(*********************************************************************************************************************************) (* 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.