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>V W X Y Z : Mon_ C<VAR>M : Bimod V W<VAR>N : Bimod W X<VAR>P : Bimod X Y<VAR>Q : Bimod Y Z<GOAL>(α_ M.X N.X P.X).hom ▷ Q.X ≫ (M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) ▷ Q.X ≫ ((α_ M.X (TensorBimod.X N P) Q.X).hom ≫ M.X ◁ coequalizer.π (TensorBimod.actRight N P ▷ Q.X) ((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q) ((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q) ((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q) (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q)) ((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)) ((M.X ⊗ W.X) ◁ coequalizer.desc ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫ coequalizer.desc ((α_ N.X P.X Q.X).hom ≫ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ coequalizer.π (N.actRight ▷ TensorBimod.X P Q) ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ⋯) ⋯) (M.X ◁ coequalizer.desc ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫ coequalizer.desc ((α_ N.X P.X Q.X).hom ≫ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ coequalizer.π (N.actRight ▷ TensorBimod.X P Q) ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ⋯) ⋯) ⋯ ⋯) = coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫ (α_ (TensorBimod.X M N) P.X Q.X).hom ≫ TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ (PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X (TensorBimod.X P Q)).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ TensorBimod.X P Q) ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q)) ((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) ⋯
<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>V W X Y Z : Mon_ C<VAR>M : Bimod V W<VAR>N : Bimod W X<VAR>P : Bimod X Y<VAR>Q : Bimod Y Z<GOAL>(α_ M.X N.X P.X).hom ▷ Q.X ≫ (α_ M.X (N.X ⊗ P.X) Q.X).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ▷ Q.X ≫ M.X ◁ coequalizer.π (TensorBimod.actRight N P ▷ Q.X) ((α_ (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Y.X Q.X).hom ≫ coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ◁ Q.actLeft) ≫ M.X ◁ coequalizer.desc ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫ coequalizer.desc ((α_ N.X P.X Q.X).hom ≫ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ⋯) ⋯ ≫ colimit.ι (parallelPair (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) WalkingParallelPair.one = coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫ (α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫ coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ (PreservesCoequalizer.iso (tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫ coequalizer.π (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) ⋯
