mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
this : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + 1 = q₂ → IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)
k : ℕ
h : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + k = q₂ → IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)
q₁ q₂ : ℕ
hq : q₁ ≤ q₂
hi : i ≤ n₀ + ↑q₁
hk : q₁ + (k + 1) = q₂
⊢ q₁ + k ≤ q₂

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
this : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + 1 = q₂ → IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)
k : ℕ
h : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + k = q₂ → IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)
q₁ q₂ : ℕ
hq : q₁ ≤ q₂
hi : i ≤ n₀ + ↑q₁
hk : q₁ + (k + 1) = q₂
⊢ i ≤ n₀ + ↑(q₁ + k)

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
this : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + 1 = q₂ → IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)
k : ℕ
h : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + k = q₂ → IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)
q₁ q₂ : ℕ
hq : q₁ ≤ q₂
hi : i ≤ n₀ + ↑q₁
hk : q₁ + (k + 1) = q₂
⊢ q₁ + k + 1 = q₂

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
this : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + 1 = q₂ → IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)
k : ℕ
h : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + k = q₂ → IsIso (((functor f n₀).map (homOfLE hq).op).hom.h.f i)
q₁ q₂ : ℕ
hq : q₁ ≤ q₂
hi : i ≤ n₀ + ↑q₁
hk : q₁ + (k + 1) = q₂
⊢ i ≤ n₀ + ↑q₁

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
q : ℕ
h : i ≤ n₀ + ↑q
x✝¹ : ℕᵒᵖ
x✝ : x✝¹ ⟶ op q
⊢ i ≤ n₀ + ↑q

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
q : ℕ
⊢ 0 ≤ q

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
⊢ i + 1 ≤ n₀ + ↑(i + 1 - n₀).natAbs

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
q : ℕ
hq : i + 1 ≤ n₀ + ↑q
this : QuasiIsoAt (midπ f n₀ q) i
⊢ i ≤ n₀ + ↑q

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
i : ℤ
⊢ i ≤ n₀ + ↑(i - n₀).natAbs

Tactic:
lia

Goal state after:
Module: Mathlib.CategoryTheory.Localization.StructuredArrow

Goal state before:
pp: C : Type u_1
D : Type u_2
D' : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D'
W : MorphismProperty C
L : C ⥤ D
L' : C ⥤ D'
inst✝¹ : L.IsLocalization W
inst✝ : L'.IsLocalization W
X Y : C
f : L.obj X ⟶ L.obj Y
⊢ StructuredArrow.mk ((homEquiv W L L').symm ((homEquiv W L L') f)) = StructuredArrow.mk f

Tactic:
Equiv.symm_apply_apply

Goal state after:
C : Type u_1
D : Type u_2
D' : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D'
W : MorphismProperty C
L : C ⥤ D
L' : C ⥤ D'
inst✝¹ : L.IsLocalization W
inst✝ : L'.IsLocalization W
X Y : C
f : L.obj X ⟶ L.obj Y
⊢ StructuredArrow.mk f = StructuredArrow.mk f
Module: Mathlib.CategoryTheory.Localization.StructuredArrow

Goal state before:
pp: C : Type u_1
D : Type u_2
D' : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D'
W : MorphismProperty C
L : C ⥤ D
L' : C ⥤ D'
inst✝¹ : L.IsLocalization W
inst✝ : L'.IsLocalization W
X Y : C
f : L'.obj X ⟶ L'.obj Y
⊢ StructuredArrow.mk ((homEquiv W L' L).symm ((homEquiv W L' L) f)) = StructuredArrow.mk f

Tactic:
Equiv.symm_apply_apply

Goal state after:
C : Type u_1
D : Type u_2
D' : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D'
W : MorphismProperty C
L : C ⥤ D
L' : C ⥤ D'
inst✝¹ : L.IsLocalization W
inst✝ : L'.IsLocalization W
X Y : C
f : L'.obj X ⟶ L'.obj Y
⊢ StructuredArrow.mk f = StructuredArrow.mk f
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝² : EnoughInjectives C
n : ℤ
inst✝¹ : K.IsStrictlyGE (n + 1)
inst✝ : L.IsStrictlyGE n
⊢ n ≤ n + 1

Tactic:
lia

Goal state after:
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.OfLocalizedEquivalences

