X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FProgrammingLanguage.v;h=83b435ab06ac2b44d545a7f2a5bb6c769d0a2078;hp=dc2256c6d7ebb403520f595036e9914f456cc63c;hb=034f7e7856bebbbcb3c83946aa603c640b17f3bb;hpb=bef99d21b3f5697d6fb1871493290c8dcf9dea93 diff --git a/src/ProgrammingLanguage.v b/src/ProgrammingLanguage.v index dc2256c..83b435a 100644 --- a/src/ProgrammingLanguage.v +++ b/src/ProgrammingLanguage.v @@ -26,9 +26,7 @@ Require Import Enrichment_ch2_8. Require Import RepresentableStructure_ch7_2. Require Import FunctorCategories_ch7_7. -Require Import Enrichments. Require Import NaturalDeduction. -Require Import NaturalDeductionCategory. Section Programming_Language. @@ -47,687 +45,15 @@ Section Programming_Language. Open Scope pl_scope. Class ProgrammingLanguage := - { pl_eqv0 : @ND_Relation PLJudg Rule + { pl_eqv0 :> @ND_Relation PLJudg Rule ; pl_snd :> @SequentND PLJudg Rule _ sequent ; pl_cnd :> @ContextND PLJudg Rule T sequent pl_snd ; pl_eqv1 :> @SequentND_Relation PLJudg Rule _ sequent pl_snd pl_eqv0 ; pl_eqv :> @ContextND_Relation PLJudg Rule _ sequent pl_snd pl_cnd pl_eqv0 pl_eqv1 }. Notation "pf1 === pf2" := (@ndr_eqv _ _ pl_eqv _ _ pf1 pf2) : temporary_scope3. - - Section LanguageCategory. - - Context (PL:ProgrammingLanguage). - - (* 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. - - Existing Instance pl_eqv. - - Definition TypesL : ECategory JudgmentsL (Tree ??T) (fun x y => [x|=y]). - refine - {| eid := identityProof - ; ecomp := cutProof - |}; intros. - apply (mon_commutative(MonoidalCat:=JudgmentsL)). - apply (mon_commutative(MonoidalCat:=JudgmentsL)). - unfold identityProof; unfold cutProof; simpl; eapply cndr_inert. apply pl_eqv. auto. auto. - unfold identityProof; unfold cutProof; simpl; eapply cndr_inert. apply pl_eqv. auto. auto. - unfold identityProof; unfold cutProof; simpl; eapply cndr_inert. apply pl_eqv. auto. auto. - apply ndpc_comp; auto. - 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)). - intros. unfold ehom. unfold hom. unfold identityProof. unfold eid. simpl. unfold identityProof. - apply (cndr_inert pl_cnd); auto. - intros. unfold ehom. unfold comp. simpl. unfold cutProof. - rewrite <- (@ndr_prod_preserves_comp _ _ pl_eqv _ _ (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_eqv _ [a,, c |= b,, c]). - setoid_rewrite (@ndr_comp_left_identity _ _ pl_eqv [b |= c0]). - simpl; eapply cndr_inert. apply pl_eqv. 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)). - intros. unfold ehom. unfold hom. unfold identityProof. unfold eid. simpl. unfold identityProof. - eapply cndr_inert; auto. apply pl_eqv. - intros. unfold ehom. unfold comp. simpl. unfold cutProof. - rewrite <- (@ndr_prod_preserves_comp _ _ pl_eqv _ _ (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_eqv _ [c,,a |= c,,b]). - setoid_rewrite (@ndr_comp_left_identity _ _ pl_eqv [b |= c0]). - simpl; eapply cndr_inert. apply pl_eqv. auto. auto. - Defined. - - Definition Types_binoidal : EBinoidalCat TypesL (@T_Branch _). - refine - {| ebc_first := Types_first - ; ebc_second := Types_second - |}. - Defined. - - 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 - }. - simpl; eapply cndr_inert. unfold identityProof; apply pl_eqv. auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - auto. - simpl; eapply cndr_inert. unfold identityProof; apply pl_eqv. auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - auto. - 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 - }. - unfold eqv; unfold comp; simpl. - eapply cndr_inert. apply pl_eqv. auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - auto. - unfold eqv; unfold comp; simpl. - eapply cndr_inert. apply pl_eqv. auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - auto. - 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 - }. - unfold eqv; unfold comp; simpl. - eapply cndr_inert. apply pl_eqv. auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - auto. - unfold