<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>V W X Y Z : Mon_ C<VAR>M : Bimod V W<VAR>N : Bimod W X<VAR>P : Bimod X Y<VAR>Q : Bimod Y Z<GOAL>(α_ M.X N.X P.X).hom ▷ Q.X ≫ (α_ M.X (N.X ⊗ P.X) Q.X).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ▷ Q.X ≫ M.X ◁ ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫ coequalizer.desc ((α_ N.X P.X Q.X).hom ≫ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ⋯) ≫ colimit.ι (parallelPair (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) WalkingParallelPair.one = coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫ (α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫ coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ (PreservesCoequalizer.iso (tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫ coequalizer.π (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) ⋯
<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>V W X Y Z : Mon_ C<VAR>M : Bimod V W<VAR>N : Bimod W X<VAR>P : Bimod X Y<VAR>Q : Bimod Y Z<GOAL>(α_ M.X N.X P.X).hom ▷ Q.X ≫ (α_ M.X (N.X ⊗ P.X) Q.X).hom ≫ (M.X ◁ (α_ N.X P.X Q.X).hom ≫ M.X ◁ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ≫ colimit.ι (parallelPair (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) WalkingParallelPair.one = coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫ (α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)) P.X Q.X).hom ≫ coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ (PreservesCoequalizer.iso (tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫ coequalizer.π (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) ⋯
<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>V W X Y Z : Mon_ C<VAR>M : Bimod V W<VAR>N : Bimod W X<VAR>P : Bimod X Y<VAR>Q : Bimod Y Z<GOAL>(α_ M.X N.X P.X).hom ▷ Q.X ≫ (α_ M.X (N.X ⊗ P.X) Q.X).hom ≫ (M.X ◁ (α_ N.X P.X Q.X).hom ≫ M.X ◁ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ≫ colimit.ι (parallelPair (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) WalkingParallelPair.one = ((((α_ (M.X ⊗ N.X) P.X Q.X).hom ≫ coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ (P.X ⊗ Q.X)) ≫ coequalizer (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫ (PreservesCoequalizer.iso (tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv) ≫ coequalizer.desc ((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫ coequalizer.π (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) ⋯
<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>V W X Y Z : Mon_ C<VAR>M : Bimod V W<VAR>N : Bimod W X<VAR>P : Bimod X Y<VAR>Q : Bimod Y Z<GOAL>(α_ M.X N.X P.X).hom ▷ Q.X ≫ (α_ M.X (N.X ⊗ P.X) Q.X).hom ≫ (M.X ◁ (α_ N.X P.X Q.X).hom ≫ M.X ◁ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ≫ colimit.ι (parallelPair (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) WalkingParallelPair.one = (α_ (M.X ⊗ N.X) P.X Q.X).hom ≫ (((M.X ⊗ N.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫ (PreservesCoequalizer.iso (tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv) ≫ coequalizer.desc ((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫ coequalizer.π (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) ⋯
<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>V W X Y Z : Mon_ C<VAR>M : Bimod V W<VAR>N : Bimod W X<VAR>P : Bimod X Y<VAR>Q : Bimod Y Z<GOAL>(α_ M.X N.X P.X).hom ▷ Q.X ≫ (α_ M.X (N.X ⊗ P.X) Q.X).hom ≫ (M.X ◁ (α_ N.X P.X Q.X).hom ≫ M.X ◁ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ≫ colimit.ι (parallelPair (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) WalkingParallelPair.one = (α_ (M.X ⊗ N.X) P.X Q.X).hom ≫ (M.X ⊗ N.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ (α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫ coequalizer.π (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
<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>V W X Y Z : Mon_ C<VAR>M : Bimod V W<VAR>N : Bimod W X<VAR>P : Bimod X Y<VAR>Q : Bimod Y Z<GOAL>(α_ M.X N.X P.X).hom ▷ Q.X ≫ (α_ M.X (N.X ⊗ P.X) Q.X).hom ≫ (M.X ◁ (α_ N.X P.X Q.X).hom ≫ M.X ◁ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ≫ colimit.ι (parallelPair (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) WalkingParallelPair.one = (α_ (M.X ⊗ N.X) P.X Q.X).hom ≫ (((α_ M.X N.X (P.X ⊗ Q.X)).hom ≫ M.X ◁ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫ M.X ◁ coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ≫ coequalizer.π (M.actRight ▷ coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ((α_ M.X W.X (coequalizer (N.actRight ▷ (P.tensorBimod Q).X) ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
<VAR>θ : ℂ<GOAL>cos θ ≠ 0 ↔ ∀ (k : ℤ), θ ≠ (2 * ↑k + 1) * ↑π / 2
<VAR>p : ℕ<VAR>G : Type u_1<VAR>inst✝² : Group G<VAR>hG : IsPGroup p G<VAR>hp : Fact (Nat.Prime p)<VAR>α : Type u_2<VAR>inst✝¹ : MulAction G α<VAR>a : α<VAR>inst✝ : Finite ↑(orbit G a)<VAR>ϕ : ↑(orbit G a) ≃ G ⧸ stabilizer G a := orbitEquivQuotientStabilizer G a<GOAL>∃ n, Nat.card ↑(orbit G a) = p ^ n
<VAR>p : ℕ<VAR>G : Type u_1<VAR>inst✝² : Group G<VAR>hG : IsPGroup p G<VAR>hp : Fact (Nat.Prime p)<VAR>α : Type u_2<VAR>inst✝¹ : MulAction G α<VAR>a : α<VAR>inst✝ : Finite ↑(orbit G a)<VAR>ϕ : ↑(orbit G a) ≃ G ⧸ stabilizer G a := orbitEquivQuotientStabilizer G a<VAR>this : Finite (G ⧸ stabilizer G a)<GOAL>∃ n, Nat.card ↑(orbit G a) = p ^ n
<VAR>p : ℕ<VAR>G : Type u_1<VAR>inst✝² : Group G<VAR>hG : IsPGroup p G<VAR>hp : Fact (Nat.Prime p)<VAR>α : Type u_2<VAR>inst✝¹ : MulAction G α<VAR>a : α<VAR>inst✝ : Finite ↑(orbit G a)<VAR>ϕ : ↑(orbit G a) ≃ G ⧸ stabilizer G a := orbitEquivQuotientStabilizer G a<VAR>this✝ : Finite (G ⧸ stabilizer G a)<VAR>this : (stabilizer G a).FiniteIndex<GOAL>∃ n, Nat.card ↑(orbit G a) = p ^ n
<VAR>p : ℕ<VAR>G : Type u_1<VAR>inst✝² : Group G<VAR>hG : IsPGroup p G<VAR>hp : Fact (Nat.Prime p)<VAR>α : Type u_2<VAR>inst✝¹ : MulAction G α<VAR>a : α<VAR>inst✝ : Finite ↑(orbit G a)<VAR>ϕ : ↑(orbit G a) ≃ G ⧸ stabilizer G a := orbitEquivQuotientStabilizer G a<VAR>this✝ : Finite (G ⧸ stabilizer G a)<VAR>this : (stabilizer G a).FiniteIndex<GOAL>∃ n, Nat.card (G ⧸ stabilizer G a) = p ^ n
<VAR>F : Type w<VAR>R : Type u<VAR>S : Type v<VAR>T : Type u_1<VAR>inst✝⁴ : NonUnitalNonAssocRing R<VAR>inst✝³ : NonUnitalNonAssocRing S<VAR>inst✝² : NonUnitalNonAssocRing T<VAR>inst✝¹ : FunLike F R S<VAR>inst✝ : NonUnitalRingHomClass F R S<VAR>g : S →ₙ+* T<VAR>f : R →ₙ+* S<VAR>s : Set R<VAR>p : ↥(closure s) → Prop<VAR>a : ↥(closure s)<VAR>mem : ∀ (x : R) (hx : x ∈ s), p ⟨x, ⋯⟩<VAR>zero : p 0<VAR>add : ∀ (x y : ↥(closure s)), p x → p y → p (x + y)<VAR>neg : ∀ (x : ↥(closure s)), p x → p (-x)<VAR>mul : ∀ (x y : ↥(closure s)), p x → p y → p (x * y)<VAR>b : R<VAR>hb : b ∈ closure s<GOAL>p ⟨b, hb⟩
<VAR>F : Type w<VAR>R : Type u<VAR>S : Type v<VAR>T : Type u_1<VAR>inst✝⁴ : NonUnitalNonAssocRing R<VAR>inst✝³ : NonUnitalNonAssocRing S<VAR>inst✝² : NonUnitalNonAssocRing T<VAR>inst✝¹ : FunLike F R S<VAR>inst✝ : NonUnitalRingHomClass F R S<VAR>g : S →ₙ+* T<VAR>f : R →ₙ+* S<VAR>s : Set R<VAR>p : ↥(closure s) → Prop<VAR>a : ↥(closure s)<VAR>mem : ∀ (x : R) (hx : x ∈ s), p ⟨x, ⋯⟩<VAR>zero : p 0<VAR>add : ∀ (x y : ↥(closure s)), p x → p y → p (x + y)<VAR>neg : ∀ (x : ↥(closure s)), p x → p (-x)<VAR>mul : ∀ (x y : ↥(closure s)), p x → p y → p (x * y)<VAR>b : R<VAR>hb : b ∈ closure s<GOAL>∃ (x : b ∈ closure s), p ⟨b, x⟩
<VAR>F : Type w<VAR>R : Type u<VAR>S : Type v<VAR>T : Type u_1<VAR>inst✝⁴ : NonUnitalNonAssocRing R<VAR>inst✝³ : NonUnitalNonAssocRing S<VAR>inst✝² : NonUnitalNonAssocRing T<VAR>inst✝¹ : FunLike F R S<VAR>inst✝ : NonUnitalRingHomClass F R S<VAR>g : S →ₙ+* T<VAR>f : R →ₙ+* S<VAR>s : Set R<VAR>p : ↥(closure s) → Prop<VAR>a : ↥(closure s)<VAR>mem : ∀ (x : R) (hx : x ∈ s), p ⟨x, ⋯⟩<VAR>zero : p 0<VAR>add : ∀ (x y : ↥(closure s)), p x → p y → p (x + y)<VAR>neg : ∀ (x : ↥(closure s)), p x → p (-x)<VAR>mul : ∀ (x y : ↥(closure s)), p x → p y → p (x * y)<VAR>b : R<VAR>hb : b ∈ closure s<GOAL>∀ (x y : R), (∃ (x_1 : x ∈ closure s), p ⟨x, x_1⟩) → (∃ (x : y ∈ closure s), p ⟨y, x⟩) → ∃ (x_1 : x + y ∈ closure s), p ⟨x + y, x_1⟩
<VAR>F : Type w<VAR>R : Type u<VAR>S : Type v<VAR>T : Type u_1<VAR>inst✝⁴ : NonUnitalNonAssocRing R<VAR>inst✝³ : NonUnitalNonAssocRing S<VAR>inst✝² : NonUnitalNonAssocRing T<VAR>inst✝¹ : FunLike F R S<VAR>inst✝ : NonUnitalRingHomClass F R S<VAR>g : S →ₙ+* T<VAR>f : R →ₙ+* S<VAR>s : Set R<VAR>p : ↥(closure s) → Prop<VAR>a : ↥(closure s)<VAR>mem : ∀ (x : R) (hx : x ∈ s), p ⟨x, ⋯⟩<VAR>zero : p 0<VAR>add : ∀ (x y : ↥(closure s)), p x → p y → p (x + y)<VAR>neg : ∀ (x : ↥(closure s)), p x → p (-x)<VAR>mul : ∀ (x y : ↥(closure s)), p x → p y → p (x * y)<VAR>b : R<VAR>hb : b ∈ closure s<GOAL>∀ (x : R), (∃ (x_1 : x ∈ closure s), p ⟨x, x_1⟩) → ∃ (x_1 : -x ∈ closure s), p ⟨-x, x_1⟩
<VAR>F : Type w<VAR>R : Type u<VAR>S : Type v<VAR>T : Type u_1<VAR>inst✝⁴ : NonUnitalNonAssocRing R<VAR>inst✝³ : NonUnitalNonAssocRing S<VAR>inst✝² : NonUnitalNonAssocRing T<VAR>inst✝¹ : FunLike F R S<VAR>inst✝ : NonUnitalRingHomClass F R S<VAR>g : S →ₙ+* T<VAR>f : R →ₙ+* S<VAR>s : Set R<VAR>p : ↥(closure s) → Prop<VAR>a : ↥(closure s)<VAR>mem : ∀ (x : R) (hx : x ∈ s), p ⟨x, ⋯⟩<VAR>zero : p 0<VAR>add : ∀ (x y : ↥(closure s)), p x → p y → p (x + y)<VAR>neg : ∀ (x : ↥(closure s)), p x → p (-x)<VAR>mul : ∀ (x y : ↥(closure s)), p x → p y → p (x * y)<VAR>b : R<VAR>hb : b ∈ closure s<GOAL>∀ (x y : R), (∃ (x_1 : x ∈ closure s), p ⟨x, x_1⟩) → (∃ (x : y ∈ closure s), p ⟨y, x⟩) → ∃ (x_1 : x * y ∈ closure s), p ⟨x * y, x_1⟩
<VAR>α : Type u<VAR>β : Type v<VAR>X : Type u_1<VAR>inst✝² : PseudoEMetricSpace α<VAR>π : β → Type u_2<VAR>inst✝¹ : Fintype β<VAR>inst✝ : (b : β) → EDist (π b)<VAR>f g : (b : β) → π b<VAR>d : ℝ≥0∞<GOAL>(∀ b ∈ Finset.univ, edist (f b) (g b) ≤ d) ↔ ∀ (b : β), edist (f b) (g b) ≤ d