l3lab/ntp-mathlib huggingface dataset:
theorem snd_map_prod_mk₀ {X : α → β} {Y : α → γ} {μ : Measure α} (hX : AEMeasurable X μ) :
    (μ.map fun a => (X a, Y a)).snd = μ.map Y := by
  by_cases hY : AEMeasurable Y μ
  · ext1 s hs
    rw [Measure.snd_apply hs, Measure.map_apply_of_aemeasurable (hX.prod_mk hY) (measurable_snd hs),
      Measure.map_apply_of_aemeasurable hY hs, ← univ_prod, mk_preimage_prod, preimage_univ,
      univ_inter]
  · have : ¬AEMeasurable (fun x ↦ (X x, Y x)) μ := by
      contrapose! hY; exact measurable_snd.comp_aemeasurable hY
    

Proof state:
case neg
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ✝ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
ρ : Measure (α × β)
X : α → β
Y : α → γ
μ : Measure α
hX : AEMeasurable X
hY : ¬AEMeasurable Y
this : ¬AEMeasurable fun x => (X x, Y x)
⊢ snd (map (fun a => (X a, Y a)) μ) = map Y μ

Next tactic:
simp [map_of_not_aemeasurable, hY, this]
@[simp]
theorem dotProduct_stdBasis_eq_mul [DecidableEq n] (v : n → R) (c : R) (i : n) :
    dotProduct v (LinearMap.stdBasis R (fun _ => R) i c) = v i * c := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝² : Semiring R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → R
c : R
i : n
⊢ v ⬝ᵥ (LinearMap.stdBasis R (fun x => R) i) c = v i * c

Next tactic:
rw [dotProduct, Finset.sum_eq_single i, LinearMap.stdBasis_same]
@[simp]
theorem dotProduct_stdBasis_eq_mul [DecidableEq n] (v : n → R) (c : R) (i : n) :
    dotProduct v (LinearMap.stdBasis R (fun _ => R) i c) = v i * c := by
  rw [dotProduct, Finset.sum_eq_single i, LinearMap.stdBasis_same]
  

Proof state:
case h₀
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝² : Semiring R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → R
c : R
i : n
⊢ ∀ b ∈ Finset.univ, b ≠ i → v b * (LinearMap.stdBasis R (fun x => R) i) c b = 0
case h₁
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝² : Semiring R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → R
c : R
i : n
⊢ i ∉ Finset.univ → v i * (LinearMap.stdBasis R (fun x => R) i) c i = 0

Next tactic:
exact fun _ _ hb => by rw [LinearMap.stdBasis_ne _ _ _ _ hb, mul_zero]
@[simp]
theorem dotProduct_stdBasis_eq_mul [DecidableEq n] (v : n → R) (c : R) (i : n) :
    dotProduct v (LinearMap.stdBasis R (fun _ => R) i c) = v i * c := by
  rw [dotProduct, Finset.sum_eq_single i, LinearMap.stdBasis_same]
  exact fun _ _ hb => by 

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝² : Semiring R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → R
c : R
i x✝¹ : n
x✝ : x✝¹ ∈ Finset.univ
hb : x✝¹ ≠ i
⊢ v x✝¹ * (LinearMap.stdBasis R (fun x => R) i) c x✝¹ = 0

Next tactic:
rw [LinearMap.stdBasis_ne _ _ _ _ hb, mul_zero]
@[simp]
theorem dotProduct_stdBasis_eq_mul [DecidableEq n] (v : n → R) (c : R) (i : n) :
    dotProduct v (LinearMap.stdBasis R (fun _ => R) i c) = v i * c := by
  rw [dotProduct, Finset.sum_eq_single i, LinearMap.stdBasis_same]
  exact fun _ _ hb => by rw [LinearMap.stdBasis_ne _ _ _ _ hb, mul_zero]
  

Proof state:
case h₁
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝² : Semiring R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → R
c : R
i : n
⊢ i ∉ Finset.univ → v i * (LinearMap.stdBasis R (fun x => R) i) c i = 0

