(*********************************************************************************************************************************)
(* NaturalDeductionCategory: *)
(* *)
-(* Natural Deduction proofs form a category (under mild assumptions, see below) *)
+(* Natural Deduction proofs form a category *)
(* *)
(*********************************************************************************************************************************)
Require Import Categories_ch1_3.
Require Import Functors_ch1_4.
Require Import Isomorphisms_ch1_5.
-Require Import ProductCategories_ch1_6_1.
Require Import OppositeCategories_ch1_6_2.
Require Import Enrichment_ch2_8.
Require Import Subcategories_ch7_1.
| unfold Symmetric; intros; symmetry; auto
| unfold Transitive; intros; transitivity y; auto ].
unfold Proper; unfold respectful; intros; simpl; apply ndr_comp_respects; auto.
- intros; apply ndr_comp_left_identity.
- intros; apply ndr_comp_right_identity.
+ intros; apply (ndr_builtfrom_structural f); auto.
+ intros; apply (ndr_builtfrom_structural f); auto.
intros; apply ndr_comp_associativity.
Defined.
- (* it is monoidal, with the judgment-tree-formation operator as its tensor *)
- Definition Judgments_Category_monoidal_endofunctor_fobj : Judgments_Category ×× Judgments_Category -> Judgments_Category :=
- fun xy => (fst_obj _ _ xy),,(snd_obj _ _ xy).
- Definition Judgments_Category_monoidal_endofunctor_fmor :
- forall a b, (a~~{Judgments_Category ×× Judgments_Category}~~>b) ->
- ((Judgments_Category_monoidal_endofunctor_fobj a)
- ~~{Judgments_Category}~~>
- (Judgments_Category_monoidal_endofunctor_fobj b)).
- intros.
- destruct a.
- destruct b.
- destruct X.
- simpl in *.
- exact (h**h0).
- Defined.
- Definition Judgments_Category_monoidal_endofunctor
- : Functor (Judgments_Category ×× Judgments_Category) Judgments_Category Judgments_Category_monoidal_endofunctor_fobj.
- refine {| fmor := Judgments_Category_monoidal_endofunctor_fmor |}; intros; simpl.
- abstract (destruct a; destruct b; destruct f; destruct f'; auto; destruct H; simpl in *; apply ndr_prod_respects; auto).
- abstract (destruct a; simpl in *; reflexivity).
- abstract (destruct a; destruct b; destruct c; destruct f; destruct g; symmetry; simpl in *; apply ndr_prod_preserves_comp).
+ (* Judgments form a binoidal category *)
+ Instance jud_first (a:Judgments_Category) : Functor Judgments_Category Judgments_Category (fun x => x,,a) :=
+ { fmor := fun b c (f:b /⋯⋯/ c) => f ** (nd_id a) }.
+ intros; unfold eqv; simpl; apply ndr_prod_respects; auto.
+ intros; unfold eqv in *; simpl in *; reflexivity.
+ intros; unfold eqv in *; simpl in *; apply (ndr_builtfrom_structural (nd_id a)); auto.
+ intros; unfold eqv in *; simpl in *.
+ setoid_rewrite <- ndr_prod_preserves_comp.
+ apply (ndr_builtfrom_structural (f;;g)); auto.
Defined.
-
+ Instance jud_second (a:Judgments_Category) : Functor Judgments_Category Judgments_Category (fun x => a,,x) :=
+ { fmor := fun b c (f:b /⋯⋯/ c) => (nd_id a) ** f }.
+ intros; unfold eqv; simpl; apply ndr_prod_respects; auto.
+ intros; unfold eqv in *; simpl in *; reflexivity.
+ intros; unfold eqv in *; simpl in *; apply (ndr_builtfrom_structural (nd_id a)); auto.
+ intros; unfold eqv in *; simpl in *.
+ setoid_rewrite <- ndr_prod_preserves_comp.
+ apply (ndr_builtfrom_structural (f;;g)); auto.
+ Defined.
+ Instance Judgments_Category_binoidal : BinoidalCat Judgments_Category (@T_Branch (??Judgment)) :=
+ { bin_first := jud_first
+ ; bin_second := jud_second }.
+
+ (* and that category is commutative (all morphisms central) *)
+ Instance Judgments_Category_Commutative : CommutativeCat Judgments_Category_binoidal.
