(garrow_functor >>>> bin_second(BinoidalCat:=enr_c_bin C) (R' a))
<~~~> (bin_second(BinoidalCat:=enr_v_pmon K) a >>>> garrow_functor) :=
{ ni_iso := fun a => iso_id _ }.
- admit.
+
+ intros.
+ etransitivity. apply iso_id_lemma1. symmetry.
+ etransitivity. apply iso_id_lemma2. symmetry.
+
+ Opaque Underlying.
+ unfold garrow_functor.
+ unfold functor_comp at 1.
+ unfold functor_comp at 1.
+ Opaque functor_comp. simpl. Transparent functor_comp.
+
+ symmetry.
+ eapply transitivity.
+ apply (functor_comp_assoc (RestrictToImage reification) (R' >>>> step2_functor) (ebc_second (R' a)) f).
+ unfold functor_comp at 1.
+ unfold functor_comp at 1.
+ Opaque functor_comp. simpl. Transparent functor_comp.
+
+ symmetry.
+ eapply transitivity.
+ set (ni_commutes (mf_second(PreMonoidalFunctor:=reification_rstar reification) a) f) as qq.
+ unfold functor_comp in qq.
+ simpl in qq.
+ apply iso_shift_right' in qq.
+ apply (fmor_respects(R' >>>> step2_functor) _ _ qq).
+
+ apply (me_faithful(MonicEnrichment:=CMon)).
+ symmetry.
+ unfold fmor at 1.
+ eapply transitivity.
+ set (ni_commutes (mf_second(PreMonoidalFunctor:=CM) (R' a))) as zz.
+ unfold functor_comp in zz; unfold functor_fobj in zz; simpl in zz.
+ set (zz _ _ ((R' >>>> step2_functor) \ (reification \ f))) as zz'.
+ apply iso_shift_right' in zz'.
+ apply zz'.
+
+ unfold functor_comp; simpl.
+
+ symmetry.
+ eapply transitivity.
+ set full_roundtrip as full_roundtrip'.
+ unfold fmor in full_roundtrip'.
+ simpl in full_roundtrip'.
+ apply full_roundtrip'.
+
+ set (@iso_shift_right') as q. simpl in q. apply q. clear q.
+
+ set (@iso_shift_left) as q. simpl in q. apply q. clear q.
+
+ symmetry.
+ eapply transitivity.
+ set full_roundtrip as full_roundtrip'.
+ unfold fmor in full_roundtrip'.
+ simpl in full_roundtrip'.
+ apply (fun a' b' f z => fmor_respects (bin_second(BinoidalCat:=enr_v_bin C) z) _ _ (full_roundtrip' a' b' f)).
+ symmetry.
+
+ intros.
+ unfold bin_obj.
+ unfold senr_v_bobj.
+
+ setoid_rewrite <- associativity.
+ setoid_rewrite <- associativity.
+ setoid_rewrite <- associativity.
+ setoid_rewrite <- associativity.
+ simpl.
+ setoid_rewrite <- associativity.
+ etransitivity.
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity; [ idtac | apply right_identity ].
+ apply comp_respects; [ reflexivity | idtac ].
+ etransitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ apply (mf_consistent(PreMonoidalFunctor:=CM)).
+ apply iso_comp1.
+
+ eapply transitivity.
+ eapply comp_respects; [ idtac | reflexivity ].
+ eapply comp_respects; [ idtac | reflexivity ].
+ eapply comp_respects; [ idtac | reflexivity ].
+ eapply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ eapply symmetry.
+ eapply associativity.
+ eapply transitivity; [ idtac | apply left_identity ].
+ eapply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ eapply comp_respects; [ idtac | reflexivity ].
+ eapply symmetry.
+ apply (mf_consistent(PreMonoidalFunctor:=CM)).
+ apply iso_comp1.
+
+ eapply symmetry.
+ eapply transitivity.
+ setoid_rewrite <- fmor_preserves_comp.
