mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCone

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
⊢ -1 + n = m

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + -1 = p₂
h₂₃ : p₂ + n = p₃
⊢ -1 + n = m

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + -1 = p₂
h₂₃ : p₂ + n = p₃
⊢ p₁ + -1 = p₂

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + -1 = p₂
h₂₃ : p₂ + n = p₃
⊢ p₂ + n = p₃

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + -1 = p₂
h₂₃ : p₂ + n = p₃
⊢ p₁ + m = p₃

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ : ℤ
h₁₂ : p₁ + n = p₂
⊢ p₁ + n = p₂

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
n' : ℤ
hn' : n + 1 = n'
⊢ 1 + n = n'

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
n' : ℤ
hn' : n + 1 = n'
⊢ 1 + 1 = 2

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
n' : ℤ
hn' : n + 1 = n'
⊢ m + 1 = n

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
α : Cochain F K (-1)
β : G ⟶ K
eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)
p q : ℤ
hpq : p + 1 = q
⊢ q + -1 = p

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
φ₁ φ₂ : F ⟶ G
z : { z // ofHom φ₁ = δ (-1) 0 z + ofHom φ₂ }
i j : ℤ
hij : j + 1 ≠ i
x✝ : i + -1 = j
⊢ j + 1 = i

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
h : n + 1 = m
⊢ m + -1 = n

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
h : n + 1 = m
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + 1 = p₃
⊢ p₁ + m = p₃

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
h : n + 1 = m
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + 1 = p₃
⊢ p₁ + m = p₃

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
h : n + 1 = m
p₁ p₂ : ℤ
h₁₂ : p₁ + n = p₂
⊢ p₁ + n = p₂

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
h : n + 1 = m
m' : ℤ
hm' : m + 1 = m'
⊢ m' + -1 = m

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
α : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
p q : ℤ
hpq : p + 0 = q
⊢ q = p

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
φ₁ φ₂ : F ⟶ G
ho : Homotopy φ₁ φ₂
i j : ℤ
h : ¬i + -1 = j
h' : j + 1 = i
⊢ i + -1 = j

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
⊢ m + m' = p

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
⊢ n + n' = p

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
⊢ -1 + n' = m'

Tactic:
lia

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

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

Tactic:
lia

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

Goal state before:
pp: case hnc
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ L.X q
n p' q' : ℤ
hpq' : p' + n = q'
hp' : p' ≠ p
h : p = p' ∧ q = q'
⊢ False

Tactic:
exact hp' (by lia)

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + n' = p₃
q : ℤ
hq : p₁ + m = q
⊢ p₁ + p = p₃

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + n' = p₃
q : ℤ
hq : p₁ + m = q
eq : ((liftCochain φ α β h).comp (descCochain φ α' β' h') hp).v p₁ p₃ ⋯ = (α.comp α' ⋯ + β.comp β' ⋯).v p₁ p₃ ⋯
⊢ q + m' = p₃

Tactic:
lia

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

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

Tactic:
lia

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

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

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v, u_1} C
inst✝⁵ : Category.{v', u_2} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝² : HasHomotopyCofiber φ
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)
n m : ℤ
hnm : n + 1 = m
⊢ n + 1 = m

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v, u_1} C
inst✝⁵ : Category.{v', u_2} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝² : HasHomotopyCofiber φ
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)
n m : ℤ
hnm : n + 1 = m
⊢ m + -1 = n

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n m : ℤ
hm : n + 1 = m
p' q' p'' q'' : ℤ
hpq'' : p'' + m = q''
hp' : p' + 1 = p''
hq' : q'' - 1 + 1 = q'
f : K.X p'' ⟶ L.X (q'' - 1)
⊢ q' = q''

Tactic:
lia

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v, u_1} C
inst✝⁵ : Category.{v', u_2} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝² : HasHomotopyCofiber φ
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)
n : ℤ
⊢ n + 1 + 1 = n + 2

Tactic:
lia

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

Goal state before:
pp: case neg.hq'
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
q n m : ℤ
hm : n + 1 = m
p' q' : ℤ
hq' : q + 1 = q'
p'' q'' : ℤ
hpq'' : p'' + m = q''
f : K.X p'' ⟶ L.X q
hp' : p' + 1 = p''
h : ¬q = q'' - 1
⊢ q'' ≠ q'

Tactic:
lia

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

Goal state before:
pp: case neg.hq'
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
q n m : ℤ
hm : n + 1 = m
p' q' : ℤ
hq' : q + 1 = q'
p'' q'' : ℤ
hpq'' : p'' + m = q''
f : K.X p'' ⟶ L.X q
hp' : p' + 1 = p''
h : ¬q = q'' - 1
⊢ q'' - 1 ≠ q

Tactic:
lia

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

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

Tactic:
lia

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

Goal state before:
pp: case neg.hp'
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
p n p' q' : ℤ
hp' : p' + 1 = p
p'' q'' : ℤ
hpq'' : p'' + (n + 1) = q''
f : K.X p ⟶ L.X q''
hq' : q'' + 1 = q'
h : ¬p'' = p'
⊢ p'' + 1 ≠ p

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
F G K : CochainComplex C ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
Φ : C ⥤ D
inst✝ : Φ.Additive
p q : ℤ
hpq : p + n₁₂ = q
⊢ p + n₁ + n₂ = q

Tactic:
lia

Goal state after:
Module: Mathlib.CategoryTheory.Quotient.Preadditive

Goal state before:
pp: C : Type ?u.2
inst✝² : Category.{v_1, ?u.2} C
inst✝¹ : Preadditive C
r : HomRel C
inst✝ : Congruence r
hr : ∀ ⦃X Y : C⦄ (f₁ f₂ g₁ g₂ : X ⟶ Y), r f₁ f₂ → r g₁ g₂ → r (f₁ + g₁) (f₂ + g₂)
X Y : Quotient r
f✝ : X ⟶ Y
f g : X.as ⟶ Y.as
⊢ HomRel.CompClosure r f g → Quot.mk (HomRel.CompClosure r) (-f) = Quot.mk (HomRel.CompClosure r) (-g)

Tactic:
intro hfg

Goal state after:
C : Type ?u.2
inst✝² : Category.{v_1, ?u.2} C
inst✝¹ : Preadditive C
r : HomRel C
inst✝ : Congruence r
hr : ∀ ⦃X Y : C⦄ (f₁ f₂ g₁ g₂ : X ⟶ Y), r f₁ f₂ → r g₁ g₂ → r (f₁ + g₁) (f₂ + g₂)
X Y : Quotient r
f✝ : X ⟶ Y
f g : X.as ⟶ Y.as
hfg : HomRel.CompClosure r f g
⊢ Quot.mk (HomRel.CompClosure r) (-f) = Quot.mk (HomRel.CompClosure r) (-g)
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: case a
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n : ℤ
K : CochainComplex C ℤ
i j : ℤ
hij : ¬(ComplexShape.up ℤ).Rel i j
hij' : i + n + 1 = j + n
⊢ i + 1 = j

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n : ℤ
h : n = 0
K : CochainComplex C ℤ
i : ℤ
⊢ i = i + n

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n₁ n₂ n₁₂ : ℤ
h : n₁ + n₂ = n₁₂
K : CochainComplex C ℤ
i : ℤ
⊢ i + n₂ + n₁ = i + n₁₂

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
⊢ shiftFunctorZero (CochainComplex C ℤ) ℤ = shiftFunctorZero' C 0 ⋯

Tactic:
ext
  rw [shiftFunctorZero_hom_app_f, shiftFunctorZero'_hom_app_f]

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

Goal state before:
pp: case a
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K L : CochainComplex C ℤ
φ₁ φ₂ : K ⟶ L
h : Homotopy φ₁ φ₂
n i j : ℤ
hij : ¬j + 1 = i
hij' : j + n + 1 = i + n
⊢ False

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n : ℤ
γ γ₁ γ₂ : Cochain K L n
a n' : ℤ
hn' : n' + a = n
p q : ℤ
hpq : p + n' = q
⊢ p + n = q + a

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K L : CochainComplex C ℤ
φ₁ φ₂ : K ⟶ L
h : Homotopy φ₁ φ₂
n i : ℤ
⊢ i + n + 1 = i + 1 + n

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
K L : CochainComplex C ℤ
φ₁ φ₂ : K ⟶ L
h : Homotopy φ₁ φ₂
n i : ℤ
⊢ i - 1 + n + 1 = i + n

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n : ℤ
γ γ₁ γ₂ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
p q : ℤ
hpq : p + n' = q
⊢ p + a + n = q

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
p q : ℤ
hpq : p + n' = q
p' : ℤ
hp' : p' + n = q
⊢ p' = p + a

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n' a : ℤ
γ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'
n : ℤ
hn : n' + a = n
p q : ℤ
hpq : p + n = q
⊢ q = p + n' + a

Tactic:
rw [← hpq, add_assoc, hn]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n' a : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn : n + a = n'
p q : ℤ
hpq : p + n = q
⊢ p = p - a + a

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n' a : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn : n + a = n'
p q : ℤ
hpq : p + n = q
⊢ p - a + n' = q

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n' a : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn : n + a = n'
p q : ℤ
hpq : p + n = q
p' : ℤ
hp' : p' + n' = q
⊢ p = p' + a

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n' a : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn : n + a = n'
p q : ℤ
hpq : p + n = q
p' : ℤ
hp' : p' + n' = q
⊢ p' + n' = q

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n' a : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn : n + a = n'
p q : ℤ
hpq : p + n = q
p' : ℤ
hp' : p' + n' = q
⊢ p' = p - a

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n : ℤ
γ γ₁ γ₂ : Cochain K L n
a p q : ℤ
hpq : p + n = q
⊢ p + a + n = q + a

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
p q : ℤ
hpq : p + n = q
⊢ q - n' + n' = q

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
a n' : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn' : n + a = n'
p q : ℤ
hpq : p + n' = q
⊢ q - n + n = q

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ₁ γ₂ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
p q : ℤ
hpq : p + n' = q
⊢ p + a + n = q

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
x : R
p q : ℤ
hpq : p + n' = q
⊢ p + a + n = q

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n : ℤ
γ γ₁ γ₂ : Cochain K L n
m a : ℤ
γ' : Cochain L ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj M) m
nm : ℤ
hnm : n + m = nm
nm' : ℤ
hnm' : nm + a = nm'
m' : ℤ
hm' : m + a = m'
⊢ n + m' = nm'

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L M : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
m a : ℤ
γ' : Cochain L ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj M) m
nm : ℤ
hnm : n + m = nm
nm' : ℤ
hnm' : nm + a = nm'
m' : ℤ
hm' : m + a = m'
p q : ℤ
hpq : p + nm' = q
⊢ p + nm = p + n + m

Tactic:
lia

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L M : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
m a : ℤ
γ' : Cochain L ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj M) m
nm : ℤ
hnm : n + m = nm
nm' : ℤ
hnm' : nm + a = nm'
m' : ℤ
hm' : m + a = m'
p q : ℤ
hpq : p + nm' = q
⊢ n + m' = nm'

Tactic:
lia

Goal state after: