ω[S⁄R]
⊢ x ∈ ⊤
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_1
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
f f' : Ω[S⁄R] →ₗ[S] M
hf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S)
x✝ : Ω[S⁄R]
this : x✝ ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
x : S
⊢ f ((KaehlerDifferential.D R S) x) = f' ((KaehlerDifferential.D R S) x)

Tactic:
;

Goal state after:
case refine_1
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
f f' : Ω[S⁄R] →ₗ[S] M
hf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S)
x✝ : Ω[S⁄R]
this : x✝ ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
x : S
⊢ f ((KaehlerDifferential.D R S) x) = f' ((KaehlerDifferential.D R S) x)
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_3
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
f f' : Ω[S⁄R] →ₗ[S] M
hf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S)
x✝ : Ω[S⁄R]
this : x✝ ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
x y : Ω[S⁄R]
hx✝ : x ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
hy✝ : y ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
hx : f x = f' x
hy : f y = f' y
⊢ f (x + y) = f' (x + y)

Tactic:
;

Goal state after:
case refine_3
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
f f' : Ω[S⁄R] →ₗ[S] M
hf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S)
x✝ : Ω[S⁄R]
this : x✝ ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
x y : Ω[S⁄R]
hx✝ : x ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
hy✝ : y ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
hx : f x = f' x
hy : f y = f' y
⊢ f (x + y) = f' (x + y)
Module: Mathlib.Algebra.Homology.HomologicalComplex

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : HasZeroMorphisms V
α : Type u_2
inst✝² : AddRightCancelSemigroup α
inst✝¹ : One α
inst✝ : DecidableEq α
X✝ : α → V
d_X : (n : α) → X✝ n ⟶ X✝ (n + 1)
sq_X : ∀ (n : α), d_X n ≫ d_X (n + 1) = 0
Y✝ : α → V
d_Y : (n : α) → Y✝ n ⟶ Y✝ (n + 1)
sq_Y : ∀ (n : α), d_Y n ≫ d_Y (n + 1) = 0
X Y : CochainComplex V α
f : (i : α) → X.X i ⟶ Y.X i
comm : ∀ (i : α), f i ≫ Y.d i (i + 1) = X.d i (i + 1) ≫ f (i + 1)
n m : α
⊢ (ComplexShape.up α).Rel n m → f n ≫ Y.d n m = X.d n m ≫ f m

Tactic:
simp only [ComplexShape.up_Rel]

Goal state after:
ι : Type u_1
V : Type u
inst✝⁴ : Category.{v, u} V
inst✝³ : HasZeroMorphisms V
α : Type u_2
inst✝² : AddRightCancelSemigroup α
inst✝¹ : One α
inst✝ : DecidableEq α
X✝ : α → V
d_X : (n : α) → X✝ n ⟶ X✝ (n + 1)
sq_X : ∀ (n : α), d_X n ≫ d_X (n + 1) = 0
Y✝ : α → V
d_Y : (n : α) → Y✝ n ⟶ Y✝ (n + 1)
sq_Y : ∀ (n : α), d_Y n ≫ d_Y (n + 1) = 0
X Y : CochainComplex V α
f : (i : α) → X.X i ⟶ Y.X i
comm : ∀ (i : α), f i ≫ Y.d i (i + 1) = X.d i (i + 1) ≫ f (i + 1)
n m : α
⊢ n + 1 = m → f n ≫ Y.d n m = X.d n m ≫ f m
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_4
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
f f' : Ω[S⁄R] →ₗ[S] M
hf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S)
x✝ : Ω[S⁄R]
this : x✝ ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
a : S
x : Ω[S⁄R]
hx✝ : x ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
e : f x = f' x
⊢ f (a • x) = f' (a • x)

Tactic:
;

Goal state after:
case refine_4
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
f f' : Ω[S⁄R] →ₗ[S] M
hf : f.compDer (KaehlerDifferential.D R S) = f'.compDer (KaehlerDifferential.D R S)
x✝ : Ω[S⁄R]
this : x✝ ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
a : S
x : Ω[S⁄R]
hx✝ : x ∈ Submodule.span S (Set.range ⇑(KaehlerDifferential.D R S))
e : f x = f' x
⊢ f (a • x) = f' (a • x)
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
x : Ω[S⁄R]
⊢ ∃ a, (D R S).tensorProductTo a = x

