--- /dev/null
+(*********************************************************************************************************************************)
+(* ProgrammingLanguageCategory *)
+(* *)
+(* The category Types(L) *)
+(* *)
+(*********************************************************************************************************************************)
+
+Generalizable All Variables.
+Require Import Preamble.
+Require Import General.
+Require Import Categories_ch1_3.
+Require Import InitialTerminal_ch2_2.
+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 BinoidalCategories.
+Require Import PreMonoidalCategories.
+Require Import MonoidalCategories_ch7_8.
+Require Import Coherence_ch7_8.
+Require Import Enrichment_ch2_8.
+Require Import RepresentableStructure_ch7_2.
+Require Import FunctorCategories_ch7_7.
+
+Require Import NaturalDeduction.
+Require Import ProgrammingLanguage.
+ Export ProgrammingLanguage.
+
+Require Import NaturalDeductionCategory.
+
+Open Scope nd_scope.
+(* I am at a loss to explain why "auto" can't handle this *)
+Ltac ndpc_tac :=
+ match goal with
+ | [ |- NDPredicateClosure ?P (?A ;; ?B) ] => apply ndpc_comp; ndpc_tac
+ | [ |- NDPredicateClosure ?P (?A ** ?B) ] => apply ndpc_prod; ndpc_tac
+ | _ => auto
+ end.
+
+(* this tactical searches the environment; setoid_rewrite doesn't seem to be able to do that properly sometimes *)
+Ltac nd_swap_ltac P EQV :=
+ match goal with
+ [ |- context [ (?F ** nd_id _) ;; (nd_id _ ** ?G) ] ] =>
+ set (@nd_swap _ _ EQV _ _ _ _ F G) as P
+ end.
+
+(* I still wish I knew why "Hint Constructors" doesn't work *)
+Hint Extern 5 => apply snd_inert_initial.
+Hint Extern 5 => apply snd_inert_cut.
+Hint Extern 5 => apply snd_inert_structural.
+Hint Extern 5 => apply cnd_inert_initial.
+Hint Extern 5 => apply cnd_inert_cut.
+Hint Extern 5 => apply cnd_inert_structural.
+Hint Extern 5 => apply cnd_inert_cnd_ant_assoc.
+Hint Extern 5 => apply cnd_inert_cnd_ant_cossa.
+Hint Extern 5 => apply cnd_inert_cnd_ant_cancell.
+Hint Extern 5 => apply cnd_inert_cnd_ant_cancelr.
+Hint Extern 5 => apply cnd_inert_cnd_ant_llecnac.
+Hint Extern 5 => apply cnd_inert_cnd_ant_rlecnac.
+Hint Extern 5 => apply cnd_inert_se_expand_left.
+Hint Extern 5 => apply cnd_inert_se_expand_right.
+
+Hint Extern 2 (@Structural _ _ _ _ (@nd_id _ _ [] )) => simpl; auto.
+Hint Extern 2 (@Structural _ _ _ _ (@nd_id _ _ [ _ ])) => simpl; auto.
+
+Section ProgrammingLanguageCategory.
+
+ Context {T : Type}. (* types of the language *)
+
+ Context {Rule : Tree ??(@PLJudg T) -> Tree ??(@PLJudg T) -> Type}.
+ Notation "cs |= ss" := (@sequent T cs ss) : pl_scope.
+
+ Notation "H /⋯⋯/ C" := (ND Rule H C) : pl_scope.
+
+ Open Scope pf_scope.
+ Open Scope nd_scope.
+ Open Scope pl_scope.
+
+ Context (PL:@ProgrammingLanguage T Rule).
+
+ (* category of judgments in a fixed type/coercion context *)
+ Definition Judgments_cartesian := @Judgments_Category_CartesianCat _ Rule pl_eqv.
+
+ Definition JudgmentsL := Judgments_cartesian.
+
+ Definition identityProof t : [] ~~{JudgmentsL}~~> [t |= t].
+ unfold hom; simpl.
+ apply snd_initial.
+ Defined.
+
+ Definition cutProof a b c : [a |= b],,[b |= c] ~~{JudgmentsL}~~> [a |= c].
+ unfold hom; simpl.
+ apply snd_cut.
+ Defined.
+
+ Instance TypesL : ECategory JudgmentsL (Tree ??T) (fun x y => [x|=y]) :=
+ { eid := identityProof
+ ; ecomp := cutProof
+ }.
+ intros; apply (mon_commutative(MonoidalCat:=JudgmentsL)).
+ intros; apply (mon_commutative(MonoidalCat:=JudgmentsL)).
+ abstract (intros; unfold identityProof; unfold cutProof; simpl; eapply cndr_inert; auto; apply PL).
+ abstract (intros; unfold identityProof; unfold cutProof; simpl; eapply cndr_inert; auto; apply PL).
+ abstract (intros; unfold identityProof; unfold cutProof; simpl; eapply cndr_inert;
+ [ apply PL | idtac | idtac ]; apply ndpc_comp; auto).
+ Defined.
+
+ Instance Types_first c : EFunctor TypesL TypesL (fun x => x,,c ) :=
+ { efunc := fun x y => cnd_expand_right(ContextND:=pl_cnd) x y c }.
+ intros; apply (mon_commutative(MonoidalCat:=JudgmentsL)).
+ abstract (intros; simpl; apply (cndr_inert pl_cnd); auto).
+ abstract (intros; unfold ehom; unfold comp; simpl; unfold cutProof;
+ rewrite <- (@ndr_prod_preserves_comp _ _ PL _ _ (cnd_expand_right _ _ c) _ _ (nd_id1 (b|=c0))
+ _ (nd_id1 (a,,c |= b,,c)) _ (cnd_expand_right _ _ c));
+ setoid_rewrite (@ndr_comp_right_identity _ _ PL _ [a,, c |= b,, c]);
+ setoid_rewrite (@ndr_comp_left_identity _ _ PL [b |= c0]);
+ simpl; eapply cndr_inert; [ apply PL | auto | auto ]).
+ Defined.
+
+ Instance Types_second c : EFunctor TypesL TypesL (fun x => c,,x) :=
+ { efunc := fun x y => ((@cnd_expand_left _ _ _ _ _ _ x y c)) }.
+ intros; apply (mon_commutative(MonoidalCat:=JudgmentsL)).
+ abstract (intros; simpl; apply (cndr_inert pl_cnd); auto).
+ intros; unfold ehom; unfold comp; simpl; unfold cutProof;
+ abstract (rewrite <- (@ndr_prod_preserves_comp _ _ PL _ _ (cnd_expand_left _ _ c) _ _ (nd_id1 (b|=c0))
+ _ (nd_id1 (c,,a |= c,,b)) _ (cnd_expand_left _ _ c));
+ setoid_rewrite (@ndr_comp_right_identity _ _ PL _ [c,,a |= c,,b]);
+ setoid_rewrite (@ndr_comp_left_identity _ _ PL [b |= c0]);
+ simpl; eapply cndr_inert; [ apply PL | auto | auto ]).
+ Defined.
+
+ Instance Types_binoidal : EBinoidalCat TypesL (@T_Branch _) :=
+ { ebc_first := Types_first
+ ; ebc_second := Types_second
+ }.
+
+ Instance Types_assoc_iso a b c : Isomorphic(C:=TypesL) ((a,,b),,c) (a,,(b,,c)) :=
+ { iso_forward := snd_initial _ ;; cnd_ant_cossa _ a b c
+ ; iso_backward := snd_initial _ ;; cnd_ant_assoc _ a b c
+ }.
+ abstract (simpl; eapply cndr_inert; unfold identityProof; [ apply PL | idtac | idtac ]; ndpc_tac).
+ abstract (simpl; eapply cndr_inert; unfold identityProof; [ apply PL | idtac | idtac ]; ndpc_tac).
+ Defined.
+
+ Instance Types_cancelr_iso a : Isomorphic(C:=TypesL) (a,,[]) a :=
+ { iso_forward := snd_initial _ ;; cnd_ant_rlecnac _ a
+ ; iso_backward := snd_initial _ ;; cnd_ant_cancelr _ a
+ }.
+ abstract (simpl; eapply cndr_inert; unfold identityProof; [ apply PL | idtac | idtac ]; ndpc_tac).
+ abstract (simpl; eapply cndr_inert; unfold identityProof; [ apply PL | idtac | idtac ]; ndpc_tac).
+ Defined.
+
+ Instance Types_cancell_iso a : Isomorphic(C:=TypesL) ([],,a) a :=
+ { iso_forward := snd_initial _ ;; cnd_ant_llecnac _ a
+ ; iso_backward := snd_initial _ ;; cnd_ant_cancell _ a
+ }.
+ abstract (simpl; eapply cndr_inert; unfold identityProof; [ apply PL | idtac | idtac ]; ndpc_tac).
+ abstract (simpl; eapply cndr_inert; unfold identityProof; [ apply PL | idtac | idtac ]; ndpc_tac).
+ Defined.
+
+ Lemma coincide' : nd_llecnac === nd_rlecnac.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ apply qq.
+ Qed.
+
+ Lemma Types_assoc_lemma : ∀a b X Y : TypesL,
+ ∀f : X ~~{ TypesL }~~> Y,
+ #(Types_assoc_iso a X b) >>> (Types_first b >>>> Types_second a) \ f ~~
+ (Types_second a >>>> Types_first b) \ f >>> #(Types_assoc_iso a Y b).
+ intros.
+ Opaque nd_id.
+ simpl.
+ Transparent nd_id.
+ unfold ehom.
+
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ clear p.
+
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) f) as q.
+ simpl in q.
+ setoid_rewrite <- q.
+ clear q.
+
+ set (ni_commutes' (jud_mon_cancell PL) f) as q.
+ simpl in q.
+ setoid_rewrite coincide' in q.
+ setoid_rewrite <- q.
+ clear q.
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ apply ndr_comp_respects; try reflexivity.
+
+ apply (cndr_inert pl_cnd); auto; ndpc_tac; auto.
+ Qed.
+
+ Instance Types_assoc a b : Types_second a >>>> Types_first b <~~~> Types_first b >>>> Types_second a :=
+ { ni_iso := fun c => Types_assoc_iso a c b }.
+ apply Types_assoc_lemma.
+ Defined.
+
+ Lemma Types_assoc_ll_lemma :
+ ∀a b X Y : TypesL,
+ ∀f : X ~~{ TypesL }~~> Y,
+ #(Types_assoc_iso a b X) >>> (Types_second b >>>> Types_second a) \ f ~~
+ Types_second (a,, b) \ f >>> #(Types_assoc_iso a b Y).
+
+ intros.
+ Opaque nd_id.
+ simpl.
+ Transparent nd_id.
+ unfold ehom.
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ clear p.
+
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) f) as q.
+ Opaque nd_id.
+ simpl in q.
+ setoid_rewrite <- q.
+
+ clear q.
+ set (ni_commutes' (jud_mon_cancell PL) f) as q.
+ simpl in q.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ simpl in qq.
+ setoid_rewrite qq in q.
+ clear q' qq.
+ setoid_rewrite <- q.
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ apply ndr_comp_respects; try reflexivity.
+
+ Transparent nd_id.
+ apply (cndr_inert pl_cnd); auto; ndpc_tac.
+ Qed.
+
+ Instance Types_assoc_ll a b : Types_second (a,,b) <~~~> Types_second b >>>> Types_second a :=
+ { ni_iso := fun c => Types_assoc_iso a b c }.
+ apply Types_assoc_ll_lemma.
+ Defined.
+
+ Lemma Types_assoc_rr_lemma :
+ ∀a b A B : TypesL,
+ ∀f : A ~~{ TypesL }~~> B,
+ #(Types_assoc_iso A a b) ⁻¹ >>> (Types_first a >>>> Types_first b) \ f ~~
+ Types_first (a,, b) \ f >>> #(Types_assoc_iso B a b) ⁻¹.
+ intros.
+ Opaque nd_id.
+ simpl.
+ Transparent nd_id.
+
+ rename A into X.
+ rename B into Y.
+ unfold ehom.
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ clear p.
+
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) f) as q.
+ Opaque nd_id.
+ simpl in q.
+ setoid_rewrite <- q.
+
+ clear q.
+ set (ni_commutes' (jud_mon_cancell PL) f) as q.
+ simpl in q.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ simpl in qq.
+ setoid_rewrite qq in q.
+ clear q' qq.
+ setoid_rewrite <- q.
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ apply ndr_comp_respects; try reflexivity.
+
+ Transparent nd_id.
+ apply (cndr_inert pl_cnd); auto; ndpc_tac.
+ Qed.
+
+ Instance Types_assoc_rr a b : Types_first (a,,b) <~~~> Types_first a >>>> Types_first b :=
+ { ni_iso := fun c => iso_inv _ _ (Types_assoc_iso c a b) }.
+ apply Types_assoc_rr_lemma.
+ Defined.
+
+ Lemma Types_cancelr_lemma :
+ ∀A B : TypesL,
+ ∀f : A ~~{ TypesL }~~> B,
+ #(Types_cancelr_iso A) >>> functor_id TypesL \ f ~~
+ Types_first [] \ f >>> #(Types_cancelr_iso B).
+ intros.
+ Opaque nd_id.
+ simpl.
+ unfold ehom.
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) f) as q.
+ Opaque nd_id.
+ simpl in q.
+ setoid_rewrite <- q.
+ clear q.
+
+ set (ni_commutes' (jud_mon_cancell PL) f) as q.
+ simpl in q.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ simpl in qq.
+ setoid_rewrite qq in q.
+ clear q' qq.
+ setoid_rewrite <- q.
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ apply ndr_comp_respects; try reflexivity.
+ Transparent nd_id.
+ simpl.
+ apply (cndr_inert pl_cnd); auto; ndpc_tac.
+ Qed.
+
+ Instance Types_cancelr : Types_first [] <~~~> functor_id _ :=
+ { ni_iso := Types_cancelr_iso }.
+ apply Types_cancelr_lemma.
+ Defined.
+
+ Lemma Types_cancell_lemma :
+ ∀A B : TypesL,
+ ∀f : A ~~{ TypesL }~~> B,
+ #(Types_cancell_iso A) >>> functor_id TypesL \ f ~~
+ Types_second [] \ f >>> #(Types_cancell_iso B).
+ intros.
+ Opaque nd_id.
+ simpl.
+ unfold ehom.
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) f) as q.
+ Opaque nd_id.
+ simpl in q.
+ setoid_rewrite <- q.
+ clear q.
+
+ set (ni_commutes' (jud_mon_cancell PL) f) as q.
+ simpl in q.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ simpl in qq.
+ setoid_rewrite qq in q.
+ clear q' qq.
+ setoid_rewrite <- q.
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+
+ apply ndr_comp_respects; try reflexivity.
+ Transparent nd_id.
+ simpl.
+ apply (cndr_inert pl_cnd); auto; ndpc_tac.
+ Qed.
+
+ Instance Types_cancell : Types_second [] <~~~> functor_id _ :=
+ { ni_iso := Types_cancell_iso }.
+ apply Types_cancell_lemma.
+ Defined.
+
+ Lemma TypesL_assoc_central a b c : CentralMorphism(H:=Types_binoidal) #((Types_assoc a b) c).
+ intros.
+ apply Build_CentralMorphism.
+ intros.
+ unfold bin_obj.
+ unfold ebc_bobj.
+ Opaque nd_id.
+ simpl.
+ unfold ehom.
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ clear p.
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) g) as q.
+ Opaque nd_id.
+ simpl in q.
+ setoid_rewrite <- q.
+ clear q.
+
+ set (ni_commutes' (jud_mon_cancell PL) g) as q.
+ simpl in q.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ simpl in qq.
+ setoid_rewrite qq in q.
+ clear q' qq.
+ setoid_rewrite <- q.
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ apply ndr_comp_respects.
+ reflexivity.
+
+ Transparent nd_id.
+ apply (cndr_inert pl_cnd); auto; ndpc_tac.
+
+ Opaque nd_id.
+ intros.
+ unfold bin_obj.
+ unfold ebc_bobj.
+ simpl.
+ unfold ehom.
+ symmetry.
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ clear p.
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) g) as q.
+ Opaque nd_id.
+ simpl in q.
+ setoid_rewrite <- q.
+ clear q.
+
+ set (ni_commutes' (jud_mon_cancell PL) g) as q.
+ simpl in q.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ simpl in qq.
+ setoid_rewrite qq in q.
+ clear q' qq.
+ setoid_rewrite <- q.
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ apply ndr_comp_respects.
+ reflexivity.
+
+ Transparent nd_id.
+ apply (cndr_inert pl_cnd); auto; ndpc_tac.
+ Qed.
+
+ Lemma TypesL_cancell_central a : CentralMorphism(H:=Types_binoidal) #(Types_cancell a).
+ intros.
+ apply Build_CentralMorphism.
+ Opaque nd_id.
+ intros.
+ unfold bin_obj.
+ unfold ebc_bobj.
+ simpl.
+ unfold ehom.
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ clear p.
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) g) as q.
+ Opaque nd_id.
+ simpl in q.
+ setoid_rewrite <- q.
+ clear q.
+
+ set (ni_commutes' (jud_mon_cancell PL) g) as q.
+ simpl in q.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ simpl in qq.
+ setoid_rewrite qq in q.
+ clear q' qq.
+ setoid_rewrite <- q.
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ apply ndr_comp_respects.
+ reflexivity.
+
+ Transparent nd_id.
+ apply (cndr_inert pl_cnd); auto; ndpc_tac.
+
+ Opaque nd_id.
+ intros.
+ unfold bin_obj.
+ unfold ebc_bobj.
+ simpl.
+ unfold ehom.
+ symmetry.
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ clear p.
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) g) as q.
+ Opaque nd_id.
+ simpl in q.
+ setoid_rewrite <- q.
+ clear q.
+
+ set (ni_commutes' (jud_mon_cancell PL) g) as q.
+ simpl in q.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ simpl in qq.
+ setoid_rewrite qq in q.
+ clear q' qq.
+ setoid_rewrite <- q.
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ apply ndr_comp_respects.
+ reflexivity.
+
+ Transparent nd_id.
+ apply (cndr_inert pl_cnd); auto; ndpc_tac.
+ Qed.
+
+ Lemma TypesL_cancelr_central a : CentralMorphism(H:=Types_binoidal) #(Types_cancelr a).
+ intros.
+ apply Build_CentralMorphism.
+ Opaque nd_id.
+ intros.
+ unfold bin_obj.
+ unfold ebc_bobj.
+ simpl.
+ unfold ehom.
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ clear p.
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) g) as q.
+ Opaque nd_id.
+ simpl in q.
+ setoid_rewrite <- q.
+ clear q.
+
+ set (ni_commutes' (jud_mon_cancell PL) g) as q.
+ simpl in q.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ simpl in qq.
+ setoid_rewrite qq in q.
+ clear q' qq.
+ setoid_rewrite <- q.
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ apply ndr_comp_respects.
+ reflexivity.
+
+ Transparent nd_id.
+ apply (cndr_inert pl_cnd); auto; ndpc_tac.
+
+ Opaque nd_id.
+ intros.
+ unfold bin_obj.
+ unfold ebc_bobj.
+ simpl.
+ unfold ehom.
+ symmetry.
+ nd_swap_ltac p PL.
+ setoid_rewrite p.
+ clear p.
+ setoid_rewrite (@nd_prod_split_left _ Rule PL _ _ _ []).
+ setoid_rewrite (@nd_prod_split_right _ Rule PL _ _ _ []).
+ repeat setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ setoid_rewrite <- (@ndr_comp_associativity _ Rule PL).
+
+ set (ni_commutes' (jud_mon_cancelr PL) g) as q.
+ Opaque nd_id.
+ simpl in q.
+ setoid_rewrite <- q.
+ clear q.
+
+ set (ni_commutes' (jud_mon_cancell PL) g) as q.
+ simpl in q.
+ set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal PL)))) as q'.
+ set (isos_forward_equal_then_backward_equal _ _ q') as qq.
+ simpl in qq.
+ setoid_rewrite qq in q.
+ clear q' qq.
+ setoid_rewrite <- q.
+
+ setoid_rewrite (@ndr_comp_associativity _ Rule PL).
+ apply ndr_comp_respects.
+ reflexivity.
+
+ Transparent nd_id.
+ apply (cndr_inert pl_cnd); auto; ndpc_tac.
+ Qed.
+
+ Instance TypesL_PreMonoidal : PreMonoidalCat Types_binoidal [] :=
+ { pmon_assoc := Types_assoc
+ ; pmon_cancell := Types_cancell
+ ; pmon_cancelr := Types_cancelr
+ ; pmon_assoc_rr := Types_assoc_rr
+ ; pmon_assoc_ll := Types_assoc_ll
+ }.
+ abstract (apply Build_Pentagon; intros; simpl; eapply cndr_inert; try apply PL; ndpc_tac).
+ abstract (apply Build_Triangle; intros; simpl; eapply cndr_inert; try apply PL; ndpc_tac).
+ intros; simpl; reflexivity.
+ intros; simpl; reflexivity.
+ apply TypesL_assoc_central.
+ apply TypesL_cancelr_central.
+ apply TypesL_cancell_central.
+ Defined.
+
+End ProgrammingLanguageCategory.
+
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