+ setoid_rewrite <- fmor_preserves_comp.
+ eapply reflexivity.
+ eapply symmetry.
+
+ apply comp_respects; try reflexivity.
+
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity.
+ apply associativity.
+ apply comp_respects; try reflexivity.
+
+ eapply transitivity.
+ eapply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ eapply comp_respects; [ idtac | reflexivity ].
+ apply mf_consistent.
+ eapply transitivity.
+ eapply comp_respects; [ reflexivity | idtac ].
+ apply associativity.
+ apply iso_comp1_right.
+
+ eapply transitivity.
+ eapply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ apply associativity.
+ eapply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ eapply transitivity; [ idtac | apply left_identity ].
+ eapply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ eapply comp_respects; [ idtac | reflexivity ].
+ eapply symmetry.
+ eapply (mf_consistent(PreMonoidalFunctor:=reification)).
+ apply iso_comp1.
+
+ eapply transitivity.
+ eapply comp_respects; [ reflexivity | idtac ].
+ eapply symmetry.
+ apply (centralmor_first(CentralMorphism:=commutative_central(CommutativeCat:=enr_v_mon C) _)).
+ eapply transitivity; [ idtac | apply right_identity ].
+
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ eapply transitivity.
+ eapply comp_respects; [ idtac | reflexivity ].
+ unfold functor_fobj.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=enr_v_bin C) (reification_rstar_obj reification A))).
+
+ apply symmetry.
+ eapply transitivity.
+ apply right_identity.
+ apply symmetry.
+ eapply transitivity; [ idtac | apply left_identity ].
+ apply comp_respects; [ idtac | reflexivity ].
+
+ eapply transitivity.
+ Focus 2.
+ eapply (fmor_preserves_id (bin_first(BinoidalCat:=enr_v_bin C) (reification_rstar_obj reification A))).
+ idtac.
+ apply (fmor_respects (bin_first(BinoidalCat:=enr_v_bin C) (reification_rstar_obj reification A))).
+ apply iso_comp2.
Defined.
Implicit Arguments mf_first [[Ob] [Hom] [C1] [bin_obj'] [bc] [I1] [PM1] [Ob0] [Hom0] [C2] [bin_obj'0] [bc0] [I2] [PM2] [fobj] [F]].
Implicit Arguments mf_second [[Ob] [Hom] [C1] [bin_obj'] [bc] [I1] [PM1] [Ob0] [Hom0] [C2] [bin_obj'0] [bc0] [I2] [PM2] [fobj] [F]].
Implicit Arguments mf_i [[Ob] [Hom] [C1] [bin_obj'] [bc] [I1] [PM1] [Ob0] [Hom0] [C2] [bin_obj'0] [bc0] [I2] [PM2] [fobj] [F]].
+ Lemma assoc_lemma1 : forall a b c,
+ iso_backward ((mf_first reification c) (a ⊗ b)) >>>
+ iso_backward ((mf_second reification a) b) ⋉ reification c >>>
+ #(pmon_assoc(PreMonoidalCat:=enr_v_mon C) (reification a) (reification c) (reification_rstar_obj reification b)) >>>
+ reification a ⋊ #((mf_first reification c) b) >>>
+ #((mf_second reification a) (b ⊗ c)) ~~
+ reification \ #((pmon_assoc(PreMonoidalCat:=enr_v_mon K) a c) b).
+
+ intros.
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity.
+ apply associativity.
+ apply symmetry.
+ set (@iso_shift_right') as q.
+ simpl in q.
+ apply q.
+ clear q.
+
+ eapply symmetry.
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ set (mf_assoc(PreMonoidalFunctor:=reification)) as q.
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply q.
+
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply comp_respects; try reflexivity.
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ eapply transitivity; [ idtac | apply left_identity ].
+ apply comp_respects; try reflexivity.
+
+ eapply transitivity.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity; [ idtac |
+ apply (fmor_preserves_id (bin_first(BinoidalCat:=enr_v_bin C) _)) ].