Tactic:
;

Goal state after:
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : Ω[S⁄R]
⊢ ∃ a, (D R S).tensorProductTo a = x
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
x : S
⊢ ↑(TensorProduct.lmul' R) (TensorProduct.includeLeft x) = x

Tactic:
;

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
x : S
⊢ ↑(TensorProduct.lmul' R) (TensorProduct.includeLeft x) = x
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
this : Function.RightInverse ⇑TensorProduct.includeLeft ⇑↑(TensorProduct.lmul' R)
x✝ : S ⊗[R] S
⊢ x✝ ∈ ideal R S ↔ x✝ ∈ RingHom.ker ↑(TensorProduct.lmul' R)

Tactic:
;

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
this : Function.RightInverse ⇑TensorProduct.includeLeft ⇑↑(TensorProduct.lmul' R)
x✝ : S ⊗[R] S
⊢ x✝ ∈ ideal R S ↔ x✝ ∈ RingHom.ker ↑(TensorProduct.lmul' R)
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
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
y : S ⊗[R] S ⧸ ideal R S ^ 2
⊢ (TensorProduct.lmul' R).kerSquareLift y =
    (quotientCotangentIdealRingEquiv R S) ((Ideal.Quotient.mk (ideal R S).cotangentIdeal) y)

Tactic:
;

Goal state after:
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
y : S ⊗[R] S ⧸ ideal R S ^ 2
⊢ (TensorProduct.lmul' R).kerSquareLift y =
    (quotientCotangentIdealRingEquiv R S) ((Ideal.Quotient.mk (ideal R S).cotangentIdeal) y)
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
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
y : S ⊗[R] S
⊢ (TensorProduct.lmul' R).kerSquareLift ((Ideal.Quotient.mk (ideal R S ^ 2)) y) =
    (quotientCotangentIdealRingEquiv R S)
      ((Ideal.Quotient.mk (ideal R S).cotangentIdeal) ((Ideal.Quotient.mk (ideal R S ^ 2)) y))

Tactic:
;

Goal state after:
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : S →ₐ[R] S ⊗[R] S ⧸ ideal R S ^ 2
x : S
y : S ⊗[R] S
⊢ (TensorProduct.lmul' R).kerSquareLift ((Ideal.Quotient.mk (ideal R S ^ 2)) y) =
    (quotientCotangentIdealRingEquiv R S)
      ((Ideal.Quotient.mk (ideal R S).cotangentIdeal) ((Ideal.Quotient.mk (ideal R S ^ 2)) y))
Module: Mathlib.Algebra.Homology.HomologicalComplexLimits

Goal state before:
pp: C : Type u_1
ι : Type u_2
J : Type u_3
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_3} J
c : ComplexShape ι
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroObject C
inst✝² : DecidableEq ι
i : ι
J✝ : Type
inst✝¹ : SmallCategory J✝
inst✝ : FinCategory J✝
⊢ PreservesLimitsOfShape J✝ (single C c i)

Tactic:
;

Goal state after:
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_3} J
c : ComplexShape ι
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroObject C
inst✝² : DecidableEq ι
i : ι
J✝ : Type
inst✝¹ : SmallCategory J✝
inst✝ : FinCategory J✝
⊢ PreservesLimitsOfShape J✝ (single C c i)
Module: Mathlib.Algebra.Homology.HomologicalComplexLimits

Goal state before:
pp: C : Type u_1
ι : Type u_2
J : Type u_3
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_3} J
c : ComplexShape ι
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroObject C
inst✝² : DecidableEq ι
i : ι
J✝ : Type
inst✝¹ : SmallCategory J✝
inst✝ : FinCategory J✝
⊢ PreservesColimitsOfShape J✝ (single C c i)

Tactic:
;

Goal state after:
C : Type u_1
ι : Type u_2
J : Type u_3
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_3} J
c : ComplexShape ι
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroObject C
inst✝² : DecidableEq ι
i : ι
J✝ : Type
inst✝¹ : SmallCategory J✝
inst✝ : FinCategory J✝
⊢ PreservesColimitsOfShape J✝ (single C c i)