Datasets:

phanerozoic
/

Coq-CategoryTheory

fact stringlengths 11 3.53k	type stringclasses 19 values	library stringclasses 11 values	imports listlengths 1 19	filename stringclasses 222 values	symbolic_name stringlengths 1 52
Adjunction_Hom := { hom_adj : Hom C ◯ F^op ∏⟶ Id ≅[[D^op ∏ C, Sets]] Hom D ◯ Id^op ∏⟶ U }. Context `{Adjunction_Hom}.	Class	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Adjunction_Hom
Definition hom_unit : Id ⟹ U ◯ F := {\| transform := fun x => @morphism _ _ _ _ (to hom_adj (x, F x)) id \|}.	Program	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Definition
Obligation . spose (naturality[to hom_adj] (x, F x) (x, F y) (id, fmap[F] f) id) as X. rewrite id_right in X. rewrites. spose (naturality[to hom_adj] (y, F y) (x, F y) (f, id) id) as X. rewrite fmap_id, id_left in X. rewrites. apply proper_morphism; cat. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . symmetry. apply hom_unit_obligation_1. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Definition hom_counit : F ◯ U ⟹ Id := {\| transform := fun x => @morphism _ _ _ _ (from hom_adj (U x, x)) id \|}.	Program	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Definition
Obligation . spose (naturality[from hom_adj] (U x, x) (U x, y) (id, f) id) as X. rewrite fmap_id, id_right in X. rewrites. spose (naturality[from hom_adj] (U y, y) (U x, y) (fmap[U] f, id) id) as X. rewrite id_left in X. rewrites. apply proper_morphism; cat. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . symmetry. apply hom_counit_obligation_1. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
hom_unit_naturality_consequence {x y} (f : F x ~> y) : to hom_adj (x, y) f ≈ fmap[U] f ∘ hom_unit _. Proof. unfold hom_unit; simpl. spose (naturality[to hom_adj] (x, F x) (x, y) (id, f) id) as X. rewrite id_right in X. rewrites. apply proper_morphism; cat. Qed.	Theorem	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	hom_unit_naturality_consequence
hom_counit_naturality_consequence {x y} (f : x ~> U y) : from hom_adj (x, y) f ≈ hom_counit _ ∘ fmap[F] f. Proof. unfold hom_counit; simpl. spose (naturality[from hom_adj] (U y, y) (x, y) (f, id) id) as X. rewrite id_left in X. rewrites. apply proper_morphism; cat. Qed.	Theorem	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	hom_counit_naturality_consequence
hom_counit_fmap_unit {x} : hom_counit (F x) ∘ fmap[F] (hom_unit x) ≈ id. Proof. spose (@hom_counit_naturality_consequence x (F x) (hom_unit x)) as X. rewrites. unfold hom_unit; simpl. srewrite (iso_from_to hom_adj (x, F x) id); cat. Qed.	Theorem	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	hom_counit_fmap_unit
hom_fmap_counit_unit {x} : fmap[U] (hom_counit x) ∘ hom_unit (U x) ≈ id. Proof. spose (@hom_unit_naturality_consequence (U x) x (hom_counit x)) as X. rewrites. unfold hom_unit; simpl. srewrite (iso_to_from hom_adj (U x, x) id); cat. Qed.	Theorem	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	hom_fmap_counit_unit
Definition Adjunction_Hom_to_Transform : F ∹ U := {\| unit := hom_unit; counit := hom_counit; counit_fmap_unit := @hom_counit_fmap_unit; fmap_counit_unit := @hom_fmap_counit_unit \|}.	Program	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Definition
Definition Adjunction_Transform_to_Hom (A : F ∹ U) : Adjunction_Hom := {\| hom_adj := {\| to := {\| transform := fun _ => {\| morphism := fun f => fmap[U] f ∘ unit _ \|} \|} ; from := {\| transform := fun _ => {\| morphism := fun f => counit _ ∘ fmap[F] f \|} \|} \|} \|}.	Program	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Definition
Obligation . proper; rewrites; reflexivity. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . simpl. rewrite <- !comp_assoc. srewrite_r (@naturality _ _ _ _ unit _ _ h). rewrite !comp_assoc. rewrite <- !fmap_comp. reflexivity. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . simpl. rewrite <- !comp_assoc. srewrite_r (@naturality _ _ _ _ unit _ _ h). rewrite !comp_assoc. rewrite <- !fmap_comp. reflexivity. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . proper; rewrites; reflexivity. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . simpl. rewrite !comp_assoc. srewrite (@naturality _ _ _ _ counit _ _ h0). rewrite <- !comp_assoc. rewrite <- !fmap_comp. reflexivity. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . simpl. rewrite !comp_assoc. srewrite (@naturality _ _ _ _ counit _ _ h0). rewrite <- !comp_assoc. rewrite <- !fmap_comp. reflexivity. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . simpl; cat. rewrite fmap_comp. rewrite <- comp_assoc. srewrite (@naturality _ _ _ _ unit _ _ x0). rewrite comp_assoc. srewrite (@fmap_counit_unit _ _ _ _ A); cat. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . simpl; cat. rewrite fmap_comp. rewrite comp_assoc. srewrite_r (@naturality _ _ _ _ counit _ _ x0). rewrite <- comp_assoc. srewrite (@counit_fmap_unit _ _ _ _ A); cat. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Definition Adjunction_Hom_to_Universal : F ⊣ U := {\| adj := fun a b => {\| to := transform (to hom_adj) (a, b) ; from := transform (from hom_adj) (a, b) \|} \|}.	Program	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Definition
Obligation . simpl; srewrite (iso_to_from hom_adj (a, b) x); cat. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . simpl; srewrite (iso_from_to hom_adj (a, b) x); cat. Defined.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . spose (naturality (to hom_adj) (y, z) (x, z) (g, id) f) as X. rewrite fmap_id, id_left in X. rewrites. apply proper_morphism; cat. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . spose (naturality (to hom_adj) (x, y) (x, z) (id, f) g) as X. rewrite id_right in X. rewrites. apply proper_morphism; cat. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . spose (naturality (from hom_adj) (y, z) (x, z) (g, id) f) as X. rewrite id_left in X. rewrites. apply proper_morphism; cat. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . spose (naturality (from hom_adj) (x, y) (x, z) (id, f) g) as X. rewrite fmap_id, id_right in X. rewrites. apply proper_morphism; cat. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Definition Adjunction_Universal_to_Hom (A : F ⊣ U) : Adjunction_Hom := {\| hom_adj := {\| to := {\| transform := fun _ => {\| morphism := to adj \|} \|} ; from := {\| transform := fun _ => {\| morphism := from adj \|} \|} \|} \|}.	Program	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Definition
Obligation . rewrite <- comp_assoc. rewrite to_adj_nat_l. rewrite comp_assoc. rewrite to_adj_nat_r. reflexivity. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . rewrite <- comp_assoc. rewrite to_adj_nat_r. rewrite <- comp_assoc. rewrite to_adj_nat_l. reflexivity. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . rewrite <- comp_assoc. rewrite from_adj_nat_l. rewrite comp_assoc. rewrite from_adj_nat_r. reflexivity. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . rewrite <- comp_assoc. rewrite from_adj_nat_r. rewrite <- comp_assoc. rewrite from_adj_nat_l. reflexivity. Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . simpl; cat. apply (iso_to_from adj _). Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Obligation . simpl; cat. apply (iso_from_to adj _). Qed.	Next	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import ...	Adjunction/Hom.v	Obligation
Definition Opposite_Adjunction `(F : D ⟶ C) `(U : C ⟶ D) (A : F ⊣ U) : U^op ⊣ F^op := {\| adj := fun x y => {\| to := from (@adj _ _ _ _ A y x) ; from := to (@adj _ _ _ _ A y x) ; iso_to_from := iso_from_to (@adj _ _ _ _ A y x) ; iso_from_to := iso_to_from (@adj _ _ _ _ A y x) \|}; to_adj_nat_l := fun _ _ _ f g => @from_a...	Program	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Adjunction/Opposite.v	Definition
Opposite_Adjunction_invol `(F : D ⟶ C) `(U : C ⟶ D) (A : F ⊣ U) : (A^op)^op = A. Proof. reflexivity. Qed.	Corollary	Adjunction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Adjunction/Opposite.v	Opposite_Adjunction_invol
Arrow {C : Category} : Category := (Id[C] ↓ Id[C]).	Definition	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Arrow.v	Arrow
Instance Cayley : Category := { obj := C; hom := fun x y => { f : ∀ r, (y ~> r) → (x ~> r) & Proper (forall_relation (fun _ => respectful equiv equiv)) f ∧ ∀ (r : C) (k : y ~> r), f r k ≈ k ∘ f _ id }; homset := fun x y => {\| equiv := fun f g => ∀ r k, `1 f r k ≈ `1 g r k \|}; id := fun _ => (fun _ => Datatypes.id; _); ...	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Instance