+ apply (fmor_respects (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ apply iso_comp2.
+ Qed.
+
+ Lemma assoc_lemma2 : forall a b c,
+ iso_backward ((mf_first CM (garrow_fobj c)) _) >>>
+ (bin_first(BinoidalCat:=enr_v_bin C) _ \ iso_backward ((mf_second CM (garrow_fobj a)) (garrow_fobj b))) >>>
+ (#((pmon_assoc(PreMonoidalCat:=enr_v_mon C) _ _) _) >>>
+ (bin_second(BinoidalCat:=enr_v_bin C) _ \
+ #((mf_first CM (garrow_fobj c)) (garrow_fobj b)))) >>>
+ #((mf_second CM (garrow_fobj a)) _) ~~
+ HomFunctor C _ \ #((pmon_assoc(PreMonoidalCat:=enr_c_pm C) (garrow_fobj a) (garrow_fobj c)) (garrow_fobj b)).
+ intros.
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity.
+ apply associativity.
+ set (@iso_shift_right') as q.
+ apply symmetry.
+ simpl in q.
+ apply q.
+ clear q.
+
+ eapply symmetry.
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ set (mf_assoc(PreMonoidalFunctor:=CM)) as q.
+ apply q.
+
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply comp_respects; try reflexivity.
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ eapply transitivity; [ idtac | apply left_identity ].
+ apply comp_respects; try reflexivity.
+
+ eapply transitivity.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity; [ idtac |
+ apply (fmor_preserves_id (bin_first(BinoidalCat:=enr_v_bin C) _)) ].
+ apply (fmor_respects (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ apply iso_comp2.
+ Qed.
+
+ Lemma assoc_coherent (a b c : enr_v K) :
+ (#((pmon_assoc(PreMonoidalCat:=enr_c_pm C)
+ (garrow_functor a) (garrow_functor c)) (garrow_fobj b)) >>> garrow_functor a ⋊ #((garrow_first c) b)) >>>
+ #((garrow_second a) (b ⊗ c)) ~~
+ (#((garrow_second a) b) ⋉ garrow_functor c >>>
+ #((garrow_first c) (a ⊗ b))) >>> garrow_functor \ #((pmon_assoc(PreMonoidalCat:=enr_v_mon K) a c) b).
+ Opaque Underlying.
+ eapply transitivity.
+ eapply transitivity; [ idtac | apply right_identity ].
+ eapply comp_respects; [ eapply reflexivity | idtac ].
+ reflexivity.
+
+ apply symmetry.
+ eapply transitivity.
+ eapply transitivity; [ idtac | apply left_identity ].
+ eapply comp_respects; [ idtac | eapply reflexivity ].
+ eapply transitivity; [ idtac | apply right_identity ].
+ apply comp_respects.
+ simpl.
+ apply (fmor_preserves_id (ebc_first (garrow_functor c))).
+ reflexivity.
+
+ symmetry.
+ eapply transitivity.
+ eapply transitivity; [ idtac | apply right_identity ].
+ apply comp_respects.
+ reflexivity.
+ simpl.
+ apply (fmor_preserves_id (ebc_second (garrow_functor a))).
+ symmetry.
+
+ unfold functor_fobj.
+ unfold garrow_functor.
+ unfold step2_functor.
+ Opaque R'.
+ Opaque ff_functor_section_functor.
+ unfold functor_comp.
+ simpl.
+ Transparent R'.
+ Transparent ff_functor_section_functor.
+ idtac.
+ apply (me_faithful(MonicEnrichment:=CMon)).
+ eapply transitivity; [ eapply full_roundtrip | idtac ].
+
+ unfold fmor at 1.
+ unfold R'.
+ unfold me_homfunctor.
+ eapply transitivity.
+ eapply comp_respects; [ idtac | apply reflexivity ].
+ eapply comp_respects; [ apply reflexivity | idtac ].
