Loogle!

Result

Found 50 declarations mentioning CategoryTheory.Limits.PushoutCocone.mk.

CategoryTheory.Limits.PushoutCocone.mk 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : CategoryTheory.CategoryStruct.comp f inl = CategoryTheory.CategoryStruct.comp g inr) : CategoryTheory.Limits.PushoutCocone f g
CategoryTheory.Limits.PushoutCocone.mk_pt 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : CategoryTheory.CategoryStruct.comp f inl = CategoryTheory.CategoryStruct.comp g inr) : (CategoryTheory.Limits.PushoutCocone.mk inl inr eq).pt = W
CategoryTheory.Limits.PushoutCocone.eta 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (t : CategoryTheory.Limits.PushoutCocone f g) : t ≅ CategoryTheory.Limits.PushoutCocone.mk t.inl t.inr ⋯
CategoryTheory.Limits.PushoutCocone.mkSelfIsColimit 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {t : CategoryTheory.Limits.PushoutCocone f g} (ht : CategoryTheory.Limits.IsColimit t) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk t.inl t.inr ⋯)
CategoryTheory.Limits.PushoutCocone.mk_inl 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : CategoryTheory.CategoryStruct.comp f inl = CategoryTheory.CategoryStruct.comp g inr) : (CategoryTheory.Limits.PushoutCocone.mk inl inr eq).inl = inl
CategoryTheory.Limits.PushoutCocone.mk_inr 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : CategoryTheory.CategoryStruct.comp f inl = CategoryTheory.CategoryStruct.comp g inr) : (CategoryTheory.Limits.PushoutCocone.mk inl inr eq).inr = inr
CategoryTheory.Limits.PushoutCocone.mk_ι_app_left 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : CategoryTheory.CategoryStruct.comp f inl = CategoryTheory.CategoryStruct.comp g inr) : (CategoryTheory.Limits.PushoutCocone.mk inl inr eq).ι.app CategoryTheory.Limits.WalkingSpan.left = inl
CategoryTheory.Limits.PushoutCocone.mk_ι_app_right 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : CategoryTheory.CategoryStruct.comp f inl = CategoryTheory.CategoryStruct.comp g inr) : (CategoryTheory.Limits.PushoutCocone.mk inl inr eq).ι.app CategoryTheory.Limits.WalkingSpan.right = inr
CategoryTheory.Limits.PushoutCocone.mk_ι_app_zero 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : CategoryTheory.CategoryStruct.comp f inl = CategoryTheory.CategoryStruct.comp g inr) : (CategoryTheory.Limits.PushoutCocone.mk inl inr eq).ι.app CategoryTheory.Limits.WalkingSpan.zero = CategoryTheory.CategoryStruct.comp f inl
CategoryTheory.Limits.PushoutCocone.eta_hom_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (t : CategoryTheory.Limits.PushoutCocone f g) : t.eta.hom.hom = CategoryTheory.CategoryStruct.id t.pt
CategoryTheory.Limits.PushoutCocone.eta_inv_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (t : CategoryTheory.Limits.PushoutCocone f g) : t.eta.inv.hom = CategoryTheory.CategoryStruct.id t.pt
CategoryTheory.Limits.PushoutCocone.mk_ι_app 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {W : C} (inl : Y ⟶ W) (inr : Z ⟶ W) (eq : CategoryTheory.CategoryStruct.comp f inl = CategoryTheory.CategoryStruct.comp g inr) (j : CategoryTheory.Limits.WalkingSpan) : (CategoryTheory.Limits.PushoutCocone.mk inl inr eq).ι.app j = Option.casesOn j (CategoryTheory.CategoryStruct.comp f inl) fun j' => CategoryTheory.Limits.WalkingPair.casesOn j' inl inr
CategoryTheory.Limits.PushoutCocone.IsColimit.mk 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {W : C} {inl : Y ⟶ W} {inr : Z ⟶ W} (eq : CategoryTheory.CategoryStruct.comp f inl = CategoryTheory.CategoryStruct.comp g inr) (desc : (s : CategoryTheory.Limits.PushoutCocone f g) → W ⟶ s.pt) (fac_left : ∀ (s : CategoryTheory.Limits.PushoutCocone f g), CategoryTheory.CategoryStruct.comp inl (desc s) = s.inl) (fac_right : ∀ (s : CategoryTheory.Limits.PushoutCocone f g), CategoryTheory.CategoryStruct.comp inr (desc s) = s.inr) (uniq : ∀ (s : CategoryTheory.Limits.PushoutCocone f g) (m : W ⟶ s.pt), CategoryTheory.CategoryStruct.comp inl m = s.inl → CategoryTheory.CategoryStruct.comp inr m = s.inr → m = desc s) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk inl inr eq)
CategoryTheory.Limits.PushoutCocone.isoMk 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor CategoryTheory.Limits.WalkingSpan C} (t : CategoryTheory.Limits.Cocone F) : (CategoryTheory.Limits.Cocone.precompose (CategoryTheory.Limits.diagramIsoSpan F).inv).obj t ≅ CategoryTheory.Limits.PushoutCocone.mk (t.ι.app CategoryTheory.Limits.WalkingSpan.left) (t.ι.app CategoryTheory.Limits.WalkingSpan.right) ⋯
CategoryTheory.Limits.PushoutCocone.isoMk_hom_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor CategoryTheory.Limits.WalkingSpan C} (t : CategoryTheory.Limits.Cocone F) : (CategoryTheory.Limits.PushoutCocone.isoMk t).hom.hom = CategoryTheory.CategoryStruct.id t.pt
CategoryTheory.Limits.PushoutCocone.isoMk_inv_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
{C : Type u} [CategoryTheory.Category.{v, u} C] {F : CategoryTheory.Functor CategoryTheory.Limits.WalkingSpan C} (t : CategoryTheory.Limits.Cocone F) : (CategoryTheory.Limits.PushoutCocone.isoMk t).inv.hom = CategoryTheory.CategoryStruct.id t.pt
CategoryTheory.Limits.pushoutIsPushout 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [CategoryTheory.Limits.HasPushout f g] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Limits.pushout.inl f g) (CategoryTheory.Limits.pushout.inr f g) ⋯)
CategoryTheory.Limits.PushoutCocone.epi_of_isColimitMkIdId 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y) (t : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.CategoryStruct.id Y) (CategoryTheory.CategoryStruct.id Y) ⋯)) : CategoryTheory.Epi f
CategoryTheory.Limits.PushoutCocone.isColimitMkIdId 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [CategoryTheory.Epi f] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.CategoryStruct.id Y) (CategoryTheory.CategoryStruct.id Y) ⋯)
CategoryTheory.Limits.pushoutIsPushoutOfEpiComp 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
{C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) (h : W ⟶ X) [CategoryTheory.Epi h] [CategoryTheory.Limits.HasPushout f g] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Limits.pushout.inl f g) (CategoryTheory.Limits.pushout.inr f g) ⋯)
CategoryTheory.Limits.PushoutCocone.isColimitOfEpiComp 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
{C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) (h : W ⟶ X) [CategoryTheory.Epi h] (s : CategoryTheory.Limits.PushoutCocone f g) (H : CategoryTheory.Limits.IsColimit s) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk s.inl s.inr ⋯)
CategoryTheory.Limits.PushoutCocone.isColimitOfFactors 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
{C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) (h : X ⟶ W) [CategoryTheory.Epi h] (x : W ⟶ Y) (y : W ⟶ Z) (hhx : CategoryTheory.CategoryStruct.comp h x = f) (hhy : CategoryTheory.CategoryStruct.comp h y = g) (s : CategoryTheory.Limits.PushoutCocone f g) (hs : CategoryTheory.Limits.IsColimit s) : have reassoc₁ := ⋯; have reassoc₂ := ⋯; CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk s.inl s.inr ⋯)
CategoryTheory.Limits.isColimitMapCoconePushoutCoconeEquiv 📋 Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {W X Y Z : C} {h : X ⟶ Z} {k : Y ⟶ Z} {f : W ⟶ X} {g : W ⟶ Y} (comm : CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g k) : CategoryTheory.Limits.IsColimit (G.mapCocone (CategoryTheory.Limits.PushoutCocone.mk h k comm)) ≃ CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (G.map h) (G.map k) ⋯)
CategoryTheory.Limits.isColimitOfIsColimitPushoutCoconeMap 📋 Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {W X Y Z : C} {h : X ⟶ Z} {k : Y ⟶ Z} {f : W ⟶ X} {g : W ⟶ Y} (comm : CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g k) [CategoryTheory.Limits.ReflectsColimit (CategoryTheory.Limits.span f g) G] (l : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (G.map h) (G.map k) ⋯)) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk h k comm)
CategoryTheory.Limits.isColimitPushoutCoconeMapOfIsColimit 📋 Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {W X Y Z : C} {h : X ⟶ Z} {k : Y ⟶ Z} {f : W ⟶ X} {g : W ⟶ Y} (comm : CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g k) [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span f g) G] (l : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk h k comm)) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (G.map h) (G.map k) ⋯)
CategoryTheory.Limits.isColimitOfHasPushoutOfPreservesColimit 📋 Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {W X Y : C} (f : W ⟶ X) (g : W ⟶ Y) [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span f g) G] [i : CategoryTheory.Limits.HasPushout f g] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (G.map (CategoryTheory.Limits.pushout.inl f g)) (G.map (CategoryTheory.Limits.pushout.inr f g)) ⋯)
isCoproductOfIsInitialIsPushout 📋 Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
{C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ X) (k : W ⟶ Y) (H₁ : CategoryTheory.Limits.IsInitial W) (H₂ : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk f g ⋯)) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk f g)
isPushoutOfIsInitialIsCoproduct 📋 Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
{C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ X) (k : W ⟶ Y) (H₁ : CategoryTheory.Limits.IsInitial W) (H₂ : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk f g)) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk f g ⋯)
isBinaryCoproductOfIsInitialIsPushout 📋 Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
{C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) C) (c : CategoryTheory.Limits.Cocone F) {X : C} (hX : CategoryTheory.Limits.IsInitial X) (f : X ⟶ F.obj { as := CategoryTheory.Limits.WalkingPair.left }) (g : X ⟶ F.obj { as := CategoryTheory.Limits.WalkingPair.right }) (hc : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (c.ι.app { as := CategoryTheory.Limits.WalkingPair.left }) (c.ι.app { as := CategoryTheory.Limits.WalkingPair.right }) ⋯)) : CategoryTheory.Limits.IsColimit c
CategoryTheory.IsPushout.isColimit' 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Z X Y P : C} {f : Z ⟶ X} {g : Z ⟶ Y} {inl : X ⟶ P} {inr : Y ⟶ P} (self : CategoryTheory.IsPushout f g inl inr) : Nonempty (CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk inl inr ⋯))
CategoryTheory.IsPushout.mk 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {Z X Y P : C} {f : Z ⟶ X} {g : Z ⟶ Y} {inl : X ⟶ P} {inr : Y ⟶ P} (toCommSq : CategoryTheory.CommSq f g inl inr) (isColimit' : Nonempty (CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk inl inr ⋯))) : CategoryTheory.IsPushout f g inl inr
CategoryTheory.regularOfIsPushoutFstOfRegular 📋 Mathlib.CategoryTheory.Limits.Shapes.RegularMono
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S} (hf : CategoryTheory.RegularEpi f) (comm : CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g k) (t : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk h k comm)) : CategoryTheory.RegularEpi k
CategoryTheory.regularOfIsPushoutSndOfRegular 📋 Mathlib.CategoryTheory.Limits.Shapes.RegularMono
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S} (gr : CategoryTheory.RegularEpi g) (comm : CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g k) (t : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk h k comm)) : CategoryTheory.RegularEpi h
CategoryTheory.Limits.pushoutAssocSymmIsPushout 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Assoc
{C : Type u} [CategoryTheory.Category.{v, u} C] {X₁ X₂ X₃ Z₁ Z₂ : C} (g₁ : Z₁ ⟶ X₁) (g₂ : Z₁ ⟶ X₂) (g₃ : Z₂ ⟶ X₂) (g₄ : Z₂ ⟶ X₃) [CategoryTheory.Limits.HasPushout g₁ g₂] [CategoryTheory.Limits.HasPushout g₃ g₄] [CategoryTheory.Limits.HasPushout g₁ (CategoryTheory.CategoryStruct.comp g₂ (CategoryTheory.Limits.pushout.inl g₃ g₄))] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Limits.pushout.desc (CategoryTheory.Limits.pushout.inl g₁ (CategoryTheory.CategoryStruct.comp g₂ (CategoryTheory.Limits.pushout.inl g₃ g₄))) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inl g₃ g₄) (CategoryTheory.Limits.pushout.inr g₁ (CategoryTheory.CategoryStruct.comp g₂ (CategoryTheory.Limits.pushout.inl g₃ g₄)))) ⋯) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inr g₃ g₄) (CategoryTheory.Limits.pushout.inr g₁ (CategoryTheory.CategoryStruct.comp g₂ (CategoryTheory.Limits.pushout.inl g₃ g₄)))) ⋯)
CategoryTheory.Limits.pushoutPushoutRightIsPushout 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Assoc
{C : Type u} [CategoryTheory.Category.{v, u} C] {X₁ X₂ X₃ Z₁ Z₂ : C} (g₁ : Z₁ ⟶ X₁) (g₂ : Z₁ ⟶ X₂) (g₃ : Z₂ ⟶ X₂) (g₄ : Z₂ ⟶ X₃) [CategoryTheory.Limits.HasPushout g₁ g₂] [CategoryTheory.Limits.HasPushout g₃ g₄] [CategoryTheory.Limits.HasPushout g₁ (CategoryTheory.CategoryStruct.comp g₂ (CategoryTheory.Limits.pushout.inl g₃ g₄))] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Limits.pushout.desc (CategoryTheory.Limits.pushout.inl g₁ (CategoryTheory.CategoryStruct.comp g₂ (CategoryTheory.Limits.pushout.inl g₃ g₄))) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inl g₃ g₄) (CategoryTheory.Limits.pushout.inr g₁ (CategoryTheory.CategoryStruct.comp g₂ (CategoryTheory.Limits.pushout.inl g₃ g₄)))) ⋯) (CategoryTheory.Limits.pushout.inr g₁ (CategoryTheory.CategoryStruct.comp g₂ (CategoryTheory.Limits.pushout.inl g₃ g₄))) ⋯)
CategoryTheory.Limits.pushoutAssocIsPushout 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Assoc
{C : Type u} [CategoryTheory.Category.{v, u} C] {X₁ X₂ X₃ Z₁ Z₂ : C} (g₁ : Z₁ ⟶ X₁) (g₂ : Z₁ ⟶ X₂) (g₃ : Z₂ ⟶ X₂) (g₄ : Z₂ ⟶ X₃) [CategoryTheory.Limits.HasPushout g₁ g₂] [CategoryTheory.Limits.HasPushout g₃ g₄] [CategoryTheory.Limits.HasPushout (CategoryTheory.CategoryStruct.comp g₃ (CategoryTheory.Limits.pushout.inr g₁ g₂)) g₄] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inl g₁ g₂) (CategoryTheory.Limits.pushout.inl (CategoryTheory.CategoryStruct.comp g₃ (CategoryTheory.Limits.pushout.inr g₁ g₂)) g₄)) (CategoryTheory.Limits.pushout.desc (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inr g₁ g₂) (CategoryTheory.Limits.pushout.inl (CategoryTheory.CategoryStruct.comp g₃ (CategoryTheory.Limits.pushout.inr g₁ g₂)) g₄)) (CategoryTheory.Limits.pushout.inr (CategoryTheory.CategoryStruct.comp g₃ (CategoryTheory.Limits.pushout.inr g₁ g₂)) g₄) ⋯) ⋯)
CategoryTheory.Limits.pushoutPushoutLeftIsPushout 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.Assoc
{C : Type u} [CategoryTheory.Category.{v, u} C] {X₁ X₂ X₃ Z₁ Z₂ : C} (g₁ : Z₁ ⟶ X₁) (g₂ : Z₁ ⟶ X₂) (g₃ : Z₂ ⟶ X₂) (g₄ : Z₂ ⟶ X₃) [CategoryTheory.Limits.HasPushout g₁ g₂] [CategoryTheory.Limits.HasPushout g₃ g₄] [CategoryTheory.Limits.HasPushout (CategoryTheory.CategoryStruct.comp g₃ (CategoryTheory.Limits.pushout.inr g₁ g₂)) g₄] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Limits.pushout.inl (CategoryTheory.CategoryStruct.comp g₃ (CategoryTheory.Limits.pushout.inr g₁ g₂)) g₄) (CategoryTheory.Limits.pushout.desc (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inr g₁ g₂) (CategoryTheory.Limits.pushout.inl (CategoryTheory.CategoryStruct.comp g₃ (CategoryTheory.Limits.pushout.inr g₁ g₂)) g₄)) (CategoryTheory.Limits.pushout.inr (CategoryTheory.CategoryStruct.comp g₃ (CategoryTheory.Limits.pushout.inr g₁ g₂)) g₄) ⋯) ⋯)
CategoryTheory.normalOfIsPushoutFstOfNormal 📋 Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S} [CategoryTheory.NormalEpi f] (comm : CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g k) (t : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk h k comm)) : CategoryTheory.NormalEpi k
CategoryTheory.normalOfIsPushoutSndOfNormal 📋 Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroMorphisms C] {P Q R S : C} {f : P ⟶ Q} {g : P ⟶ R} {h : Q ⟶ S} {k : R ⟶ S} [gn : CategoryTheory.NormalEpi g] (comm : CategoryTheory.CategoryStruct.comp f h = CategoryTheory.CategoryStruct.comp g k) (t : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk h k comm)) : CategoryTheory.NormalEpi h
CategoryTheory.IsPushout.isVanKampen_iff 📋 Mathlib.CategoryTheory.Adhesive.Basic
{C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z} (H : CategoryTheory.IsPushout f g h i) : H.IsVanKampen ↔ CategoryTheory.IsVanKampenColimit (CategoryTheory.Limits.PushoutCocone.mk h i ⋯)
CategoryTheory.Limits.pushoutCoconeEquivBinaryCofan_inverse_obj 📋 Mathlib.CategoryTheory.Limits.Constructions.Over.Products
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : CategoryTheory.Limits.BinaryCofan (CategoryTheory.Under.mk f) (CategoryTheory.Under.mk g)) : CategoryTheory.Limits.pushoutCoconeEquivBinaryCofan.inverse.obj c = CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Under.Hom.right c.inl) (CategoryTheory.Under.Hom.right c.inr) ⋯
CategoryTheory.Limits.IsColimit.pushoutCoconeEquivBinaryCofanFunctor_desc_right 📋 Mathlib.CategoryTheory.Limits.Constructions.Over.Products
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {c : CategoryTheory.Limits.PushoutCocone f g} (hc : CategoryTheory.Limits.IsColimit c) (s : CategoryTheory.Limits.BinaryCofan (CategoryTheory.Under.mk f) (CategoryTheory.Under.mk g)) : (hc.pushoutCoconeEquivBinaryCofanFunctor.desc s).right = hc.desc (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Under.Hom.right s.inl) (CategoryTheory.Under.Hom.right s.inr) ⋯)
CategoryTheory.Limits.pushoutCoconeEquivBinaryCofan_inverse_map_hom 📋 Mathlib.CategoryTheory.Limits.Constructions.Over.Products
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} {c₁ c₂ : CategoryTheory.Limits.BinaryCofan (CategoryTheory.Under.mk f) (CategoryTheory.Under.mk g)} (a : c₁ ⟶ c₂) : (CategoryTheory.Limits.pushoutCoconeEquivBinaryCofan.inverse.map a).hom = CategoryTheory.Under.Hom.right a.hom
CategoryTheory.Limits.pushoutCoconeEquivBinaryCofan_unitIso 📋 Mathlib.CategoryTheory.Limits.Constructions.Over.Products
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} : CategoryTheory.Limits.pushoutCoconeEquivBinaryCofan.unitIso = CategoryTheory.NatIso.ofComponents (fun c => c.eta) ⋯
CategoryTheory.Limits.pushoutCoconeEquivBinaryCofan_counitIso 📋 Mathlib.CategoryTheory.Limits.Constructions.Over.Products
{C : Type u} [CategoryTheory.Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} : CategoryTheory.Limits.pushoutCoconeEquivBinaryCofan.counitIso = CategoryTheory.NatIso.ofComponents (fun X_1 => CategoryTheory.Limits.BinaryCofan.ext (CategoryTheory.Under.isoMk (CategoryTheory.Iso.refl (({ obj := fun c => CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Under.Hom.right c.inl) (CategoryTheory.Under.Hom.right c.inr) ⋯, map := fun {c₁ c₂} a => { hom := CategoryTheory.Under.Hom.right a.hom, w := ⋯ }, map_id := ⋯, map_comp := ⋯ }.comp { obj := fun c => CategoryTheory.Limits.BinaryCofan.mk (CategoryTheory.Under.homMk c.inl ⋯) (CategoryTheory.Under.homMk c.inr ⋯), map := fun {c₁ c₂} a => { hom := CategoryTheory.Under.homMk a.hom ⋯, w := ⋯ }, map_id := ⋯, map_comp := ⋯ }).obj X_1).pt.right) ⋯) ⋯ ⋯) ⋯
CategoryTheory.CommSq.isColimitEquivIsColimitCokernelCofork 📋 Mathlib.Algebra.Homology.CommSq
{C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Preadditive C] {X₁ X₂ X₃ X₄ : C} [CategoryTheory.Limits.HasBinaryBiproduct X₂ X₃] {f : X₁ ⟶ X₂} {g : X₁ ⟶ X₃} {inl : X₂ ⟶ X₄} {inr : X₃ ⟶ X₄} (sq : CategoryTheory.CommSq f g inl inr) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk inl inr ⋯) ≃ CategoryTheory.Limits.IsColimit sq.cokernelCofork
CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.associator_hom_left 📋 Mathlib.CategoryTheory.Monoidal.PushoutProduct
{C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPushouts C] [CategoryTheory.MonoidalCategory C] (X₁ X₂ X₃ : CategoryTheory.Arrow C) [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) (CategoryTheory.MonoidalCategory.tensorRight X₃.left)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) (CategoryTheory.MonoidalCategory.tensorRight X₃.right)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom)) (CategoryTheory.MonoidalCategory.tensorLeft X₁.left)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom)) (CategoryTheory.MonoidalCategory.tensorLeft X₁.right)] : (CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.associator X₁ X₂ X₃).hom.left = CategoryTheory.Limits.pushout.desc (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator X₁.right X₂.right X₃.left).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right (CategoryTheory.Limits.pushout.inl (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) (CategoryTheory.Limits.pushout.inl (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.right X₃.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.right) ⋯))))) (CategoryTheory.CategoryStruct.comp ⋯.isoPushout.hom (CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) X₃.right) (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom) X₃.right)) ((CategoryTheory.Limits.Cocone.precompose (CategoryTheory.Limits.spanExt (CategoryTheory.MonoidalCategoryStruct.associator X₁.left X₂.left X₃.right) (CategoryTheory.MonoidalCategoryStruct.associator X₁.right X₂.left X₃.right) (CategoryTheory.MonoidalCategoryStruct.associator X₁.left X₂.right X₃.right) ⋯ ⋯).hom).obj (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right (CategoryTheory.Limits.pushout.inr (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) (CategoryTheory.Limits.pushout.inl (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.right X₃.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.right) ⋯)))) (CategoryTheory.Limits.pushout.inr (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.right X₃.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.right) ⋯))) ⋯)))) ⋯
CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.associator_inv_left 📋 Mathlib.CategoryTheory.Monoidal.PushoutProduct
{C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasPushouts C] [CategoryTheory.MonoidalCategory C] (X₁ X₂ X₃ : CategoryTheory.Arrow C) [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) (CategoryTheory.MonoidalCategory.tensorRight X₃.left)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) (CategoryTheory.MonoidalCategory.tensorRight X₃.right)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom)) (CategoryTheory.MonoidalCategory.tensorLeft X₁.left)] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom)) (CategoryTheory.MonoidalCategory.tensorLeft X₁.right)] : (CategoryTheory.MonoidalCategory.Arrow.PushoutProduct.associator X₁ X₂ X₃).inv.left = CategoryTheory.Limits.pushout.desc (CategoryTheory.CategoryStruct.comp ⋯.isoPushout.hom (CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.span (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₂.hom X₃.left)) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₂.left X₃.hom))) ((CategoryTheory.Limits.Cocone.precompose (CategoryTheory.Limits.spanExt (CategoryTheory.MonoidalCategoryStruct.associator X₁.right X₂.left X₃.left).symm (CategoryTheory.MonoidalCategoryStruct.associator X₁.right X₂.right X₃.left).symm (CategoryTheory.MonoidalCategoryStruct.associator X₁.right X₂.left X₃.right).symm ⋯ ⋯).hom).obj (CategoryTheory.Limits.PushoutCocone.mk (CategoryTheory.Limits.pushout.inl (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right X₂.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.right) ⋯) X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) X₃.hom)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.pushout.inl (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) X₃.right) (CategoryTheory.Limits.pushout.inr (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right X₂.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.right) ⋯) X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) X₃.hom))) ⋯)))) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.associator X₁.left X₂.right X₃.right).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.pushout.inr (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) X₃.right) (CategoryTheory.Limits.pushout.inr (CategoryTheory.MonoidalCategoryStruct.whiskerRight (CategoryTheory.Limits.pushout.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.right X₂.hom) (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.right) ⋯) X₃.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (CategoryTheory.Limits.pushout (CategoryTheory.MonoidalCategoryStruct.whiskerRight X₁.hom X₂.left) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X₁.left X₂.hom)) X₃.hom)))) ⋯
CategoryTheory.BicartesianSq.isColimit' 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z} (self : CategoryTheory.BicartesianSq f g h i) : Nonempty (CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk h i ⋯))
CategoryTheory.BicartesianSq.mk 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq
{C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z} (toIsPullback : CategoryTheory.IsPullback f g h i) (isColimit' : Nonempty (CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.PushoutCocone.mk h i ⋯))) : CategoryTheory.BicartesianSq f g h i

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c