Require Import RepresentableStructure_ch7_2.
Require Import FunctorCategories_ch7_7.
-Require Import Enrichments.
Require Import NaturalDeduction.
-Require Import NaturalDeductionCategory.
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 ].
+
projects / coq-hetmet.git / blobdiff