ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>(α_ R.X (P.tensorBimod Q).X L.X).inv ≫ (P.tensorBimod Q).actLeft ▷ L.X ≫ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ TensorBimod.X Q L) ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯ = R.X ◁ coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X) ((α_ (P.tensorBimod Q).X T.X L.X).hom ≫ (P.tensorBimod Q).X ◁ L.actLeft) ≫ R.X ◁ coequalizer.desc ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ TensorBimod.X Q L) ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ⋯ ≫ TensorBimod.actLeft P (Q.tensorBimod L)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>(α_ R.X (P.tensorBimod Q).X L.X).inv ≫ (P.tensorBimod Q).actLeft ▷ L.X ≫ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ TensorBimod.X Q L) ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯ = (R.X ◁ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ R.X ◁ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ TensorBimod.X Q L) ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ≫ TensorBimod.actLeft P (Q.tensorBimod L)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>(tensorRight L.X ⋙ tensorLeft R.X).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫ (α_ R.X (P.tensorBimod Q).X L.X).inv ≫ (P.tensorBimod Q).actLeft ▷ L.X ≫ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ TensorBimod.X Q L) ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯ = (tensorRight L.X ⋙ tensorLeft R.X).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫ (R.X ◁ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ R.X ◁ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ TensorBimod.X Q L) ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ≫ TensorBimod.actLeft P (Q.tensorBimod L)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫ (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) L.X).inv ≫ TensorBimod.actLeft P Q ▷ L.X ≫ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯ = R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫ (R.X ◁ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ R.X ◁ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ≫ TensorBimod.actLeft P (Q.tensorBimod L)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>((((α_ R.X (P.X ⊗ Q.X) L.X).inv ≫ (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ L.X) ≫ TensorBimod.actLeft P Q ▷ L.X) ≫ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv) ≫ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯ = R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫ (R.X ◁ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ R.X ◁ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ≫ TensorBimod.actLeft P (Q.tensorBimod L)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>(α_ R.X (P.X ⊗ Q.X) L.X).inv ≫ (((α_ R.X P.X Q.X).inv ▷ L.X ≫ P.actLeft ▷ Q.X ▷ L.