Next tactic:
exact fun hi => False.elim (hi <| Finset.mem_univ _)
theorem dotProduct_stdBasis_one [DecidableEq n] (v : n → R) (i : n) :
    dotProduct v (LinearMap.stdBasis R (fun _ => R) i 1) = v i := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝² : Semiring R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
v : n → R
i : n
⊢ v ⬝ᵥ (LinearMap.stdBasis R (fun x => R) i) 1 = v i

Next tactic:
rw [dotProduct_stdBasis_eq_mul, mul_one]
theorem dotProduct_eq (v w : n → R) (h : ∀ u, dotProduct v u = dotProduct w u) : v = w := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝¹ : Semiring R
inst✝ : Fintype n
v w : n → R
h : ∀ (u : n → R), v ⬝ᵥ u = w ⬝ᵥ u
⊢ v = w

Next tactic:
funext x
theorem dotProduct_eq (v w : n → R) (h : ∀ u, dotProduct v u = dotProduct w u) : v = w := by
  funext x
  

Proof state:
case h
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝¹ : Semiring R
inst✝ : Fintype n
v w : n → R
h : ∀ (u : n → R), v ⬝ᵥ u = w ⬝ᵥ u
x : n
⊢ v x = w x

Next tactic:
classical rw [← dotProduct_stdBasis_one v x, ← dotProduct_stdBasis_one w x, h]
theorem dotProduct_eq (v w : n → R) (h : ∀ u, dotProduct v u = dotProduct w u) : v = w := by
  funext x
  classical 

Proof state:
case h
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝¹ : Semiring R
inst✝ : Fintype n
v w : n → R
h : ∀ (u : n → R), v ⬝ᵥ u = w ⬝ᵥ u
x : n
⊢ v x = w x

Next tactic:
rw [← dotProduct_stdBasis_one v x, ← dotProduct_stdBasis_one w x, h]
@[simp]
theorem dotProduct_self_eq_zero [LinearOrderedRing R] {v : n → R} : dotProduct v v = 0 ↔ v = 0 :=
  (Finset.sum_eq_zero_iff_of_nonneg fun i _ => mul_self_nonneg (v i)).trans <| by
    

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : Fintype p
inst✝ : LinearOrderedRing R
v : n → R
⊢ (∀ i ∈ Finset.univ, v i * v i = 0) ↔ v = 0

Next tactic:
simp [Function.funext_iff]
/-- Note that this applies to `ℂ` via `Complex.strictOrderedCommRing`. -/
@[simp]
theorem dotProduct_star_self_eq_zero {v : n → R} : dotProduct (star v) v = 0 ↔ v = 0 :=
  (Finset.sum_eq_zero_iff_of_nonneg fun i _ => (@star_mul_self_nonneg _ _ _ _ (v i) : _)).trans <|
    by 

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
v : n → R
⊢ (∀ i ∈ Finset.univ, star (v i) * v i = 0) ↔ v = 0

Next tactic:
simp [Function.funext_iff, mul_eq_zero]
/-- Note that this applies to `ℂ` via `Complex.strictOrderedCommRing`. -/
@[simp]
theorem dotProduct_self_star_eq_zero {v : n → R} : dotProduct v (star v) = 0 ↔ v = 0 :=
  (Finset.sum_eq_zero_iff_of_nonneg fun i _ => (@mul_star_self_nonneg _ _ _ _ (v i) : _)).trans <|
    by 

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
v : n → R
⊢ (∀ i ∈ Finset.univ, v i * star (v i) = 0) ↔ v = 0

Next tactic:
simp [Function.funext_iff, mul_eq_zero]
lemma conjTranspose_mul_self_mul_eq_zero (A : Matrix m n R) (B : Matrix n p R) :
    (Aᴴ * A) * B = 0 ↔ A * B = 0 := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
B : Matrix n p R
⊢ Aᴴ * A * B = 0 ↔ A * B = 0

Next tactic:
refine ⟨fun h => ?_, fun h => by simp only [Matrix.mul_assoc, h, Matrix.mul_zero]⟩
lemma conjTranspose_mul_self_mul_eq_zero (A : Matrix m n R) (B : Matrix n p R) :
    (Aᴴ * A) * B = 0 ↔ A * B = 0 := by
  refine ⟨fun h => ?_, fun h => by 

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
B : Matrix n p R
h : A * B = 0
⊢ Aᴴ * A * B = 0