+ apply Build_CommutativeCat.
+ intros; apply Build_CentralMorphism; intros; unfold eqv; simpl in *.
+ setoid_rewrite <- (ndr_prod_preserves_comp (nd_id a) g f (nd_id d)).
+ setoid_rewrite <- (ndr_prod_preserves_comp f (nd_id _) (nd_id _) g).
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ reflexivity.
+ setoid_rewrite <- (ndr_prod_preserves_comp (nd_id _) f g (nd_id _)).
+ setoid_rewrite <- (ndr_prod_preserves_comp g (nd_id _) (nd_id _) f).
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ reflexivity.
+ Defined.
+
+ (* Judgments form a premonoidal category *)
Definition jud_assoc_iso (a b c:Judgments_Category) : @Isomorphic _ _ Judgments_Category ((a,,b),,c) (a,,(b,,c)).
- refine
- {| iso_forward := nd_assoc
- ; iso_backward := nd_cossa |};
- abstract (simpl; auto).
+ refine {| iso_forward := nd_assoc ; iso_backward := nd_cossa |}.
+ unfold eqv; unfold comp; simpl; apply (ndr_builtfrom_structural nd_id0); auto.
+ unfold eqv; unfold comp; simpl; apply (ndr_builtfrom_structural nd_id0); auto.
Defined.
Definition jud_cancelr_iso (a:Judgments_Category) : @Isomorphic _ _ Judgments_Category (a,,[]) a.
- refine
- {| iso_forward := nd_cancelr
- ; iso_backward := nd_rlecnac |};
- abstract (simpl; auto).
+ refine {| iso_forward := nd_cancelr ; iso_backward := nd_rlecnac |};
+ unfold eqv; unfold comp; simpl; apply (ndr_builtfrom_structural nd_id0); auto.
Defined.
Definition jud_cancell_iso (a:Judgments_Category) : @Isomorphic _ _ Judgments_Category ([],,a) a.
- refine
- {| iso_forward := nd_cancell
- ; iso_backward := nd_llecnac |};
- abstract (simpl; auto).
+ refine {| iso_forward := nd_cancell ; iso_backward := nd_llecnac |};
+ unfold eqv; unfold comp; simpl; apply (ndr_builtfrom_structural nd_id0); auto.
Defined.
-
- Definition jud_mon_cancelr : (func_rlecnac [] >>>> Judgments_Category_monoidal_endofunctor) <~~~> functor_id Judgments_Category.
- refine {| ni_iso := fun x => jud_cancelr_iso x |}; intros; simpl.
- abstract (setoid_rewrite (ndr_prod_right_identity f);
- repeat setoid_rewrite ndr_comp_associativity;
- apply ndr_comp_respects; try reflexivity;
- symmetry;
- eapply transitivity; [ idtac | apply ndr_comp_right_identity ];
- apply ndr_comp_respects; try reflexivity; simpl; auto).
+ Instance jud_mon_cancelr : jud_first [] <~~~> functor_id Judgments_Category :=
+ { ni_iso := jud_cancelr_iso }.
+ intros; unfold eqv; unfold comp; simpl; apply (ndr_builtfrom_structural f); auto.
Defined.
- Definition jud_mon_cancell : (func_llecnac [] >>>> Judgments_Category_monoidal_endofunctor) <~~~> functor_id Judgments_Category.
- eapply Build_NaturalIsomorphism.
- instantiate (1:=fun x => jud_cancell_iso x).
- abstract (intros; simpl;
- setoid_rewrite (ndr_prod_left_identity f);
- repeat setoid_rewrite ndr_comp_associativity;
- apply ndr_comp_respects; try reflexivity;
- symmetry;
- eapply transitivity; [ idtac | apply ndr_comp_right_identity ];
- apply ndr_comp_respects; try reflexivity; simpl; auto).
+ Instance jud_mon_cancell : jud_second [] <~~~> functor_id Judgments_Category :=
+ { ni_iso := jud_cancell_iso }.
+ intros; unfold eqv; unfold comp; simpl; apply (ndr_builtfrom_structural f); auto.
Defined.
- Definition jud_mon_assoc_iso : forall X,
- (((Judgments_Category_monoidal_endofunctor **** (functor_id _)) >>>>
- Judgments_Category_monoidal_endofunctor) X) ≅
- (func_cossa >>>> ((((functor_id _) **** Judgments_Category_monoidal_endofunctor) >>>>
- Judgments_Category_monoidal_endofunctor))) X.
- intros.
- destruct X as [a c].
- destruct a as [a b].
- exact (jud_assoc_iso a b c).
+ Instance jud_mon_assoc : forall a b, a ⋊- >>>> - ⋉b <~~~> - ⋉b >>>> a ⋊- :=
+ { ni_iso := fun c => jud_assoc_iso a c b }.
+ intros; unfold eqv; unfold comp; simpl; apply (ndr_builtfrom_structural f); auto.
Defined.
- Definition jud_mon_assoc :
- ((Judgments_Category_monoidal_endofunctor **** (functor_id _)) >>>> Judgments_Category_monoidal_endofunctor)
- <~~~>
- func_cossa >>>> ((((functor_id _) **** Judgments_Category_monoidal_endofunctor) >>>> Judgments_Category_monoidal_endofunctor)).
- refine {| ni_iso := jud_mon_assoc_iso |}.
+ Instance jud_mon_assoc_rr : forall a b, - ⋉(a ⊗ b) <~~~> - ⋉a >>>> - ⋉b.
intros.
- unfold hom in *.
- destruct A as [a1 a3]. destruct a1 as [a1 a2]. simpl in *.
- destruct B as [b1 b3]. destruct b1 as [b1 b2]. simpl in *.
- destruct f as [f1 f3]. destruct f1 as [f1 f2]. simpl in *.
- unfold hom in *.
- unfold ob in *.
- unfold functor_fobj; unfold fmor; unfold functor_product_fobj; unfold Judgments_Category_monoidal_endofunctor_fobj; simpl.
- setoid_rewrite ndr_prod_associativity.
- setoid_rewrite ndr_comp_associativity.
- setoid_rewrite ndr_comp_associativity.
- apply ndr_comp_respects; try reflexivity.
- etransitivity.
- symmetry.
- apply ndr_comp_right_identity.
- apply ndr_comp_respects; try reflexivity.
- apply ndr_structural_indistinguishable; auto.
+ apply ni_inv.
+ refine {| ni_iso := fun c => (jud_assoc_iso _ _ _) |}.
+ intros; unfold eqv; unfold comp; simpl; apply (ndr_builtfrom_structural f); auto.
Defined.
-
- Instance Judgments_Category_monoidal : MonoidalCat _ _ Judgments_Category_monoidal_endofunctor [ ] :=
- { mon_cancelr := jud_mon_cancelr
- ; mon_cancell := jud_mon_cancell
- ; mon_assoc := jud_mon_assoc }.
- abstract
- (unfold functor_fobj; unfold fmor; unfold functor_product_fobj; unfold Judgments_Category_monoidal_endofunctor_fobj; simpl;
- apply Build_Pentagon; simpl; intros; apply ndr_structural_indistinguishable; auto).
- abstract
- (unfold functor_fobj; unfold fmor; unfold functor_product_fobj; unfold Judgments_Category_monoidal_endofunctor_fobj; simpl;
- apply Build_Triangle; simpl; intros; apply ndr_structural_indistinguishable; auto).
+ Instance jud_mon_assoc_ll : forall a b, (a ⊗ b) ⋊- <~~~> b ⋊- >>>> a ⋊- :=
+ { ni_iso := fun c => jud_assoc_iso _ _ _ }.
+ intros; unfold eqv; unfold comp; simpl; apply (ndr_builtfrom_structural f); auto.
Defined.
-
- Instance Judgments_Category_Terminal : Terminal Judgments_Category.
- refine {| one := [] ; drop := nd_weak' ; drop_unique := _ |}.
+ Instance Judgments_Category_premonoidal : PreMonoidalCat Judgments_Category_binoidal [] :=
+ { pmon_cancelr := jud_mon_cancelr
+ ; pmon_cancell := jud_mon_cancell
+ ; pmon_assoc := jud_mon_assoc
+ ; pmon_assoc_rr := jud_mon_assoc_rr
+ ; pmon_assoc_ll := jud_mon_assoc_ll
+ }.
+ unfold functor_fobj; unfold fmor; simpl;
+ apply Build_Pentagon; simpl; intros; apply (ndr_builtfrom_structural nd_id0); auto 10.
+ unfold functor_fobj; unfold fmor; simpl;
+ apply Build_Triangle; simpl; intros; apply (ndr_builtfrom_structural nd_id0); auto 10.
+ intros; unfold eqv; simpl; auto; reflexivity.
+ intros; unfold eqv; simpl; auto; reflexivity.
+ intros; unfold eqv; simpl; apply Judgments_Category_Commutative.
+ intros; unfold eqv; simpl; apply Judgments_Category_Commutative.
+ intros; unfold eqv; simpl; apply Judgments_Category_Commutative.
+ Defined.
+
+ (* commutative premonoidal categories are monoidal *)
+ Instance Judgments_Category_monoidal : MonoidalCat Judgments_Category_premonoidal :=
+ { mon_commutative := Judgments_Category_Commutative }.
+
+ (* Judgments also happens to have a terminal object - the empty list of judgments *)
+ Instance Judgments_Category_Terminal : TerminalObject Judgments_Category [].
+ refine {| drop := nd_weak ; drop_unique := _ |}.
abstract (intros; unfold eqv; simpl; apply ndr_void_proofs_irrelevant).
Defined.
+ (* Judgments is also a diagonal category via nd_copy *)
Instance Judgments_Category_Diagonal : DiagonalCat Judgments_Category_monoidal.
- refine {| copy_nt := _ |}.
intros.
- refine {| nt_component := nd_copy |}.
- intros.
- unfold hom in *; unfold ob in *; unfold eqv.
- simpl.
- rewrite ndr_prod_preserves_copy; auto.
- reflexivity.
- Defined.
-
- Instance Judgments_Category_CartesianCat : CartesianCat Judgments_Category_monoidal.
- refine {| car_terminal := Judgments_Category_Terminal ; car_diagonal := Judgments_Category_Diagonal
- ; car_one := iso_id [] |}
- ; intros; unfold hom; simpl
- ; unfold functor_fobj; unfold fmor; unfold functor_product_fobj; unfold Judgments_Category_monoidal_endofunctor_fobj; simpl.
-
- apply ndr_structural_indistinguishable; auto.
- apply nd_structural_comp; auto.
- apply nd_structural_comp; auto.
- unfold copy; simpl; apply nd_structural_copy; auto.
- apply nd_structural_prod; auto.
- apply nd_structural_comp; auto.
- apply weak'_structural.
- apply nd_structural_id0; auto.
- apply nd_structural_cancell; auto.
-
- apply ndr_structural_indistinguishable; auto.
- apply nd_structural_comp; auto.
- apply nd_structural_comp; auto.
- unfold copy; simpl; apply nd_structural_copy; auto.
- apply nd_structural_prod; auto.
- apply nd_structural_comp; auto.
- apply weak'_structural.
- apply nd_structural_id0; auto.
- apply nd_structural_cancelr; auto.
+ refine {| copy := nd_copy |}; intros; simpl.
+ setoid_rewrite ndr_comp_associativity.
+ setoid_rewrite <- (ndr_prod_preserves_copy f).
+ apply ndr_comp_respects; try reflexivity.
+ etransitivity.
+ symmetry.
+ apply ndr_prod_preserves_comp.
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ reflexivity.
+ setoid_rewrite ndr_comp_associativity.
+ setoid_rewrite <- (ndr_prod_preserves_copy f).
+ apply ndr_comp_respects; try reflexivity.
+ etransitivity.
+ symmetry.
+ apply ndr_prod_preserves_comp.
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ reflexivity.
+ Defined.
+
+ (* Judgments is a cartesian category: it has a terminal object, diagonal morphisms, and the right naturalities *)
+ Instance Judgments_Category_CartesianCat : CartesianCat Judgments_Category_monoidal :=
+ { car_terminal := Judgments_Category_Terminal ; car_diagonal := Judgments_Category_Diagonal }.
+ intros; unfold eqv; simpl; symmetry; apply ndr_copy_then_weak_left.
+ intros; unfold eqv; simpl; symmetry; apply ndr_copy_then_weak_right.
Defined.
End Judgments_Category.
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