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X) ≫ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv) ≫ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯ = R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫ (R.X ◁ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ R.X ◁ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ≫ TensorBimod.actLeft P (Q.tensorBimod L)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>(α_ R.X (P.X ⊗ Q.X) L.X).inv ≫ (α_ R.X P.X Q.X).inv ▷ L.X ≫ P.actLeft ▷ Q.X ▷ L.X ≫ (α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L) = R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫ (R.X ◁ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ R.X ◁ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ≫ TensorBimod.actLeft P (Q.tensorBimod L)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>(α_ R.X (P.X ⊗ Q.X) L.X).inv ≫ (α_ R.X P.X Q.X).inv ▷ L.X ≫ (((α_ (R.X ⊗ P.X) Q.X L.X).hom ≫ P.actLeft ▷ (Q.X ⊗ L.X)) ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L) = R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫ (R.X ◁ (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ R.X ◁ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ≫ TensorBimod.actLeft P (Q.tensorBimod L)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>(α_ R.X (P.X ⊗ Q.X) L.X).inv ≫ (α_ R.X P.X Q.X).inv ▷ L.X ≫ (((α_ (R.X ⊗ P.X) Q.X L.X).hom ≫ P.actLeft ▷ (Q.X ⊗ L.X)) ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L) = (R.X ◁ (α_ P.X Q.X L.X).hom ≫ R.X ◁ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ R.X ◁ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ≫ TensorBimod.actLeft P (Q.tensorBimod L)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>(α_ R.X (P.X ⊗ Q.X) L.X).inv ≫ (α_ R.X P.X Q.X).inv ▷ L.X ≫ (((α_ (R.X ⊗ P.X) Q.X L.X).hom ≫ P.actLeft ▷ (Q.X ⊗ L.X)) ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L) = R.X ◁ (α_ P.X Q.X L.X).hom ≫ R.X ◁ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ (α_ R.X P.X (Q.tensorBimod L).X).inv ≫ P.actLeft ▷ (Q.tensorBimod L).X ≫ coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X) ((α_ P.X S.X (Q.tensorBimod L).X).hom ≫ P.X ◁ (Q.tensorBimod L).actLeft)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>(α_ R.X (P.X ⊗ Q.X) L.X).inv ≫ (α_ R.X P.X Q.X).inv ▷ L.X ≫ (((α_ (R.X ⊗ P.X) Q.X L.X).hom ≫ P.actLeft ▷ (Q.X ⊗ L.X)) ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L) = R.X ◁ (α_ P.X Q.X L.X).hom ≫ (((α_ R.X P.X (Q.X ⊗ L.X)).inv ≫ (R.X ⊗ P.X) ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫ P.actLeft ▷ (Q.tensorBimod L).X) ≫ coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X) ((α_ P.X S.X (Q.tensorBimod L).X).hom ≫ P.X ◁ (Q.tensorBimod L).actLeft)