Next tactic:
simp only [Matrix.mul_assoc, h, Matrix.mul_zero]
lemma conjTranspose_mul_self_mul_eq_zero (A : Matrix m n R) (B : Matrix n p R) :
    (Aᴴ * A) * B = 0 ↔ A * B = 0 := by
  refine ⟨fun h => ?_, fun h => by simp only [Matrix.mul_assoc, h, Matrix.mul_zero]⟩
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
B : Matrix n p R
h : Aᴴ * A * B = 0
⊢ A * B = 0

Next tactic:
apply_fun (Bᴴ * ·) at h
lemma conjTranspose_mul_self_mul_eq_zero (A : Matrix m n R) (B : Matrix n p R) :
    (Aᴴ * A) * B = 0 ↔ A * B = 0 := by
  refine ⟨fun h => ?_, fun h => by simp only [Matrix.mul_assoc, h, Matrix.mul_zero]⟩
  apply_fun (Bᴴ * ·) at h
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
B : Matrix n p R
h : Bᴴ * (Aᴴ * A * B) = Bᴴ * 0
⊢ A * B = 0

Next tactic:
rwa [Matrix.mul_zero, Matrix.mul_assoc, ← Matrix.mul_assoc, ← conjTranspose_mul,
    conjTranspose_mul_self_eq_zero] at h
lemma self_mul_conjTranspose_mul_eq_zero (A : Matrix m n R) (B : Matrix m p R) :
    (A * Aᴴ) * B = 0 ↔ Aᴴ * B = 0 := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
B : Matrix m p R
⊢ A * Aᴴ * B = 0 ↔ Aᴴ * B = 0

Next tactic:
simpa only [conjTranspose_conjTranspose] using conjTranspose_mul_self_mul_eq_zero Aᴴ _
lemma mul_self_mul_conjTranspose_eq_zero (A : Matrix m n R) (B : Matrix p m R) :
    B * (A * Aᴴ) = 0 ↔ B * A = 0 := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
B : Matrix p m R
⊢ B * (A * Aᴴ) = 0 ↔ B * A = 0

Next tactic:
rw [← conjTranspose_eq_zero, conjTranspose_mul, conjTranspose_mul, conjTranspose_conjTranspose,
    self_mul_conjTranspose_mul_eq_zero, ← conjTranspose_mul, conjTranspose_eq_zero]
lemma mul_conjTranspose_mul_self_eq_zero (A : Matrix m n R) (B : Matrix p n R) :
    B * (Aᴴ * A) = 0 ↔ B * Aᴴ = 0 := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
B : Matrix p n R
⊢ B * (Aᴴ * A) = 0 ↔ B * Aᴴ = 0

Next tactic:
simpa only [conjTranspose_conjTranspose] using mul_self_mul_conjTranspose_eq_zero Aᴴ _
lemma conjTranspose_mul_self_mulVec_eq_zero (A : Matrix m n R) (v : n → R) :
    (Aᴴ * A).mulVec v = 0 ↔ A.mulVec v = 0 := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
v : n → R
⊢ mulVec (Aᴴ * A) v = 0 ↔ mulVec A v = 0

Next tactic:
simpa only [← Matrix.col_mulVec, col_eq_zero] using
    conjTranspose_mul_self_mul_eq_zero A (col v)
lemma self_mul_conjTranspose_mulVec_eq_zero (A : Matrix m n R) (v : m → R) :
    (A * Aᴴ).mulVec v = 0 ↔ Aᴴ.mulVec v = 0 := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
v : m → R
⊢ mulVec (A * Aᴴ) v = 0 ↔ mulVec Aᴴ v = 0

Next tactic:
simpa only [conjTranspose_conjTranspose] using conjTranspose_mul_self_mulVec_eq_zero Aᴴ _
lemma vecMul_conjTranspose_mul_self_eq_zero (A : Matrix m n R) (v : n → R) :
    vecMul v (Aᴴ * A) = 0 ↔ vecMul v Aᴴ = 0 := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
v : n → R
⊢ vecMul v (Aᴴ * A) = 0 ↔ vecMul v Aᴴ = 0