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D₁
inst✝⁸ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₁' : MorphismProperty D₁
W₂ : MorphismProperty C₂
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁷ : W₂'.RespectsIso
inst✝⁶ : L.IsLocalizedEquivalence
inst✝⁵ : R.IsLocalizedEquivalence
inst✝⁴ : R.functor.EssSurj
inst✝³ : L.functor.EssSurj
inst✝² : R.functor.Full
inst✝¹ : R.IsInduced
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : TwoSquare.GuitartExact iso.inv
x✝ : B.IsLeftDerivabilityStructure
this✝ : T.HasLeftResolutions
F : W₁'.Localization ⥤ W₂'.Localization := B.localizedFunctor W₁'.Q W₂'.Q
e' : B.functor ⋙ W₂'.Q ≅ W₁'.Q ⋙ F := CatCommSq.iso B.functor W₁'.Q W₂'.Q F
e : T.functor ⋙ R.functor ⋙ W₂'.Q ≅ (L.functor ⋙ W₁'.Q) ⋙ F :=
  (T.functor.associator R.functor W₂'.Q).symm ≪≫
    Functor.isoWhiskerRight iso W₂'.Q ≪≫
      L.functor.associator B.functor W₂'.Q ≪≫ L.functor.isoWhiskerLeft e' ≪≫ (L.functor.associator W₁'.Q F).symm
he' : TwoSquare.GuitartExact e'.inv
this : TwoSquare.hComp iso.inv e'.inv = e.inv
⊢ T.IsLeftDerivabilityStructure

Tactic:
rw [T.isLeftDerivabilityStructure_iff (L.functor ⋙ W₁'.Q)
    (R.functor ⋙ W₂'.Q) F e, ← this]

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D₁
inst✝⁸ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₁' : MorphismProperty D₁
W₂ : MorphismProperty C₂
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁷ : W₂'.RespectsIso
inst✝⁶ : L.IsLocalizedEquivalence
inst✝⁵ : R.IsLocalizedEquivalence
inst✝⁴ : R.functor.EssSurj
inst✝³ : L.functor.EssSurj
inst✝² : R.functor.Full
inst✝¹ : R.IsInduced
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : TwoSquare.GuitartExact iso.inv
x✝ : B.IsLeftDerivabilityStructure
this✝ : T.HasLeftResolutions
F : W₁'.Localization ⥤ W₂'.Localization := B.localizedFunctor W₁'.Q W₂'.Q
e' : B.functor ⋙ W₂'.Q ≅ W₁'.Q ⋙ F := CatCommSq.iso B.functor W₁'.Q W₂'.Q F
e : T.functor ⋙ R.functor ⋙ W₂'.Q ≅ (L.functor ⋙ W₁'.Q) ⋙ F :=
  (T.functor.associator R.functor W₂'.Q).symm ≪≫
    Functor.isoWhiskerRight iso W₂'.Q ≪≫
      L.functor.associator B.functor W₂'.Q ≪≫ L.functor.isoWhiskerLeft e' ≪≫ (L.functor.associator W₁'.Q F).symm
he' : TwoSquare.GuitartExact e'.inv
this : TwoSquare.hComp iso.inv e'.inv = e.inv
⊢ (TwoSquare.hComp iso.inv e'.inv).GuitartExact
Module: Mathlib.Algebra.Homology.HomotopyCategory.KProjective

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K : CochainComplex C ℤ
d : ℤ
inst✝¹ : K.IsStrictlyLE d
inst✝ : ∀ (n : ℤ), Projective (K.X n)
L : CochainComplex Cᵒᵖ ℤ := (opEquivalence C).functor.obj (op K)
this : ∀ (n : ℤ), Injective (L.X n)
i : ℤ
hi : autoParam (i < -d) isStrictlyGE_iff._auto_1
⊢ d < -i

Tactic:
lia

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

Goal state before:
pp: β : Type u_1
inst✝² : AddCommGroup β
b : β
V : Type u_2
inst✝¹ : Category.{v_1, u_2} V
inst✝ : HasZeroMorphisms V
X Y : DifferentialObject ℤ (GradedObjectWithShift b V)
f : X ⟶ Y
i : β
⊢ (f.f i ≫ if h : i + b = i + b then Y.d i ≫ Y.objEqToHom ⋯ else 0) =
    (if h : i + b = i + b then X.d i ≫ X.objEqToHom ⋯ else 0) ≫ f.f (i + b)

Tactic:
have : f.f i ≫ Y.d i = X.d i ≫ f.f _ := (congr_fun f.comm i).symm

Goal state after:
β : Type u_1
inst✝² : AddCommGroup β
b : β
V : Type u_2
inst✝¹ : Category.{v_1, u_2} V
inst✝ : HasZeroMorphisms V
X Y : DifferentialObject ℤ (GradedObjectWithShift b V)
f : X ⟶ Y
i : β
this : f.f i ≫ Y.d i = X.d i ≫ f.f ((fun b_1 ↦ b_1 + { as := 1 }.as • b) i)
⊢ (f.f i ≫ if h : i + b = i + b then Y.d i ≫ Y.objEqToHom ⋯ else 0) =
    (if h : i + b = i + b then X.d i ≫ X.objEqToHom ⋯ else 0) ≫ f.f (i + b)
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
n✝ m✝ : ℤ
n m : ℕ
hnm : ¬↑n + 1 = ↑m
⊢ ¬n + 1 = m

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
n✝ m✝ : ℤ
n m : ℕ
hnm : ¬Int.negSucc n + 1 = Int.negSucc m
⊢ ¬m + 1 = n

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
p : ℤ
n : ℕ
hmp : Int.ofNat (n + 1) + 1 = p
hnm : Int.ofNat n + 1 = Int.ofNat (n + 1)
⊢ ↑n + 1 + 1 = ↑n + 2

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
p : ℤ
n : ℕ
hmp : Int.ofNat (n + 1) + 1 = p
hnm : Int.ofNat n + 1 = Int.ofNat (n + 1)
⊢ p = Int.ofNat (n + 2)

Tactic:
simp [← hmp]; lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
hnm : Int.negSucc 0 + 1 = m
⊢ m = 0

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
p : ℤ
hmp : 0 + 1 = p
hnm : Int.negSucc 0 + 1 = 0
⊢ p = 1

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
hnm : Int.negSucc (0 + 1) + 1 = m
⊢ m = -1

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
p : ℤ
hmp : -1 + 1 = p
hnm : Int.negSucc (0 + 1) + 1 = -1
⊢ p = 0

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
m p : ℤ
hmp : m + 1 = p
n : ℕ
hnm : Int.negSucc (n + 1 + 1) + 1 = m
⊢ m = Int.negSucc (n + 1)

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
p : ℤ
n : ℕ
hmp : Int.negSucc (n + 1) + 1 = p
hnm : Int.negSucc (n + 1 + 1) + 1 = Int.negSucc (n + 1)
⊢ p = Int.negSucc n

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
n : ℕ
⊢ -1 - ↑n = Int.negSucc n

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
n : ℕ
⊢ -1 - (↑n + 1) = Int.negSucc (n + 1)

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: case zero
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
m : ℤ
inst✝² : HasHomology h.cochainComplex m
inst✝¹ : NeZero 0
hm : m = ↑0
inst✝ : HasHomology L 0
this✝ : (restriction h.cochainComplex (ComplexShape.embeddingUpIntGE 0)).HasHomology 0
this : 0 ≠ 0
⊢ 0 = m - 1

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: case succ
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
m : ℤ
inst✝² : HasHomology h.cochainComplex m
n✝ : ℕ
inst✝¹ : NeZero (n✝ + 1)
hm : m = ↑(n✝ + 1)
inst✝ : HasHomology L (n✝ + 1)
this✝ : (restriction h.cochainComplex (ComplexShape.embeddingUpIntGE 0)).HasHomology (n✝ + 1)
this : n✝ + 1 ≠ 0
⊢ ↑n✝ = m - 1

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
n : ℕ
inst✝² : NeZero n
m : ℤ
hm : m = ↑n
inst✝¹ : HasHomology h.cochainComplex m
inst✝ : HasHomology L n
this : (restriction h.cochainComplex (ComplexShape.embeddingUpIntGE 0)).HasHomology n
⊢ ↑n = m

Tactic:
lia

Goal state after:
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X : DifferentialObject S C
⊢ autoParam (((shiftFunctorAdd C n 0).hom.app X).f = (eqToHom ⋯ ≫ (shiftZero C).inv.app ((shiftFunctor C n).obj X)).f)
    ext._auto_1

Tactic:
convert! shiftFunctorAdd_add_zero_hom_app n X.obj

Goal state after:
case e'_3
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X : DifferentialObject S C
e_1✝ :
  (((shiftFunctor C (n + 0)).obj X).obj ⟶ ((shiftFunctor C n ⋙ shiftFunctor C 0).obj X).obj) =
    ((CategoryTheory.shiftFunctor C (n + 0)).obj X.obj ⟶
      (CategoryTheory.shiftFunctor C n ⋙ CategoryTheory.shiftFunctor C 0).obj X.obj)
⊢ (eqToHom ⋯ ≫ (shiftZero C).inv.app ((shiftFunctor C n).obj X)).f =
    eqToHom ⋯ ≫ (shiftFunctorZero C S).inv.app ((CategoryTheory.shiftFunctor C n).obj X.obj)
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
n : ℕ
inst✝² : NeZero n
m : ℤ
hm : m = -↑(n + 1)
inst✝¹ : HasHomology h.cochainComplex m
inst✝ : HasHomology K n
this : (restriction h.cochainComplex (ComplexShape.embeddingUpIntLE (-1))).HasHomology n
⊢ -1 - (↑n + 1) = m - 1