+ eapply symmetry.
+ apply assoc_lemma1.
+
+ apply symmetry.
+ eapply transitivity.
+ eapply symmetry.
+ apply assoc_lemma2.
+
+ simpl.
+ eapply transitivity.
+ eapply comp_respects; [ apply reflexivity | idtac ].
+ eapply symmetry.
+ eapply (mf_consistent(PreMonoidalFunctor:=CM)).
+ apply symmetry.
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply comp_respects; [ idtac | apply reflexivity ].
+
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity.
+ apply associativity.
+ symmetry.
+ eapply transitivity.
+ apply associativity.
+ apply comp_respects; [ reflexivity | idtac ].
+
+ apply symmetry.
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity.
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply symmetry.
+ apply (mf_consistent(PreMonoidalFunctor:=reification)).
+ apply comp_respects; [ reflexivity | idtac ].
+ apply iso_comp1.
+ apply right_identity.
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply associativity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply symmetry.
+ eapply (centralmor_first(CentralMorphism:=commutative_central(CommutativeCat:=enr_v_mon C) _)).
+ eapply transitivity.
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ eapply symmetry.
+ apply (fmor_preserves_comp (bin_second(BinoidalCat:=enr_v_mon C) _)).
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply symmetry.
+ apply (fmor_preserves_comp (bin_second(BinoidalCat:=enr_v_mon C) _)).
+ apply comp_respects; [ reflexivity | idtac ].
+ apply associativity.
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ eapply transitivity.
+ apply associativity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity; [ idtac | apply right_identity ].
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ apply (fmor_preserves_comp (bin_second(BinoidalCat:=enr_v_mon C) _)).
+ eapply transitivity; [ idtac |
+ apply (fmor_preserves_id (bin_second(BinoidalCat:=enr_v_mon C) _)) ].
+ apply (fmor_respects (bin_second(BinoidalCat:=enr_v_mon C) _)).
+ apply iso_comp1.
+ apply reflexivity.
+
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ eapply symmetry.
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply comp_respects; [ idtac | reflexivity ].
+
+ apply symmetry.
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply associativity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ apply associativity.
+ apply iso_comp1_right.
+
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ apply associativity.
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ eapply symmetry.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply symmetry.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ eapply symmetry.
+ apply associativity.
+ eapply symmetry.
+ apply associativity.
+ apply reflexivity.
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity; [ idtac | apply right_identity ].
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity; [ idtac | apply (fmor_preserves_id (bin_first(BinoidalCat:=enr_v_mon C) _)) ].
+ apply (fmor_respects (bin_first(BinoidalCat:=enr_v_mon C) _)).
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply mf_consistent.
+ apply iso_comp1.
+
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply symmetry.
+ eapply (centralmor_first(CentralMorphism:=commutative_central(CommutativeCat:=enr_v_mon C) _)).
+ eapply transitivity.
+ apply associativity.
+ eapply transitivity.
+ apply associativity.
+ set (fun a b => isos_forward_equal_then_backward_equal _ _ (mf_consistent(PreMonoidalFunctor:=CM) a b)) as q.
+ apply symmetry.
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ set (q (garrow_fobj b) (garrow_fobj a)) as q'.
+ simpl in q'.
+ set (fun qq => fmor_respects (bin_first(BinoidalCat:=enr_v_bin C) qq) _ _ q') as q''.
+ eapply symmetry.
+ apply q''.
+ apply comp_respects; [ reflexivity | idtac ].
+
+ apply symmetry.
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ apply comp_respects; [ reflexivity | idtac ].
+ apply comp_respects; [ idtac | reflexivity ].
+ set (ni_commutes (pmon_assoc_rr(PreMonoidalCat:=enr_v_mon C) (reification b) (reification c))
+ #(homset_tensor_iso a)) as xx.
+ unfold functor_comp in xx.
+ simpl in xx.