Next tactic:
simpa only [← Matrix.row_vecMul, row_eq_zero] using
    mul_conjTranspose_mul_self_eq_zero A (row v)
lemma vecMul_self_mul_conjTranspose_eq_zero (A : Matrix m n R) (v : m → R) :
    vecMul v (A * Aᴴ) = 0 ↔ vecMul v A = 0 := by
  

Proof state:
m : Type u_1
n : Type u_2
p : Type u_3
R : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : Fintype p
inst✝³ : PartialOrder R
inst✝² : NonUnitalRing R
inst✝¹ : StarOrderedRing R
inst✝ : NoZeroDivisors R
A : Matrix m n R
v : m → R
⊢ vecMul v (A * Aᴴ) = 0 ↔ vecMul v A = 0

Next tactic:
simpa only [conjTranspose_conjTranspose] using vecMul_conjTranspose_mul_self_eq_zero Aᴴ _
instance isScalarTower'' [SMul R M] [SMul M N] [SMul R N] [IsScalarTower R M N] :
    IsScalarTower R M (ULift N) :=
  ⟨fun x y z => show up ((x • y) • z.down) = ⟨x • y • z.down⟩ by 

Proof state:
R : Type u
M : Type v
N : Type w
inst✝³ : SMul R M
inst✝² : SMul M N
inst✝¹ : SMul R N
inst✝ : IsScalarTower R M N
x : R
y : M
z : ULift.{?u.2467, w} N
⊢ { down := (x • y) • z.down } = { down := x • y • z.down }

Next tactic:
rw [smul_assoc]
instance smulZeroClass' [Zero M] [SMulZeroClass R M] : SMulZeroClass R (ULift M) where
  smul_zero c := by 

Proof state:
R : Type u
M : Type v
N : Type w
inst✝¹ : Zero M
inst✝ : SMulZeroClass R M
c : R
⊢ c • 0 = 0

Next tactic:
{ ext; simp [smul_zero] }
instance smulZeroClass' [Zero M] [SMulZeroClass R M] : SMulZeroClass R (ULift M) where
  smul_zero c := by { 

Proof state:
R : Type u
M : Type v
N : Type w
inst✝¹ : Zero M
inst✝ : SMulZeroClass R M
c : R
⊢ c • 0 = 0

Next tactic:
ext
instance smulZeroClass' [Zero M] [SMulZeroClass R M] : SMulZeroClass R (ULift M) where
  smul_zero c := by { ext; 

Proof state:
case h
R : Type u
M : Type v
N : Type w
inst✝¹ : Zero M
inst✝ : SMulZeroClass R M
c : R
⊢ (c • 0).down = 0.down

Next tactic:
simp [smul_zero]
instance distribSMul' [AddZeroClass M] [DistribSMul R M] : DistribSMul R (ULift M) where
  smul_add c f g := by
    

Proof state:
R : Type u
M : Type v
N : Type w
inst✝¹ : AddZeroClass M
inst✝ : DistribSMul R M
c : R
f g : ULift.{?u.15250, v} M
⊢ c • (f + g) = c • f + c • g

Next tactic:
ext
instance distribSMul' [AddZeroClass M] [DistribSMul R M] : DistribSMul R (ULift M) where
  smul_add c f g := by
    ext
    

Proof state:
case h
R : Type u
M : Type v
N : Type w
inst✝¹ : AddZeroClass M
inst✝ : DistribSMul R M
c : R
f g : ULift.{?u.15250, v} M
⊢ (c • (f + g)).down = (c • f + c • g).down

Next tactic:
simp [smul_add]
instance mulDistribMulAction' [Monoid R] [Monoid M] [MulDistribMulAction R M] :
    MulDistribMulAction R (ULift M) :=
  { ULift.mulAction' with
    smul_one := fun _ => by
      ext
      simp [smul_one]
    smul_mul := fun _ _ _ => by
      

Proof state:
R : Type u
M : Type v
N : Type w
inst✝² : Monoid R
inst✝¹ : Monoid M
inst✝ : MulDistribMulAction R M
src✝ : MulAction R (ULift.{?u.24441, v} M) := mulAction'
x✝² : R
x✝¹ x✝ : ULift.{?u.24441, v} M
⊢ x✝² • (x✝¹ * x✝) = x✝² • x✝¹ * x✝² • x✝

Next tactic:
ext
instance mulDistribMulAction' [Monoid R] [Monoid M] [MulDistribMulAction R M] :
    MulDistribMulAction R (ULift M) :=
  { ULift.mulAction' with
    smul_one := fun _ => by
      ext
      simp [smul_one]
    smul_mul := fun _ _ _ => by
      ext
      

Proof state:
case h
R : Type u
M : Type v
N : Type w
inst✝² : Monoid R
inst✝¹ : Monoid M
inst✝ : MulDistribMulAction R M
src✝ : MulAction R (ULift.{?u.24441, v} M) := mulAction'
x✝² : R
x✝¹ x✝ : ULift.{?u.24441, v} M
⊢ (x✝² • (x✝¹ * x✝)).down = (x✝² • x✝¹ * x✝² • x✝).down

Next tactic:
simp [smul_mul']
instance mulDistribMulAction' [Monoid R] [Monoid M] [MulDistribMulAction R M] :
    MulDistribMulAction R (ULift M) :=
  { ULift.mulAction' with
    smul_one := fun _ => by
      

Proof state:
R : Type u
M : Type v
N : Type w
inst✝² : Monoid R
inst✝¹ : Monoid M
inst✝ : MulDistribMulAction R M
src✝ : MulAction R (ULift.{?u.24441, v} M) := mulAction'
x✝ : R
⊢ x✝ • 1 = 1

Next tactic:
ext
instance mulDistribMulAction' [Monoid R] [Monoid M] [MulDistribMulAction R M] :
    MulDistribMulAction R (ULift M) :=
  { ULift.mulAction' with
    smul_one := fun _ => by
      ext
      

Proof state:
case h
R : Type u
M : Type v
N : Type w
inst✝² : Monoid R
inst✝¹ : Monoid M
inst✝ : MulDistribMulAction R M
src✝ : MulAction R (ULift.{?u.24441, v} M) := mulAction'
x✝ : R
⊢ (x✝ • 1).down = 1.down

Next tactic:
simp [smul_one]
instance (X : C) :
    Nonempty (Octahedron (comp_id (𝟙 X)) (contractible_distinguished X)
      (contractible_distinguished X) (contractible_distinguished X)) := by
  

Proof state:
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ Nonempty
    (Octahedron (_ : 𝟙 X ≫ 𝟙 X = 𝟙 X) (_ : contractibleTriangle X ∈ distinguishedTriangles)
      (_ : contractibleTriangle X ∈ distinguishedTriangles) (_ : contractibleTriangle X ∈ distinguishedTriangles))

Next tactic:
refine' ⟨⟨0, 0, _, _, _, _, isomorphic_distinguished _ (contractible_distinguished (0 : C)) _
    (Triangle.isoMk _ _ (by rfl) (by rfl) (by rfl))⟩⟩
instance (X : C) :
    Nonempty (Octahedron (comp_id (𝟙 X)) (contractible_distinguished X)
      (contractible_distinguished X) (contractible_distinguished X)) := by
  refine' ⟨⟨0, 0, _, _, _, _, isomorphic_distinguished _ (contractible_distinguished (0 : C)) _
    (Triangle.isoMk _ _ (by 

Proof state:
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ (Triangle.mk 0 0 (0 ≫ (shiftFunctor C 1).map 0)).obj₁ ≅ (contractibleTriangle 0).obj₁

Next tactic:
rfl
instance (X : C) :
    Nonempty (Octahedron (comp_id (𝟙 X)) (contractible_distinguished X)
      (contractible_distinguished X) (contractible_distinguished X)) := by
  refine' ⟨⟨0, 0, _, _, _, _, isomorphic_distinguished _ (contractible_distinguished (0 : C)) _
    (Triangle.isoMk _ _ (by rfl) (by 

Proof state:
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ (Triangle.mk 0 0 (0 ≫ (shiftFunctor C 1).map 0)).obj₂ ≅ (contractibleTriangle 0).obj₂

Next tactic:
rfl
instance (X : C) :
    Nonempty (Octahedron (comp_id (𝟙 X)) (contractible_distinguished X)
      (contractible_distinguished X) (contractible_distinguished X)) := by
  refine' ⟨⟨0, 0, _, _, _, _, isomorphic_distinguished _ (contractible_distinguished (0 : C)) _
    (Triangle.isoMk _ _ (by rfl) (by rfl) (by 

Proof state:
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ (Triangle.mk 0 0 (0 ≫ (shiftFunctor C 1).map 0)).obj₃ ≅ (contractibleTriangle 0).obj₃

Next tactic:
rfl
instance (X : C) :
    Nonempty (Octahedron (comp_id (𝟙 X)) (contractible_distinguished X)
      (contractible_distinguished X) (contractible_distinguished X)) := by
  refine' ⟨⟨0, 0, _, _, _, _, isomorphic_distinguished _ (contractible_distinguished (0 : C)) _
    (Triangle.isoMk _ _ (by rfl) (by rfl) (by rfl))⟩⟩
  

Proof state:
case refine'_1
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ 0 ≫ 0 = 𝟙 X ≫ 0
case refine'_2
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ 0 ≫ 0 = 0
case refine'_3
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ 0 ≫ 0 = 0
case refine'_4
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ 0 ≫ (shiftFunctor C 1).map (𝟙 X) = 0 ≫ 0

Next tactic:
all_goals apply Subsingleton.elim
instance (X : C) :
    Nonempty (Octahedron (comp_id (𝟙 X)) (contractible_distinguished X)
      (contractible_distinguished X) (contractible_distinguished X)) := by
  refine' ⟨⟨0, 0, _, _, _, _, isomorphic_distinguished _ (contractible_distinguished (0 : C)) _
    (Triangle.isoMk _ _ (by rfl) (by rfl) (by rfl))⟩⟩
  all_goals 

Proof state:
case refine'_1
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ 0 ≫ 0 = 𝟙 X ≫ 0

Next tactic:
apply Subsingleton.elim
instance (X : C) :
    Nonempty (Octahedron (comp_id (𝟙 X)) (contractible_distinguished X)
      (contractible_distinguished X) (contractible_distinguished X)) := by
  refine' ⟨⟨0, 0, _, _, _, _, isomorphic_distinguished _ (contractible_distinguished (0 : C)) _
    (Triangle.isoMk _ _ (by rfl) (by rfl) (by rfl))⟩⟩
  all_goals 

Proof state:
case refine'_2
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ 0 ≫ 0 = 0

Next tactic:
apply Subsingleton.elim
instance (X : C) :
    Nonempty (Octahedron (comp_id (𝟙 X)) (contractible_distinguished X)
      (contractible_distinguished X) (contractible_distinguished X)) := by
  refine' ⟨⟨0, 0, _, _, _, _, isomorphic_distinguished _ (contractible_distinguished (0 : C)) _
    (Triangle.isoMk _ _ (by rfl) (by rfl) (by rfl))⟩⟩
  all_goals 

Proof state:
case refine'_3
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ 0 ≫ 0 = 0

Next tactic:
apply Subsingleton.elim
instance (X : C) :
    Nonempty (Octahedron (comp_id (𝟙 X)) (contractible_distinguished X)
      (contractible_distinguished X) (contractible_distinguished X)) := by
  refine' ⟨⟨0, 0, _, _, _, _, isomorphic_distinguished _ (contractible_distinguished (0 : C)) _
    (Triangle.isoMk _ _ (by rfl) (by rfl) (by rfl))⟩⟩
  all_goals 

Proof state:
case refine'_4
C : Type u_1
inst✝⁵ : Category.{?u.9345, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X : C
⊢ 0 ≫ (shiftFunctor C 1).map (𝟙 X) = 0 ≫ 0

Next tactic:
apply Subsingleton.elim
/-- The first morphism of triangles given by an octahedron. -/
@[simps]
def triangleMorphism₁ : Triangle.mk u₁₂ v₁₂ w₁₂ ⟶ Triangle.mk u₁₃ v₁₃ w₁₃
    where
  hom₁ := 𝟙 X₁
  hom₂ := u₂₃
  hom₃ := h.m₁
  comm₁ := by
    

Proof state:
C : Type u_1
inst✝⁵ : Category.{?u.26669, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron comm h₁₂ h₂₃ h₁₃
⊢ (Triangle.mk u₁₂ v₁₂ w₁₂).mor₁ ≫ u₂₃ = 𝟙 X₁ ≫ (Triangle.mk u₁₃ v₁₃ w₁₃).mor₁

Next tactic:
dsimp