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.
Notation "pf1 === pf2" := (@ndr_eqv _ _ nd_eqv _ _ pf1 pf2).
+ (* 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
intros; apply ndr_comp_associativity.
Defined.
- 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).
- Defined.
+ Hint Extern 1 => apply nd_structural_id0.
+ Hint Extern 1 => apply nd_structural_id1.
+ Hint Extern 1 => apply nd_structural_weak.
+ Hint Extern 1 => apply nd_structural_copy.
+ Hint Extern 1 => apply nd_structural_prod.
+ Hint Extern 1 => apply nd_structural_comp.
+ Hint Extern 1 => apply nd_structural_cancell.
+ Hint Extern 1 => apply nd_structural_cancelr.
+ Hint Extern 1 => apply nd_structural_llecnac.
+ Hint Extern 1 => apply nd_structural_rlecnac.
+ Hint Extern 1 => apply nd_structural_assoc.
+ Hint Extern 1 => apply nd_structural_cossa.
+ Hint Extern 1 => apply weak'_structural.
+ 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 *.
+ setoid_rewrite <- ndr_prod_preserves_comp.
+ setoid_rewrite ndr_comp_left_identity.
+ reflexivity.
+ 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 *.
+ setoid_rewrite <- ndr_prod_preserves_comp.
+ setoid_rewrite ndr_comp_left_identity.
+ reflexivity.
+ Defined.
+ Instance Judgments_Category_binoidal : BinoidalCat Judgments_Category (fun x y => x,,y) :=
+ { bin_first := jud_first
+ ; bin_second := jud_second }.
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_llecnac |};
abstract (simpl; 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 in *; simpl in *.
+ apply ndr_comp_preserves_cancelr.
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 in *; simpl in *.
+ apply ndr_comp_preserves_cancell.
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 in *; simpl in *.
+ apply ndr_comp_preserves_assoc.
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 in *; simpl in *.
+ apply ndr_comp_preserves_assoc.
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 in *; simpl in *.
+ apply ndr_comp_preserves_assoc.
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_structural_indistinguishable; auto.
+
+ unfold functor_fobj; unfold fmor; simpl;
+ apply Build_Triangle; simpl; intros; apply ndr_structural_indistinguishable; auto.
+
+ intros; unfold eqv; simpl; auto.
+
+ intros; unfold eqv; simpl; auto.
+
+ intros; unfold eqv; simpl.
+ apply Build_CentralMorphism; intros; unfold eqv; simpl in *; auto.
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ apply ndr_prod_respects; try reflexivity.
+ apply ndr_structural_indistinguishable; auto.
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ apply ndr_prod_respects; try reflexivity.
+ apply ndr_structural_indistinguishable; auto.
+
+ intros; unfold eqv; simpl.
+ apply Build_CentralMorphism; intros; unfold eqv; simpl in *; auto.
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ apply ndr_prod_respects; try reflexivity.
+ apply ndr_structural_indistinguishable; auto.
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ apply ndr_prod_respects; try reflexivity.
+ apply ndr_structural_indistinguishable; auto.
+
+ intros; unfold eqv; simpl.
+ apply Build_CentralMorphism; intros; unfold eqv; simpl in *; auto.
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ apply ndr_prod_respects; try reflexivity.
+ apply ndr_structural_indistinguishable; auto.
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ symmetry.
+ etransitivity; [ idtac | apply ndr_prod_preserves_comp ].
+ setoid_rewrite ndr_comp_left_identity.
+ setoid_rewrite ndr_comp_right_identity.
+ apply ndr_prod_respects; try reflexivity.
+ apply ndr_structural_indistinguishable; auto.
+
+ Defined.
+
+ Instance Judgments_Category_monoidal : MonoidalCat Judgments_Category_premonoidal.
+ apply Build_MonoidalCat.
+ 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.
+
+ Instance Judgments_Category_Terminal : TerminalObject Judgments_Category [].
+ refine {| drop := nd_weak' ; drop_unique := _ |}.
abstract (intros; unfold eqv; simpl; apply ndr_void_proofs_irrelevant).
Defined.
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.
- 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,,[] )]
- }.
+ 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.
- Section Sequents.
- Context {S:Type}. (* type of sequent components *)
- Context (sequent:S->S->Judgment).
- Notation "a |= b" := (sequent a b).
+ Instance Judgments_Category_CartesianCat : CartesianCat Judgments_Category_monoidal :=
+ { car_terminal := Judgments_Category_Terminal ; car_diagonal := Judgments_Category_Diagonal }.
- Class SequentCalculus :=
- { nd_seq_reflexive : forall a, ND Rule [ ] [ a |= a ]
- }.
-
- Class CutRule :=
- { nd_cutrule_seq :> SequentCalculus
- ; nd_cut : forall a b c, [ a |= b ] ,, [ b |= c ] /⋯⋯/ [ a |= c ]
- ; nd_cut_associativity : forall {a b c d},
- ((nd_cut a b c) ** (nd_id1 (c|=d))) ;; (nd_cut a c d)
- ===
- nd_assoc ;; ((nd_id1 (a|=b)) ** (nd_cut b c d) ;; (nd_cut a b d))
- ; nd_cut_left_identity : forall a b, (( (nd_seq_reflexive a)**(nd_id _));; nd_cut _ _ b) === nd_cancell
- ; nd_cut_right_identity : forall a b, (((nd_id _)**(nd_seq_reflexive a) );; nd_cut b _ _) === nd_cancelr
- }.
- End Sequents.
-
- (* Structure ExpressionAlgebra (sig:Signature) := *)
+ intros; unfold hom; simpl; unfold functor_fobj; unfold fmor; simpl.
+ apply ndr_structural_indistinguishable; auto.
+ intros; unfold hom; simpl; unfold functor_fobj; unfold fmor; simpl.
+ apply ndr_structural_indistinguishable; auto.
+ Defined.
End Judgments_Category.
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