ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Sites/ConstantSheaf.lean
C : Type u_1
inst✝² : Category.{u_3, u_1} C
J : GrothendieckTopology C
D : Type u_2
inst✝¹ : Category.{u_4, u_2} D
inst✝ : HasWeakSheafify J D
T : C
hT : IsTerminal T
F : Sheaf J D
\u22a2 toSheafify J ((const Cᵒᵖ).obj (((sheafSections J D).obj (op T)).obj F)) ≫ ((constantSheafAdj J D hT).counit.app F).val = toSheafify J ((const Cᵒᵖ).obj (((sheafSections J D).obj (op T)).obj F)) ≫ ((presheafToSheaf J D).map ((constantPresheafAdj D hT).counit.app F.val) ≫ (sheafificationAdjunction J D).counit.app F).val
-- Mathlib/CategoryTheory/Sites/ConstantSheaf.lean
C : Type u_1
inst✝² : Category.{u_3, u_1} C
J : GrothendieckTopology C
D : Type u_2
inst✝¹ : Category.{u_4, u_2} D
inst✝ : HasWeakSheafify J D
T : C
hT : IsTerminal T
F : Sheaf J D
\u22a2 toSheafify J ((const Cᵒᵖ).obj (F.val.obj (op T))) ≫ 𝟙 ((presheafToSheaf J D).obj ((const Cᵒᵖ).obj (F.val.obj (op T)))).val ≫ sheafifyLift J { app := fun x => F.val.map (hT.from (unop x)).op, naturality := ⋯ } ⋯ = { app := fun x => F.val.map (hT.from (unop x)).op, naturality := ⋯ }
-- Mathlib/Order/Bounded.lean
α : Type u_1
r : α → α → Prop
s t : Set α
inst✝ : LinearOrder α
a : α
\u22a2 Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a < b}) ↔ Bounded (fun x x_1 => x ≤ x_1) s
-- Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X Y : C
p : ExactPairing X Y
\u22a2 (leftDualIso p p).hom = (Iso.refl X).hom
-- Mathlib/MeasureTheory/Integral/CircleIntegral.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
hR : R ≠ 0
\u22a2 (∮ (z : ℂ) in C(c, R), (z - c)⁻¹) = 2 * ↑π * I
-- Mathlib/Order/SuccPred/Limit.lean
α : Type u_1
inst✝² : Preorder α
inst✝¹ : PredOrder α
inst✝ : NoMinOrder α
b : α
h : IsPredLimit (pred b)
\u22a2 False
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
\u22a2 XYIdeal W x (C (W.negY x y)) * XYIdeal W x (C y) = XIdeal W x
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw : (Y - C (C y)) * (Y - C (C (W.negY x y))) - C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = W.polynomial * 1
\u22a2 XYIdeal W x (C (W.negY x y)) * XYIdeal W x (C y) = XIdeal W x
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw : (Y - C (C y)) * (Y - C (C (W.negY x y))) - C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = W.polynomial * 1
\u22a2 span {(mk W) (C (X - C x))} * span {(mk W) (C (X - C x)), (mk W) (Y - C (C y)), (mk W) (Y - C (C (W.negY x y))), (AdjoinRoot.mk W.polynomial) (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y)} = XIdeal W x
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw : (Y - C (C y)) * (Y - C (C (W.negY x y))) - C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = W.polynomial * 1
\u22a2 span {(mk W) (C (X - C x)), (mk W) (Y - C (C y)), (mk W) (Y - C (C (W.negY x y))), (AdjoinRoot.mk W.polynomial) (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y)} = ⊤
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw : (Y - C (C y)) * (Y - C (C (W.negY x y))) - C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = W.polynomial * 1
\u22a2 Ideal.map (mk W) (span {C (X - C x), Y - C (C y), Y - C (C (W.negY x y)), C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y}) = ⊤
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw : (Y - C (C y)) * (Y - C (C (W.negY x y))) - C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = W.polynomial * 1
\u22a2 Ideal.map (mk W) (span {C (X - C x), Y - C (C y), Y - C (C (W.negY x y)), C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y}) = Ideal.map (mk W) ⊤
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw : (Y - C (C y)) * (Y - C (C (W.negY x y))) - C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = W.polynomial * 1
\u22a2 span {C (X - C x), Y - C (C y), Y - C (C (W.negY x y)), C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y} = ⊤
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw : (Y - C (C y)) * (Y - C (C (W.negY x y))) - C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = W.polynomial * 1
\u22a2 ∃ a a_1 a_2 a_3, a_3 * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = 1 + a * C (X - C x) + a_1 * (Y - C (C y)) + a_2 * (Y - C (C (W.negY x y)))
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw : (Y - C (C y)) * (Y - C (C (W.negY x y))) - C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = W.polynomial * 1
hx : W.a₁ * y - (3 * x ^ 2 + 2 * W.a₂ * x + W.a₄) ≠ 0
\u22a2 ∃ a a_1 a_2 a_3, a_3 * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = 1 + a * C (X - C x) + a_1 * (Y - C (C y)) + a_2 * (Y - C (C (W.negY x y)))
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw : (Y - C (C y)) * (Y - C (C (W.negY x y))) - C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = W.polynomial * 1
hy : 2 * y + W.a₁ * x + W.a₃ ≠ 0
\u22a2 ∃ a a_1 a_2 a_3, a_3 * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) = 1 + a * C (X - C x) + a_1 * (Y - C (C y)) + a_2 * (Y - C (C (W.negY x y)))
-- Mathlib/Algebra/Order/Kleene.lean
α : Type u_1
β : Type u_2
ι : Type u_3
π : ι → Type u_4
inst✝ : KleeneAlgebra α
a b c : α
n : ℕ
\u22a2 a ^ (n + 1) ≤ a∗
-- Mathlib/Algebra/Order/Kleene.lean
α : Type u_1
β : Type u_2
ι : Type u_3
π : ι → Type u_4
inst✝ : KleeneAlgebra α
a b c : α
n : ℕ
\u22a2 a * a ^ n ≤ a∗
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
\u22a2 Nonempty (IsColimit c)
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
t₁' : Nonempty (IsColimit (Cofan.mk (pullback c₂.inr i) fun j => pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯))
\u22a2 Nonempty (IsColimit c)
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
j : Fin n
\u22a2 (α.app { as := j.succ } ≫ c₁.inj j) ≫ c₂.inr = c.ι.app { as := j.succ } ≫ i
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
\u22a2 (Discrete.natTrans fun i => α.app { as := i.as.succ }) ≫ c₁.ι = (Cofan.mk (pullback c₂.inr i) fun j => pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯).ι ≫ (Functor.const (Discrete (Fin n))).map (pullback.fst c₂.inr i)
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
\u22a2 ∀ (j : Discrete (Fin n)), IsPullback ((Cofan.mk (pullback c₂.inr i) fun j => pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯).ι.app j) ((Discrete.natTrans fun i => α.app { as := i.as.succ }).app j) (pullback.fst c₂.inr i) (c₁.ι.app j)
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
H₁ : IsColimit (Cofan.mk (pullback c₂.inr i) fun j => pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯)
\u22a2 Nonempty (IsColimit c)
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
H₁ : IsColimit (Cofan.mk (pullback c₂.inr i) fun j => pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯)
t₂' : Nonempty (IsColimit (BinaryCofan.mk (c.ι.app { as := 0 }) (pullback.snd c₂.inr i)))
\u22a2 Nonempty (IsColimit c)
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
H₁ : IsColimit (Cofan.mk (pullback c₂.inr i) fun j => pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯)
\u22a2 mapPair (α.app { as := 0 }) (pullback.fst c₂.inr i) ≫ c₂.ι = (BinaryCofan.mk (c.ι.app { as := 0 }) (pullback.snd c₂.inr i)).ι ≫ (Functor.const (Discrete WalkingPair)).map i
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Discrete (Fin (n + 1)) ⥤ C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
H₁ : IsColimit (Cofan.mk (pullback c₂.inr i) fun j => pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯)
\u22a2 ∀ (j : Discrete WalkingPair), IsPullback ((BinaryCofan.mk (c.ι.app { as := 0 }) (pullback.snd c₂.inr i)).ι.app j) ((mapPair (α.app { as := 0 }) (pullback.fst c₂.inr i)).app j) i (c₂.ι.app j)
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F' : Fin (n + 1) → C
c : Cocone (Discrete.functor F')
α : Discrete.functor F' ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
H₁ : IsColimit (Cofan.mk (pullback c₂.inr i) fun j => pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯)
H₂ : IsColimit (BinaryCofan.mk (c.ι.app { as := 0 }) (pullback.snd c₂.inr i))
j : Fin (n + 1)
\u22a2 Fin.cases (c.ι.app { as := 0 }) (fun i_1 => pullback.lift (α.app { as := i_1.succ } ≫ c₁.inj i_1) (c.ι.app { as := i_1.succ }) ⋯ ≫ pullback.snd c₂.inr i) { as := j }.as ≫ 𝟙 c.pt = c.ι.app { as := j }
-- Mathlib/CategoryTheory/Limits/VanKampen.lean
J : Type v'
inst✝⁴ : Category.{u', v'} J
C : Type u
inst✝³ : Category.{v, u} C
K : Type u_1
inst✝² : Category.{?u.288379, u_1} K
D : Type u_2
inst✝¹ : Category.{?u.288386, u_2} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F' : Fin (n + 1) → C
c : Cocone (Discrete.functor F')
α : Discrete.functor F' ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
H₁ : IsColimit (Cofan.mk (pullback c₂.inr i) fun j => pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯)
H₂ : IsColimit (BinaryCofan.mk (c.ι.app { as := 0 }) (pullback.snd c₂.inr i))
j : Fin (n + 1)
\u22a2 Fin.cases (c.ι.app { as := 0 }) (fun i => c.ι.app { as := i.succ }) j = c.ι.app { as := j }
-- Mathlib/Algebra/Algebra/Bilinear.lean
R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : NoZeroDivisors A
x : A
hx : x ≠ 0
this : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }
\u22a2 Function.Injective ⇑((mul R A) x)
-- Mathlib/Algebra/Algebra/Bilinear.lean
R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : NoZeroDivisors A
x : A
hx : x ≠ 0
this✝ : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }
this : IsDomain A := NoZeroDivisors.to_isDomain A
\u22a2 Function.Injective ⇑((mul R A) x)
-- Mathlib/Algebra/ModEq.lean
α : Type u_1
inst✝¹ : AddCommGroup α
p a a₁ a₂ b b₁ b₂ c : α
n : ℕ
z : ℤ
inst✝ : NoZeroSMulDivisors ℕ α
hn : n ≠ 0
m : ℤ
\u22a2 n • b - n • a = m • n • p ↔ b - a = m • p
-- Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
x : CliffordAlgebra Q
\u22a2 (changeForm h') ((changeForm h) x) = (changeForm ⋯) x
-- Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
r : R
\u22a2 (changeForm h') ((changeForm h) ((algebraMap R (CliffordAlgebra Q)) r)) = (changeForm ⋯) ((algebraMap R (CliffordAlgebra Q)) r)
-- Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
x y : CliffordAlgebra Q
hx : (changeForm h') ((changeForm h) x) = (changeForm ⋯) x
hy : (changeForm h') ((changeForm h) y) = (changeForm ⋯) y
\u22a2 (changeForm h') ((changeForm h) (x + y)) = (changeForm ⋯) (x + y)
-- Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
m : CliffordAlgebra Q
x : M
hx : (changeForm h') ((changeForm h) m) = (changeForm ⋯) m
\u22a2 (changeForm h') ((changeForm h) ((ι Q) x * m)) = (changeForm ⋯) ((ι Q) x * m)
-- Mathlib/Computability/TMToPartrec.lean
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
s : Option Γ'
L₁ : List Γ'
o : Option Γ'
L₂ : List Γ'
S : K' → List Γ'
h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂
h₂ : S rev = []
e : splitAtPred p (S k₁) = (L₁, o, L₂)
\u22a2 Reaches₁ (TM2.step tr) { l := some (move₂ p k₁ k₂ q), var := s, stk := S } { l := some q, var := none, stk := update (update S k₁ (o.elim id List.cons L₂)) k₂ (L₁ ++ S k₂) }
-- Mathlib/Computability/TMToPartrec.lean
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
s : Option Γ'
L₁ : List Γ'
o : Option Γ'
L₂ : List Γ'
S : K' → List Γ'
h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂
h₂ : S rev = []
e : splitAtPred p (S k₁) = (L₁, o, L₂)
\u22a2 TransGen (fun a b => b ∈ TM2.step tr a) (TM2.stepAux (tr (Λ'.push k₁ id (Λ'.move (fun x => false) rev k₂ q))) o (update (update S k₁ L₂) rev (L₁.reverseAux (S rev)))) { l := some q, var := none, stk := update (update S k₁ (o.elim id List.cons L₂)) k₂ (L₁ ++ S k₂) }
-- Mathlib/Computability/TMToPartrec.lean
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
s : Option Γ'
L₁ : List Γ'
o : Option Γ'
L₂ : List Γ'
S : K' → List Γ'
h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂
h₂ : S rev = []
e : splitAtPred p (S k₁) = (L₁, o, L₂)
\u22a2 TransGen (fun a b => (match a with | { l := none, var := var, stk := stk } => none | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) (bif o.isSome then { l := some (Λ'.move (fun x => false) rev k₂ q), var := o, stk := update (update (update S k₁ L₂) rev (L₁.reverseAux (S rev))) k₁ (o.iget :: update (update S k₁ L₂) rev (L₁.reverseAux (S rev)) k₁) } else { l := some (Λ'.move (fun x => false) rev k₂ q), var := o, stk := update (update S k₁ L₂) rev (L₁.reverseAux (S rev)) }) { l := some q, var := none, stk := update (update S k₁ ((match (motive := Option Γ' → (List Γ' → List Γ') → (Γ' → List Γ' → List Γ') → List Γ' → List Γ') o, id, List.cons with | some x, x_1, f => f x | none, y, x => y) L₂)) k₂ (L₁ ++ S k₂) }
-- Mathlib/RingTheory/MatrixAlgebra.lean
R : Type u
inst✝⁴ : CommSemiring R
A : Type v
inst✝³ : Semiring A
inst✝² : Algebra R A
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
a : A
M : Matrix n n A
\u22a2 invFun R A n (a • M) = a ⊗ₜ[R] 1 * invFun R A n M
-- Mathlib/Analysis/Calculus/FDeriv/Add.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hg : DifferentiableOn 𝕜 g s
\u22a2 DifferentiableOn 𝕜 (fun y => f y + g y) s ↔ DifferentiableOn 𝕜 f s