<VAR>C : Type u₁<VAR>inst✝⁴ : Category.{v₁, u₁} C<VAR>inst✝³ : MonoidalCategory C<VAR>A B : Mon_ C<VAR>M : Bimod A B<VAR>inst✝² : HasCoequalizers C<VAR>inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)<VAR>inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)<VAR>R S T U : Mon_ C<VAR>P : Bimod R S<VAR>Q : Bimod S T<VAR>L : Bimod T U<GOAL>(α_ R.X (P.X ⊗ Q.X) L.X).inv ≫ (α_ R.X P.X Q.X).inv ▷ L.X ≫ (((α_ (R.X ⊗ P.X) Q.X L.X).hom ≫ P.actLeft ▷ (Q.X ⊗ L.X)) ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L) = R.X ◁ (α_ P.X Q.X L.X).hom ≫ (α_ R.X P.X (Q.X ⊗ L.X)).inv ≫ (P.actLeft ▷ (Q.X ⊗ L.X) ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫ coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X) ((α_ P.X S.X (Q.tensorBimod L).X).hom ≫ P.X ◁ (Q.tensorBimod L).actLeft)
<VAR>X : Type u_1<VAR>Y : Type u_2<VAR>ι : Type u_3<VAR>ι' : Sort u_4<VAR>inst✝ : TopologicalSpace X<VAR>s t : Set X<VAR>hs : IsGδ s<VAR>ht : IsGδ t<GOAL>IsGδ (s ∩ t)
<VAR>X : Type u_1<VAR>Y : Type u_2<VAR>ι : Type u_3<VAR>ι' : Sort u_4<VAR>inst✝ : TopologicalSpace X<VAR>s t : Set X<VAR>hs : IsGδ s<VAR>ht : IsGδ t<GOAL>IsGδ (⋂ b, bif b then s else t)
<VAR>J : Type v<VAR>inst✝ : SmallCategory J<VAR>F : J ⥤ TopCat<VAR>c : Cocone F<VAR>hc : IsColimit c<VAR>homeo : ↑c.pt ≃ₜ ↑(colimitCocone F).pt := homeoOfIso (hc.coconePointUniqueUpToIso (colimitCoconeIsColimit F))<VAR>x✝ : Set ↑c.pt<GOAL>IsOpen x✝ ↔ IsOpen x✝
<VAR>J : Type v<VAR>inst✝ : SmallCategory J<VAR>F : J ⥤ TopCat<VAR>c : Cocone F<VAR>hc : IsColimit c<VAR>homeo : ↑c.pt ≃ₜ ↑(colimitCocone F).pt := homeoOfIso (hc.coconePointUniqueUpToIso (colimitCoconeIsColimit F))<VAR>x✝ : Set ↑c.pt<GOAL>IsOpen (⇑homeo.symm ⁻¹' x✝) ↔ ∀ (i : J), IsOpen x✝
<VAR>K : Type u_1<VAR>inst✝¹ : Field K<VAR>inst✝ : NumberField K<VAR>B : ℝ<GOAL>volume (convexBodySum K B) = ↑(convexBodySumFactor K) * ENNReal.ofReal B ^ finrank ℚ K
<VAR>K : Type u_1<VAR>inst✝¹ : Field K<VAR>inst✝ : NumberField K<VAR>B : ℝ<VAR>hB : B ≤ 0<GOAL>volume (convexBodySum K B) = ↑(convexBodySumFactor K) * ENNReal.ofReal B ^ finrank ℚ K
<VAR>K : Type u_1<VAR>inst✝¹ : Field K<VAR>inst✝ : NumberField K<VAR>B : ℝ<VAR>hB : 0 < B<GOAL>volume (convexBodySum K B) = ↑(convexBodySumFactor K) * ENNReal.ofReal B ^ finrank ℚ K
<VAR>V : Type u<VAR>V' : Type v<VAR>G : SimpleGraph V<VAR>G' : SimpleGraph V'<VAR>u v w : V<VAR>p : G.Walk u v<GOAL>p.reverse.toSubgraph = p.toSubgraph
<VAR>R : Type u<VAR>inst✝¹ : CommRing R<VAR>W : WeierstrassCurve R<VAR>A : Type v<VAR>inst✝ : CommRing A<VAR>φ : R →+* A<GOAL>(W.map φ).b₆ = φ W.b₆
<VAR>R : Type u<VAR>inst✝¹ : CommRing R<VAR>W : WeierstrassCurve R<VAR>A : Type v<VAR>inst✝ : CommRing A<VAR>φ : R →+* A<GOAL>φ W.a₃ ^ 2 + 4 * φ W.a₆ = φ (W.a₃ ^ 2 + 4 * W.a₆)
<VAR>b : Board<VAR>m : ℤ × ℤ<VAR>h : m ∈ left b<GOAL>(m.1, m.2 - 1) ∈ Finset.erase b m
<VAR>b : Board<VAR>m : ℤ × ℤ<VAR>h : m ∈ b ∧ (m.1, m.2 - 1) ∈ b<GOAL>(m.1, m.2 - 1) ∈ Finset.erase b m
<VAR>b : Board<VAR>m : ℤ × ℤ<VAR>h : m ∈ b ∧ (m.1, m.2 - 1) ∈ b<GOAL>(m.1, m.2 - 1) ≠ m
<VAR>ι : Type u_1<VAR>ι' : Type u_2<VAR>α : ι → Type u_3<VAR>inst✝⁴ : Fintype ι<VAR>m : (i : ι) → OuterMeasure (α i)<VAR>inst✝³ : (i : ι) → MeasurableSpace (α i)<VAR>μ : (i : ι) → Measure (α i)<VAR>inst✝² : ∀ (i : ι), SigmaFinite (μ i)<VAR>inst✝¹ : (i : ι) → PartialOrder (α i)<VAR>inst✝ : ∀ (i : ι), NoAtoms (μ i)<VAR>f : (i : ι) → α i<GOAL>(univ.pi fun i => Iio (f i)) =ᶠ[ae (Measure.pi μ)] Iic f
<VAR>ι : Type u_1<VAR>ι' : Type u_2<VAR>α : ι → Type u_3<VAR>inst✝⁴ : Fintype ι<VAR>m : (i : ι) → OuterMeasure (α i)<VAR>inst✝³ : (i : ι) → MeasurableSpace (α i)<VAR>μ : (i : ι) → Measure (α i)<VAR>inst✝² : ∀ (i : ι), SigmaFinite (μ i)<VAR>inst✝¹ : (i : ι) → PartialOrder (α i)<VAR>inst✝ : ∀ (i : ι), NoAtoms (μ i)<VAR>f : (i : ι) → α i<GOAL>(univ.pi fun i => Iio (f i)) =ᶠ[ae (Measure.pi μ)] univ.pi fun i => Iic (f i)
<VAR>F : Type u_1<VAR>ι : Type u_2<VAR>α : Type u_3<VAR>β : Type u_4<VAR>γ : Type u_5<VAR>f : α → β → β<VAR>op : α → α → α<VAR>inst✝³ : Monoid α<VAR>inst✝² : Monoid β<VAR>inst✝¹ : FunLike F α β<VAR>inst✝ : DecidableEq α<VAR>s : Multiset α<VAR>a : α<VAR>h : a ∈ s<VAR>comm : {x | x ∈ s}.Pairwise Commute<VAR>comm' : optParam (∀ x ∈ {x | x ∈ s.erase a}, ∀ y ∈ {x | x ∈ s.erase a}, x ≠ y → Commute x y) ⋯<GOAL>a * (s.erase a).noncommProd comm' = s.noncommProd comm
<VAR>F : Type u_1<VAR>ι : Type u_2<VAR>α : Type u_3<VAR>β : Type u_4<VAR>γ : Type u_5<VAR>f : α → β → β<VAR>op : α → α → α<VAR>inst✝³ : Monoid α<VAR>inst✝² : Monoid β<VAR>inst✝¹ : FunLike F α β<VAR>inst✝ : DecidableEq α<VAR>a : α<VAR>l : List α<VAR>h : a ∈ ⟦l⟧<VAR>comm : {x | x ∈ ⟦l⟧}.Pairwise Commute<VAR>comm' : optParam (∀ x ∈ {x | x ∈ erase ⟦l⟧ a}, ∀ y ∈ {x | x ∈ erase ⟦l⟧ a}, x ≠ y → Commute x y) ⋯<GOAL>a * (erase ⟦l⟧ a).noncommProd comm' = noncommProd ⟦l⟧ comm
<VAR>F : Type u_1<VAR>ι : Type u_2<VAR>α : Type u_3<VAR>β : Type u_4<VAR>γ : Type u_5<VAR>f : α → β → β<VAR>op : α → α → α<VAR>inst✝³ : Monoid α<VAR>inst✝² : Monoid β<VAR>inst✝¹ : FunLike F α β<VAR>inst✝ : DecidableEq α<VAR>a : α<VAR>l : List α<VAR>comm✝ : {x | x ∈ ⟦l⟧}.Pairwise Commute<VAR>comm' : optParam (∀ x ∈ {x | x ∈ erase ⟦l⟧ a}, ∀ y ∈ {x | x ∈ erase ⟦l⟧ a}, x ≠ y → Commute x y) ⋯<VAR>comm : {x | x ∈ l}.Pairwise Commute<VAR>h : a ∈ l<GOAL>a * (l.erase a).prod = l.prod
<VAR>F : Type u_1<VAR>ι : Type u_2<VAR>α : Type u_3<VAR>β : Type u_4<VAR>γ : Type u_5<VAR>f : α → β → β<VAR>op : α → α → α<VAR>inst✝³ : Monoid α<VAR>inst✝² : Monoid β<VAR>inst✝¹ : FunLike F α β<VAR>inst✝ : DecidableEq α<VAR>a : α<VAR>l : List α<VAR>comm✝ : {x | x ∈ ⟦l⟧}.Pairwise Commute<VAR>comm' : optParam (∀ x ∈ {x | x ∈ erase ⟦l⟧ a}, ∀ y ∈ {x | x ∈ erase ⟦l⟧ a}, x ≠ y → Commute x y) ⋯<VAR>comm : {x | x ∈ l}.Pairwise Commute<VAR>h : a ∈ l<GOAL>∀ x ∈ l, ∀ y ∈ l, x * y = y * x