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
n : ℕ
inst✝² : NeZero n
m : ℤ
hm : m = -↑(n + 1)
inst✝¹ : HasHomology h.cochainComplex m
inst✝ : HasHomology K n
this : (restriction h.cochainComplex (ComplexShape.embeddingUpIntLE (-1))).HasHomology n
⊢ -1 - ↑n = m

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: case zero
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
m : ℤ
inst✝² : HasHomology h.cochainComplex m
inst✝¹ : NeZero 0
hm : m = -↑(0 + 1)
inst✝ : HasHomology K 0
this✝ : (restriction h.cochainComplex (ComplexShape.embeddingUpIntLE (-1))).HasHomology 0
this : 0 ≠ 0
⊢ -1 = m + 1

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: case succ
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroMorphisms C
K K' K'' : ChainComplex C ℕ
L L' L'' : CochainComplex C ℕ
h : ConnectData K L
h' : ConnectData K' L'
h'' : ConnectData K'' L''
m : ℤ
inst✝² : HasHomology h.cochainComplex m
n✝ : ℕ
inst✝¹ : NeZero (n✝ + 1)
hm : m = -↑(n✝ + 1 + 1)
inst✝ : HasHomology K (n✝ + 1)
this✝ : (restriction h.cochainComplex (ComplexShape.embeddingUpIntLE (-1))).HasHomology (n✝ + 1)
this : n✝ + 1 ≠ 0
⊢ -1 - ↑n✝ = m + 1

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
ι : Type t
c : ComplexShape ι
inst✝² : HasZeroMorphisms C
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : Small.{w, t} ι
K L : HomologicalComplex C c
emb : (K ⟶ L) → (i : Shrink.{w, t} ι) → Shrink.{w, v} (K.X ((equivShrink ι).symm i) ⟶ L.X ((equivShrink ι).symm i)) :=
  fun f i ↦ (equivShrink (K.X ((equivShrink ι).symm i) ⟶ L.X ((equivShrink ι).symm i))) (f.f ((equivShrink ι).symm i))
f g : K ⟶ L
h : emb f = emb g
i : Shrink.{w, t} ι
⊢ f.f ((equivShrink ι).symm i) = g.f ((equivShrink ι).symm i)

Tactic:
simpa [emb] using congr_fun h i

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
n' q' : ℤ
h' : p + n' = q'
hq : q + 1 = q'
⊢ n + 1 = n'

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
n' q' : ℤ
h' : p + n' = q'
hq : q + 1 = q'
⊢ p - 1 + 1 = p

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
n' q' : ℤ
h' : p + n' = q'
hq : ¬q + 1 = q'
⊢ ¬n + 1 = n'

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
n' q' : ℤ
h' : p + n' = q'
hq : ¬q + 1 = q'
⊢ ¬q + 1 = q'

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ X
n : ℤ
h : p + n = q
n' p' : ℤ
h' : p' + n' = q
hp : p' + 1 = p
⊢ n + 1 = n'

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ X
n : ℤ
h : p + n = q
n' p' : ℤ
h' : p' + n' = q
hp : p' + 1 = p
⊢ p' + 1 = p

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ X
n : ℤ
h : p + n = q
n' p' : ℤ
h' : p' + n' = q
hp : ¬p' + 1 = p
⊢ ¬n + 1 = n'

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ X
n : ℤ
h : p + n = q
n' p' : ℤ
h' : p' + n' = q
hp : ¬p' + 1 = p
⊢ ¬p' + 1 = p

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
hf : f ≫ K.d q q' = 0
⊢ p + (n + 1) = q'

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p n : ℤ
α : Cocycle ((singleFunctor C p).obj X) K n
q : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
f : X ⟶ K.X q
hf : Cochain.fromSingleMk f h = ↑α
hα : δ n (n + 1) (Cochain.fromSingleMk f h) = 0
⊢ p + (n + 1) = q'

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p n : ℤ
α : Cocycle ((singleFunctor C p).obj X) K n
q : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
f : X ⟶ K.X q
hf : Cochain.fromSingleMk f h = ↑α
hα : Cochain.fromSingleMk (f ≫ K.d q q') ⋯ = 0
⊢ p + (n + 1) = q'

Tactic:
lia

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
hf : f ≫ K.d q q' = 0
q'' : ℤ
hq'' : q'' + 1 = q
α : Cochain ((singleFunctor C p).obj X) K (n - 1)
hα : δ (n - 1) n α = ↑(fromSingleMk f h q' hq' hf)
⊢ p + (n - 1) = q''

Tactic:
lia

Goal state after: