Require Import Reification.
Require Import GeneralizedArrow.
Require Import GeneralizedArrowFromReification.
-Require Import ReificationFromGeneralizedArrow.
Section Programming_Language.
Defined.
Definition Types_first c : EFunctor TypesL TypesL (fun x => x,,c ).
- refine {| efunc := fun x y => (nd_rule (@se_expand_right _ _ _ _ _ _ _ (@pl_sequent_join PL) c x y)) |}.
+ refine {| efunc := fun x y => (@se_expand_right _ _ _ _ _ _ _ (@pl_sequent_join PL) c x y) |}.
intros; apply MonoidalCat_all_central.
intros. unfold ehom. unfold hom. unfold identityProof. unfold eid. simpl. unfold identityProof.
apply se_reflexive_right.
intros. unfold ehom. unfold comp. simpl. unfold cutProof.
- rewrite <- (@ndr_prod_preserves_comp _ _ pl_eqv _ _ [#se_expand_right _ c#] _ _ (nd_id1 (b|=c0))
- _ (nd_id1 (a,,c |= b,,c)) _ [#se_expand_right _ c#]).
+ rewrite <- (@ndr_prod_preserves_comp _ _ pl_eqv _ _ (se_expand_right _ c) _ _ (nd_id1 (b|=c0))
+ _ (nd_id1 (a,,c |= b,,c)) _ (se_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]).
apply se_cut_right.
Definition Types_second c : EFunctor TypesL TypesL (fun x => c,,x).
eapply Build_EFunctor.
- instantiate (1:=(fun x y => (nd_rule (@se_expand_left _ _ _ _ _ _ _ (@pl_sequent_join PL) c x y)))).
+ instantiate (1:=(fun x y => ((@se_expand_left _ _ _ _ _ _ _ (@pl_sequent_join PL) c x y)))).
intros; apply MonoidalCat_all_central.
intros. unfold ehom. unfold hom. unfold identityProof. unfold eid. simpl. unfold identityProof.
apply se_reflexive_left.
intros. unfold ehom. unfold comp. simpl. unfold cutProof.
- rewrite <- (@ndr_prod_preserves_comp _ _ pl_eqv _ _ [#se_expand_left _ c#] _ _ (nd_id1 (b|=c0))
- _ (nd_id1 (c,,a |= c,,b)) _ [#se_expand_left _ c#]).
+ rewrite <- (@ndr_prod_preserves_comp _ _ pl_eqv _ _ (se_expand_left _ c) _ _ (nd_id1 (b|=c0))
+ _ (nd_id1 (c,,a |= c,,b)) _ (se_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]).
apply se_cut_left.
|}.
Defined.
- Definition Types_PreMonoidal : PreMonoidalCat Types_binoidal [].
+ Definition Types_assoc a b : Types_second a >>>> Types_first b <~~~> Types_first b >>>> Types_second a.
admit.
Defined.
+ Definition Types_cancelr : Types_first [] <~~~> functor_id _.
+ admit.
+ Defined.
+
+ Definition Types_cancell : Types_second [] <~~~> functor_id _.
+ admit.
+ Defined.
+
+ Definition Types_assoc_ll a b : Types_second (a,,b) <~~~> Types_second b >>>> Types_second a.
+ admit.
+ Defined.
+
+ Definition Types_assoc_rr a b : Types_first (a,,b) <~~~> Types_first a >>>> Types_first b.
+ admit.
+ Defined.
+
+ Instance Types_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
+ }.
+ admit. (* pentagon law *)
+ admit. (* triangle law *)
+ admit. (* assoc_rr/assoc coherence *)
+ admit. (* assoc_ll/assoc coherence *)
+ Defined.
+
Definition TypesEnrichedInJudgments : Enrichment.
refine {| enr_c := TypesL |}.
Defined.