eqv; unfold comp; simpl. - eapply cndr_inert. apply pl_eqv. auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - auto. - Defined. - - (* 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. - - 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 }. - intros. - Opaque nd_id. - simpl. - Transparent nd_id. - - rename A into X. - rename B into Y. - unfold ehom. - nd_swap_ltac p pl_eqv. - setoid_rewrite p. - clear p. - - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) f) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - - clear q. - set (ni_commutes' (jud_mon_cancell pl_eqv) f) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - apply ndr_comp_respects; try reflexivity. - - Transparent nd_id. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - Defined. - - 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 }. - intros. - Opaque nd_id. - simpl. - Transparent nd_id. - - rename A into X. - rename B into Y. - unfold ehom. - nd_swap_ltac p pl_eqv. - setoid_rewrite p. - clear p. - - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) f) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - - clear q. - set (ni_commutes' (jud_mon_cancell pl_eqv) f) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - apply ndr_comp_respects; try reflexivity. - - Transparent nd_id. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - Defined. - - 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) }. - intros. - Opaque nd_id. - simpl. - Transparent nd_id. - - rename A into X. - rename B into Y. - unfold ehom. - nd_swap_ltac p pl_eqv. - setoid_rewrite p. - clear p. - - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) f) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - - clear q. - set (ni_commutes' (jud_mon_cancell pl_eqv) f) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - apply ndr_comp_respects; try reflexivity. - - Transparent nd_id. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - Defined. - - Instance Types_cancelr : Types_first [] <~~~> functor_id _ := - { ni_iso := Types_cancelr_iso }. - intros. - Opaque nd_id. - simpl. - unfold ehom. - nd_swap_ltac p pl_eqv. - setoid_rewrite p. - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) f) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - clear q. - - set (ni_commutes' (jud_mon_cancell pl_eqv) f) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - apply ndr_comp_respects; try reflexivity. - Transparent nd_id. - simpl. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - Defined. - - Instance Types_cancell : Types_second [] <~~~> functor_id _ := - { ni_iso := Types_cancell_iso }. - intros. - Opaque nd_id. - simpl. - unfold ehom. - nd_swap_ltac p pl_eqv. - setoid_rewrite p. - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) f) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - clear q. - - set (ni_commutes' (jud_mon_cancell pl_eqv) f) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - - apply ndr_comp_respects; try reflexivity. - Transparent nd_id. - simpl. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - Defined. - - Lemma TypesL_assoc_central a b c : CentralMorphism(H:=Types_binoidal) #((Types_assoc a b) c). - intros. - apply Build_CentralMorphism. - Opaque nd_id. - intros. - unfold bin_obj. - unfold ebc_bobj. - simpl. - unfold ehom. - nd_swap_ltac p pl_eqv. - setoid_rewrite p. - clear p. - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) g) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - clear q. - - set (ni_commutes' (jud_mon_cancell pl_eqv) g) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - apply ndr_comp_respects. - reflexivity. - - Transparent nd_id. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - - Opaque nd_id. - intros. - unfold bin_obj. - unfold ebc_bobj. - simpl. - unfold ehom. - symmetry. - nd_swap_ltac p pl_eqv. - setoid_rewrite p. - clear p. - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) g) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - clear q. - - set (ni_commutes' (jud_mon_cancell pl_eqv) g) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - apply ndr_comp_respects. - reflexivity. - - Transparent nd_id. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - 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_eqv. - setoid_rewrite p. - clear p. - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) g) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - clear q. - - set (ni_commutes' (jud_mon_cancell pl_eqv) g) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - apply ndr_comp_respects. - reflexivity. - - Transparent nd_id. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - - Opaque nd_id. - intros. - unfold bin_obj. - unfold ebc_bobj. - simpl. - unfold ehom. - symmetry. - nd_swap_ltac p pl_eqv. - setoid_rewrite p. - clear p. - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) g) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - clear q. - - set (ni_commutes' (jud_mon_cancell pl_eqv) g) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - apply ndr_comp_respects. - reflexivity. - - Transparent nd_id. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - 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_eqv. - setoid_rewrite p. - clear p. - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) g) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - clear q. - - set (ni_commutes' (jud_mon_cancell pl_eqv) g) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - apply ndr_comp_respects. - reflexivity. - - Transparent nd_id. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - - Opaque nd_id. - intros. - unfold bin_obj. - unfold ebc_bobj. - simpl. - unfold ehom. - symmetry. - nd_swap_ltac p pl_eqv. - setoid_rewrite p. - clear p. - setoid_rewrite (@nd_prod_split_left _ Rule pl_eqv _ _ _ []). - setoid_rewrite (@nd_prod_split_right _ Rule pl_eqv _ _ _ []). - repeat setoid_rewrite (@ndr_comp_associativity _ Rule pl_eqv). - setoid_rewrite <- (@ndr_comp_associativity _ Rule pl_eqv). - - set (ni_commutes' (jud_mon_cancelr pl_eqv) g) as q. - Opaque nd_id. - simpl in q. - setoid_rewrite <- q. - clear q. - - set (ni_commutes' (jud_mon_cancell pl_eqv) g) as q. - simpl in q. - set (coincide (pmon_triangle(PreMonoidalCat:=(Judgments_Category_premonoidal pl_eqv)))) 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_eqv). - apply ndr_comp_respects. - reflexivity. - - Transparent nd_id. - apply (cndr_inert pl_cnd); auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - apply ndpc_comp; auto. - 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 - }. - apply Build_Pentagon. - intros; simpl. - eapply cndr_inert. apply pl_eqv. - apply ndpc_comp. - apply ndpc_comp. - auto. - apply ndpc_comp. - apply ndpc_prod. - apply ndpc_comp. - apply ndpc_comp. - auto. - apply ndpc_comp. - auto. - auto. - auto. - auto. - auto. - auto. - apply ndpc_comp. - apply ndpc_comp. - auto. - apply ndpc_comp. - auto. - auto. - auto. - - apply Build_Triangle; intros; simpl. - eapply cndr_inert. apply pl_eqv. - auto. - apply ndpc_comp. - apply ndpc_comp. - auto. - apply ndpc_comp. - auto. - auto. - auto. - eapply cndr_inert. apply pl_eqv. auto. - auto. - intros; simpl; reflexivity. - intros; simpl; reflexivity. - apply TypesL_assoc_central. - apply TypesL_cancelr_central. - apply TypesL_cancell_central. - Defined. - - Definition TypesEnrichedInJudgments : SurjectiveEnrichment. - refine - {| senr_c_pm := TypesL_PreMonoidal - ; senr_v := JudgmentsL - ; senr_v_bin := Judgments_Category_binoidal _ - ; senr_v_pmon := Judgments_Category_premonoidal _ - ; senr_v_mon := Judgments_Category_monoidal _ - ; senr_c_bin := Types_binoidal - ; senr_c := TypesL - |}. - Defined. - - End LanguageCategory. + Coercion pl_eqv : ProgrammingLanguage >-> ContextND_Relation. + Coercion pl_cnd : ProgrammingLanguage >-> ContextND. End Programming_Language. -Implicit Arguments ND [ Judgment ]. +