(*********************************************************************************************************************************)
(* 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.
Require Import NaturalTransformations_ch7_4.
Require Import NaturalIsomorphisms_ch7_5.
-Require Import MonoidalCategories_ch7_8.
Require Import Coherence_ch7_8.
+Require Import InitialTerminal_ch2_2.
+Require Import BinoidalCategories.
+Require Import PreMonoidalCategories.
+Require Import MonoidalCategories_ch7_8.
Open Scope nd_scope.
Open Scope pf_scope.
Context {Rule : forall (hypotheses:Tree ??Judgment)(conclusion:Tree ??Judgment), Type}.
Context (nd_eqv : @ND_Relation Judgment Rule).
- (* actually you can use any type as the objects, so long as you give a mapping from that type to judgments *)
- Context {Ob : Type}.
- Context (ob2judgment : Ob -> Judgment).
- Coercion ob2judgment : Ob >-> Judgment.
-
Notation "pf1 === pf2" := (@ndr_eqv _ _ nd_eqv _ _ pf1 pf2).
- Instance Judgments_Category
- : Category (Tree ??Ob) (fun h c => (mapOptionTree ob2judgment h) /⋯⋯/ (mapOptionTree ob2judgment c)) :=
+ (* there is a category whose objects are judgments and whose morphisms are proofs *)
+ Instance Judgments_Category : Category (Tree ??Judgment) (fun h c => h /⋯⋯/ c) :=
{ id := fun h => nd_id _
; comp := fun a b c f g => f ;; g
; eqv := fun a b f g => f === g
| 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.
- Definition Judgments_Category_monoidal_endofunctor_fobj : Judgments_Category ×× Judgments_Category -> Judgments_Category :=
- (fun xy =>
- match xy with
- | pair_obj x y => T_Branch x y
- end).
- 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.
- 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)).
- apply (@Build_Isomorphic _ _ Judgments_Category _ _
- (@nd_assoc _ Rule (mapOptionTree ob2judgment a) (mapOptionTree ob2judgment b) (mapOptionTree ob2judgment c)
- : (a,, b),, c ~~{Judgments_Category}~~> a,, (b,, c))
- (@nd_cossa _ Rule (mapOptionTree ob2judgment a) (mapOptionTree ob2judgment b) (mapOptionTree ob2judgment c)
- : a,, (b,, c) ~~{Judgments_Category}~~> (a,, b),, c)); 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.
- apply (@Build_Isomorphic _ _ Judgments_Category _ _
- (@nd_cancelr _ Rule (mapOptionTree ob2judgment a) : a,,[] ~~{Judgments_Category}~~> a)
- (@nd_rlecnac _ Rule (mapOptionTree ob2judgment a) : a ~~{Judgments_Category}~~> a,,[])); 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.
- apply (@Build_Isomorphic _ _ Judgments_Category _ _
- (@nd_cancell _ Rule (mapOptionTree ob2judgment a) : [],,a ~~{Judgments_Category}~~> a)
- (@nd_llecnac _ Rule (mapOptionTree ob2judgment a) : a ~~{Judgments_Category}~~> [],,a)); 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.
- 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).
- 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].
- apply (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.
- destruct A as [a1 a3]. destruct a1 as [a1 a2].
- destruct B as [b1 b3]. destruct b1 as [b1 b2].
- destruct f as [f1 f3]. destruct f1 as [f1 f2].
- simpl.
- setoid_rewrite ndr_prod_associativity.
- setoid_rewrite ndr_comp_associativity.
- 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.
+ 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 }.
- apply Build_Pentagon; simpl; intros; apply ndr_structural_indistinguishable; auto.
- 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_CartesianCat : CartesianCat Judgments_Category_monoidal.
- admit.
- Defined.
-
- (* Given some mapping "rep" that turns a (Tree ??T) intoto Judgment,
- * this asserts that we have sensible structural rules with respect
- * to that mapping. Doing all of this "with respect to a mapping"
- * lets us avoid duplicating code for both the antecedent and
- * succedent of sequent deductions. *)
- Class TreeStructuralRules {T:Type}(rep:Tree ??T -> Judgment) :=
- { tsr_eqv : @ND_Relation Judgment Rule where "pf1 === pf2" := (@ndr_eqv _ _ tsr_eqv _ _ pf1 pf2)
- ; tsr_ant_assoc : forall {a b c}, Rule [rep ((a,,b),,c)] [rep ((a,,(b,,c)))]
- ; tsr_ant_cossa : forall {a b c}, Rule [rep (a,,(b,,c))] [rep (((a,,b),,c))]
- ; tsr_ant_cancell : forall {a }, Rule [rep ( [],,a )] [rep ( a )]
- ; tsr_ant_cancelr : forall {a }, Rule [rep (a,,[] )] [rep ( a )]
- ; tsr_ant_llecnac : forall {a }, Rule [rep ( a )] [rep ( [],,a )]
- ; tsr_ant_rlecnac : forall {a }, Rule [rep ( a )] [rep ( a,,[] )]
+ 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.
-
- (* Structure ExpressionAlgebra (sig:Signature) := *)
+ (* Judgments is also a diagonal category via nd_copy *)
+ Instance Judgments_Category_Diagonal : DiagonalCat Judgments_Category_monoidal.
+ intros.
+ 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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