Obligation . equivalence. now rewrite X, X0. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Obligation . split. - proper. - intros; cat. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Obligation . split. - proper. apply p. apply p0. apply X. - intros. symmetry. rewrite e. rewrite comp_assoc. rewrite <- e0. rewrite <- e. reflexivity. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Obligation . proper. simpl in *. rewrite H, H0. rewrite X. comp_left. apply X0. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Instance To_Cayley : C ⟶ Cayley := { fobj := fun x => x; fmap := fun _ _ f => (fun _ k => k ∘ f; _); }.	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Instance
Obligation . proper. proper. Defined.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Instance From_Cayley : Cayley ⟶ C := { fobj := fun x => x; fmap := fun _ y f => `1 f y (@id C y); }. Context `{Cayley}. (* No matter how we associate the mapped morphisms, the functor back from Cayley yields them left-associated. *)	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Instance
Cayley_Right (x y z w : C) (f : z ~> w) (g : y ~> z) (h : x ~> y) : (∀ a b (k : a ~{C}~> b), id[b] ∘ k = k) -> f ∘ g ∘ h = fmap[From_Cayley] (fmap[To_Cayley] f ∘ (fmap[To_Cayley] g ∘ fmap[To_Cayley] h)). Proof. intros. simpl. rewrite H0. reflexivity. Qed.	Lemma	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Cayley_Right
Cayley_Left (x y z w : C) (f : z ~> w) (g : y ~> z) (h : x ~> y) : (∀ a b (k : a ~{C}~> b), id[b] ∘ k = k) -> f ∘ g ∘ h = fmap[From_Cayley] (((fmap[To_Cayley] f ∘ fmap[To_Cayley] g) ∘ fmap[To_Cayley] h)). Proof. intros. simpl. rewrite H0. reflexivity. Qed.	Lemma	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Cayley_Left
Instance Cayley_Cartesian `{CA : @Cartesian C} : @Cartesian Cayley := { product_obj := @product_obj C CA; fork := fun x y z f g => let f' := to (Covariant_Yoneda_Embedding C x y) (_ f) in let g' := to (Covariant_Yoneda_Embedding C x z) (_ g) in _ f' g'; exl := fun x y => let f' := from (Covariant_Yoneda_Embedding C _ _...	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Instance
Obligation . construct. - construct. + apply f. exact X. + proper. rewrite e1. rewrite X. rewrite <- e1. reflexivity. - simpl. rewrite e1. rewrite comp_assoc. rewrite <- e1. reflexivity. - simpl. rewrite e1. rewrite <- comp_assoc. rewrite <- e1. reflexivity. Defined.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Obligation . construct. - construct. + apply g. exact X. + proper. rewrite e0. rewrite X. rewrite <- e0. reflexivity. - simpl. rewrite e0. rewrite comp_assoc. rewrite <- e0. reflexivity. - simpl. rewrite e0. rewrite <- comp_assoc. rewrite <- e0. reflexivity. Defined.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Obligation . exists (fun r h => h ∘ x0 △ x1). split. - proper. - intros; cat. Defined.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Obligation . destruct x0; simpl in *. exists (fun r h => transform r h). split. - proper. now apply proper_morphism. - intros. rewrite naturality. apply proper_morphism; cat. Defined.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Obligation . destruct x0; simpl in *. exists (fun r h => transform r h). split. - proper. now apply proper_morphism. - intros. rewrite naturality. apply proper_morphism; cat. Defined.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Obligation . proper; simpl in *. comp_left. apply fork_respects. - apply X. - apply X0. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Obligation . proper; simpl in *. - rewrite X. rewrite <- comp_assoc. rewrite exl_fork. rewrite <- e1. reflexivity. - rewrite X. rewrite <- comp_assoc. rewrite exr_fork. rewrite <- e0. reflexivity. - rewrite <- X, <- H0. rewrite e. comp_left. rewrite (e _ (_ ∘ exl)). rewrite (e _ (_ ∘ exr)). cat. rewrite fork_comp. rewr...	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Obligation
Instance To_Cayley_CartesianFunctor `{@Cartesian C} : @CartesianFunctor _ _ To_Cayley _ Cayley_Cartesian. #[export]	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Instance
Instance From_Cayley_CartesianFunctor `{@Cartesian C} : @CartesianFunctor _ _ From_Cayley Cayley_Cartesian _.	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Cayley.v	Instance
Definition Comma : Category := {\| obj := ∃ p : A ∏ B, S (fst p) ~{C}~> T (snd p); hom := fun x y => ∃ f : (fst (`1 x) ~{A}~> fst (`1 y)) * (snd (`1 x) ~{B}~> snd (`1 y)), `2 y ∘ fmap[S] (fst f) ≈ fmap[T] (snd f) ∘ `2 x; homset := fun _ _ => {\| equiv := fun f g => (fst `1 f ≈ fst `1 g) * (snd `1 f ≈ snd `1 g) \|}; id := ...	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Definition
Obligation . intros [[]] [[]]; simpl in *; equivalence. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . intros. simpl. rewrite !fmap_id. rewrite id_left, id_right. reflexivity. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . intros ? ? ?; simpl. intros [[]] [[]]; simpl in *. rewrite !fmap_comp. rewrite comp_assoc. rewrites. rewrite <- !comp_assoc. rewrites. reflexivity. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . intros ? ? ? ? ? [e0 e1] ? ? [e2 e3]. split. - now simpl; rewrite e0, e2. - now simpl; rewrite e1, e3. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . intros; simpl. split; now rewrite id_left. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . intros. simpl. split; now rewrite id_right. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . intros. simpl. split; apply comp_assoc. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . intros. simpl. split; apply comp_assoc_sym. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Instance comma_proj : Comma ⟶ A ∏ B := {\| fobj := fun x => ``x; fmap := fun _ _ f => ``f \|}.	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Instance
Obligation . intros ? ? ? ? [e0 e1]. now split. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . now repeat intro. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . now repeat intro. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Instance comma_proj1 : Comma ⟶ A := {\| fobj := fun x => fst ``x; fmap := fun _ _ f => fst ``f \|}.	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Instance
Obligation . now intros ? ? ? ? [e0 e1]. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . now repeat intro. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . now repeat intro. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Instance comma_proj2 : Comma ⟶ B := {\| fobj := fun x => snd ``x; fmap := fun _ _ f => snd ``f \|}.	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Instance
Obligation . now intros ? ? ? ? [e0 e1]. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . now repeat intro. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Obligation . now repeat intro. Qed. #[local] Obligation Tactic := cat_simpl.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Obligation
Instance comma_proj_nat : S ◯ comma_proj1 ⟹ T ◯ comma_proj2.	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Instance
comma_proj_mor_iso A B C (S : A ⟶ C) (T : B ⟶ C) (x y : S ↓ T) : x ≅ y → `1 x ≅[A ∏ B] `1 y. Proof. destruct 1; simpl. isomorphism. - exact (`1 to). - exact (`1 from). - apply iso_to_from. - apply iso_from_to. Defined.	Theorem	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	comma_proj_mor_iso
comma_proj_com_iso A B C (S : A ⟶ C) (T : B ⟶ C) (x y : S ↓ T) : ∀ iso : x ≅ y, `2 x ≈ fmap[T] (snd `1 (from iso)) ∘ `2 y ∘ fmap[S] (fst `1 (to iso)). Proof. intros. pose proof (iso_from_to iso); simpl in X. destruct (from iso), x0; simpl in *. rewrite <- e. rewrite <- comp_assoc. rewrite <- fmap_comp. rewrite (fst X)....	Theorem	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	comma_proj_com_iso
Cocomma {A : Category} {B : Category} {C : Category} {S : A ⟶ C} {T : B ⟶ C} := @Comma (B^op) (A^op) (C^op) (T^op) (S^op).	Definition	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Comma.v	Cocomma
Definition Coproduct : Category := {\| obj := C + D; hom := fun x y => match x return Type with \| Datatypes.inl x => match y with \| Datatypes.inl y => x ~> y \| Datatypes.inr _ => False end \| Datatypes.inr x => match y with \| Datatypes.inl _ => False \| Datatypes.inr y => x ~> y end end; homset := fun x y => {\| equiv := f...	Program	Construction	[ "Require Import Category.", "Require Import Category." ]	Construction/Coproduct.v	Definition
Obligation . destruct x. - destruct y. + exact (f ≈ g). + contradiction. - destruct y. + contradiction. + exact (f ≈ g). Defined.	Next	Construction	[ "Require Import Category.", "Require Import Category." ]	Construction/Coproduct.v	Obligation
Obligation . equivalence; destruct x, y; simpl; intuition; unfold Coproduct_obligation_1 in ; intuition; auto with . Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category." ]	Construction/Coproduct.v	Obligation
Obligation . repeat match goal with [ H : _ + _ \|- _ ] => destruct H end; intuition; exact (f ∘ g). Defined.	Next	Construction	[ "Require Import Category.", "Require Import Category." ]	Construction/Coproduct.v	Obligation
Obligation . proper. destruct x, y, z; unfold Coproduct_obligation_1 in ; unfold Coproduct_obligation_3 in ; intuition; auto with *. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category." ]	Construction/Coproduct.v	Obligation
Obligation . destruct x, y; unfold Coproduct_obligation_1 in ; unfold Coproduct_obligation_3 in ; intuition; cat. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category." ]	Construction/Coproduct.v	Obligation
Obligation . destruct x, y; unfold Coproduct_obligation_1 in ; unfold Coproduct_obligation_3 in ; intuition; cat. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category." ]	Construction/Coproduct.v	Obligation
Obligation . destruct x, y, z, w; unfold Coproduct_obligation_1 in ; unfold Coproduct_obligation_3 in ; intuition; cat. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category." ]	Construction/Coproduct.v	Obligation
Obligation . destruct x, y, z, w; unfold Coproduct_obligation_1 in ; unfold Coproduct_obligation_3 in ; intuition; cat. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category." ]	Construction/Coproduct.v	Obligation
Enriched (K : Category) `{@Monoidal K} := { eobj : Type; ehom : eobj → eobj → K where "a ⟿ b" := (ehom a b); eid {x} : I ~{K}~> (x ⟿ x); ecompose {x y z} : (y ⟿ z) ⨂ (x ⟿ y) ~{K}~> (x ⟿ z); eid_left {x y} : ecompose ∘ eid ⨂ id << I ⨂ (x ⟿ y) ~~> (x ⟿ y) >> unit_left; eid_right {x y} : ecompose ∘ id ⨂ eid << (x ⟿ y) ⨂ I...	Class	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Enriched.v	Enriched
eobj : Enriched >-> Sortclass.	Coercion	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Enriched.v	eobj
EnrichedFunctor (K : Category) `{@Monoidal K} (C : Enriched K) (D : Enriched K) := { efobj : C → D; efmap {x y} : (x ⟿ y) ~{K}~> (efobj x ⟿ efobj y); efmap_id : ∀ x, efmap ∘ eid << I ~~> (efobj x ⟿ efobj x) >> eid; efmap_comp : ∀ x y z, ecompose ∘ efmap ⨂ efmap << (y ⟿ z) ⨂ (x ⟿ y) ~~> (efobj x ⟿ efobj z) >> efmap ∘ ec...	Class	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Enriched.v	EnrichedFunctor
Category_is_Enriched_over_Set : Enriched Sets ↔ Category. Proof. split; intros. - unshelve refine {\| obj := eobj ; hom := @ehom _ _ X ; homset := @ehom _ _ X ; id := fun x => @eid _ _ X x ttt ; compose := fun x y z f g => @ecompose _ _ X x y z (f, g) \|}. + intros. proper. destruct X. simpl in *. destruct (ecompose0 x y...	Theorem	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Enriched.v	Category_is_Enriched_over_Set
Functor_is_Enriched_over_Set (C D : Category) : EnrichedFunctor Sets (snd Category_is_Enriched_over_Set C) (snd Category_is_Enriched_over_Set D) ↔ (C ⟶ D). Proof. split; intros. - destruct X; simpl in . construct. + now apply efobj0. + now apply efmap0. + proper. now apply efmap0. + now apply efmap_id0. + simpl in . ...	Theorem	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Enriched.v	Functor_is_Enriched_over_Set
Definition Free@{fo fh fp} : Category@{fo fh fp} := {\| obj := C; hom := tlist hom; homset := fun _ _ => {\| equiv := eq \|}; id := fun _ => tnil; compose := fun _ _ _ f g => g +++ f \|}.	Program	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Free.v	Definition
Obligation . equivalence; congruence. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Free.v	Obligation
Obligation . now apply tlist_app_tnil_r. Qed.	Next	Construction	[ "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category.", "Require Import Category." ]	Construction/Free.v	Obligation

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Coq-CategoryTheory

Structured dataset from category-theory — Axiom-free category theory formalization.

2,952 declarations extracted from Coq source files.

Applications

Training language models on formal proofs
Fine-tuning theorem provers
Retrieval-augmented generation for proof assistants
Learning proof embeddings and representations

Source

Repository: https://github.com/jwiegley/category-theory

Schema

Column	Type	Description
fact	string	Declaration body
type	string	Lemma, Definition, Theorem, etc.
library	string	Source module
imports	list	Required imports
filename	string	Source file path
symbolic_name	string	Identifier

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