<VAR>F : Type u_1<VAR>ι : Type u_2<VAR>α : Type u_3<VAR>β : Type u_4<VAR>γ : Type u_5<VAR>f : α → β → β<VAR>op : α → α → α<VAR>inst✝³ : Monoid α<VAR>inst✝² : Monoid β<VAR>inst✝¹ : FunLike F α β<VAR>inst✝ : DecidableEq α<VAR>a : α<VAR>l : List α<VAR>comm✝ : {x | x ∈ ⟦l⟧}.Pairwise Commute<VAR>comm' : optParam (∀ x ∈ {x | x ∈ erase ⟦l⟧ a}, ∀ y ∈ {x | x ∈ erase ⟦l⟧ a}, x ≠ y → Commute x y) ⋯<VAR>comm : {x | x ∈ l}.Pairwise Commute<VAR>h : a ∈ l<VAR>x : α<VAR>hx : x ∈ l<VAR>y : α<VAR>hy : y ∈ l<GOAL>x * y = y * x
<VAR>F : Type u_1<VAR>ι : Type u_2<VAR>α : Type u_3<VAR>β : Type u_4<VAR>γ : Type u_5<VAR>f : α → β → β<VAR>op : α → α → α<VAR>inst✝³ : Monoid α<VAR>inst✝² : Monoid β<VAR>inst✝¹ : FunLike F α β<VAR>inst✝ : DecidableEq α<VAR>a : α<VAR>l : List α<VAR>comm✝ : {x | x ∈ ⟦l⟧}.Pairwise Commute<VAR>comm' : optParam (∀ x ∈ {x | x ∈ erase ⟦l⟧ a}, ∀ y ∈ {x | x ∈ erase ⟦l⟧ a}, x ≠ y → Commute x y) ⋯<VAR>comm : {x | x ∈ l}.Pairwise Commute<VAR>h : a ∈ l<VAR>x : α<VAR>hx hy : x ∈ l<GOAL>x * x = x * x
<VAR>F : Type u_1<VAR>ι : Type u_2<VAR>α : Type u_3<VAR>β : Type u_4<VAR>γ : Type u_5<VAR>f : α → β → β<VAR>op : α → α → α<VAR>inst✝³ : Monoid α<VAR>inst✝² : Monoid β<VAR>inst✝¹ : FunLike F α β<VAR>inst✝ : DecidableEq α<VAR>a : α<VAR>l : List α<VAR>comm✝ : {x | x ∈ ⟦l⟧}.Pairwise Commute<VAR>comm' : optParam (∀ x ∈ {x | x ∈ erase ⟦l⟧ a}, ∀ y ∈ {x | x ∈ erase ⟦l⟧ a}, x ≠ y → Commute x y) ⋯<VAR>comm : {x | x ∈ l}.Pairwise Commute<VAR>h : a ∈ l<VAR>x : α<VAR>hx : x ∈ l<VAR>y : α<VAR>hy : y ∈ l<VAR>hxy : x ≠ y<GOAL>x * y = y * x
<VAR>n k p : ℕ<VAR>inst✝ : Fact (Nat.Prime p)<GOAL>n.choose k ≡ (n % p).choose (k % p) * (n / p).choose (k / p) [MOD p]
<VAR>n k p : ℕ<VAR>inst✝ : Fact (Nat.Prime p)<GOAL>↑(n.choose k) ≡ ↑((n % p).choose (k % p) * (n / p).choose (k / p)) [ZMOD ↑p]
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>m : MeasurableSpace α<VAR>μ✝ ν : Measure α<VAR>s✝ t s : SignedMeasure α<VAR>μ : Measure α<GOAL>Measurable (s.rnDeriv μ)
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>m : MeasurableSpace α<VAR>μ✝ ν : Measure α<VAR>s✝ t s : SignedMeasure α<VAR>μ : Measure α<GOAL>Measurable fun x => (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal - (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>γ : Type u_3<VAR>a a'✝ a' : α<GOAL>a' ∈ (pure a).support ↔ a' ∈ {a}
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>γ : Type u_3<VAR>ι : Sort u_4<VAR>π : α → Type u_5<VAR>s✝ s₁ s₂ : Set α<VAR>t : Set β<VAR>f✝ : α → β<VAR>inst✝ : Nonempty α<VAR>f : α → β<VAR>s : Set α<VAR>x : α<VAR>hx : x ∈ s<VAR>x' : α<VAR>hx' : x' ∈ s<VAR>he : invFunOn f s (f x) = invFunOn f s (f x')<GOAL>f x = f x'