+ set (pmon_coherent_r(PreMonoidalCat:=enr_v_mon C) (reification a) (reification b) (reification c)) as yy.
+ set (isos_forward_equal_then_backward_equal _ _ yy) as yy'.
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ apply symmetry in yy'.
+ eapply transitivity; [ idtac | apply yy' ].
+ eapply symmetry.
+ apply iso_inv_inv.
+ clear yy' yy.
+ apply iso_shift_right' in xx.
+ apply symmetry in xx.
+ idtac.
+ assert (#((pmon_assoc_rr(PreMonoidalCat:=enr_v_mon C) (reification b) (reification c)) (reification a)) ⁻¹ >>>
+ #(homset_tensor_iso a) ⋉ (reification b ⊗ reification c)
+ ~~
+ (#(homset_tensor_iso a) ⋉ reification b) ⋉ reification c >>>
+ #((pmon_assoc_rr(PreMonoidalCat:=enr_v_mon C) (reification b) (reification c)) _)⁻¹).
+ apply iso_shift_left.
+ eapply transitivity.
+ apply associativity.
+ apply xx.
+ apply H.
+ clear q.
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply symmetry.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=enr_v_mon C) _)).
+ eapply transitivity.
+ apply associativity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ eapply transitivity; [ idtac | apply left_identity ].
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity; [ idtac |
+ apply (fmor_preserves_id (bin_first(BinoidalCat:=enr_v_bin C) _)) ].
+ apply (fmor_respects (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity; [ idtac |
+ apply (fmor_preserves_id (bin_first(BinoidalCat:=enr_v_bin C) _)) ].
+ apply (fmor_respects (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ apply iso_comp2.
+
+ set (fun a' => ni_commutes (pmon_assoc(PreMonoidalCat:=enr_v_mon C) a' (reification c))
+ (iso_backward (homset_tensor_iso b))) as xx.
+ unfold fmor in xx; unfold functor_comp in xx; simpl in xx.
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ eapply symmetry.
+ eapply associativity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ eapply isos_forward_equal_then_backward_equal.
+ apply (pmon_coherent_r(PreMonoidalCat:=enr_v_mon C)).
+ apply iso_inv_inv.
+ eapply symmetry.
+ eapply xx.
+ clear xx.
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ eapply associativity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply symmetry.
+ apply (fmor_preserves_comp (bin_second(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity.
+ eapply symmetry.
+ eapply associativity.
+ eapply transitivity; [ idtac | apply left_identity ].
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply (fmor_preserves_comp (bin_second(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity; [ idtac |
+ apply (fmor_preserves_id (bin_second(BinoidalCat:=enr_v_bin C) _)) ].
+ apply fmor_respects.
+ eapply transitivity.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity; [ idtac |
+ apply (fmor_preserves_id (bin_first(BinoidalCat:=enr_v_bin C) _)) ].
+ apply (fmor_respects (bin_first(BinoidalCat:=enr_v_bin C) _)).
+ apply iso_comp2.
+
+ set (fun a b => ni_commutes (pmon_assoc_ll(PreMonoidalCat:=enr_v_mon C) a b)) as xx.
+ unfold functor_comp in xx.
+ simpl in xx.
+ eapply transitivity.
+ apply comp_respects; [ reflexivity | idtac ].
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply symmetry.
+ apply (pmon_coherent_l(PreMonoidalCat:=enr_v_mon C)).
+ apply xx.
+ eapply transitivity.
+ eapply symmetry.
+ apply associativity.
+ apply symmetry.
+ eapply transitivity.
+ eapply symmetry.
+ apply (pmon_coherent_l(PreMonoidalCat:=enr_v_mon C)).
+ apply symmetry.
+ eapply transitivity; [ idtac | apply left_identity ].
+ apply comp_respects; [ idtac | reflexivity ].
+ eapply transitivity.
+ apply (fmor_preserves_comp (bin_second(BinoidalCat:=enr_v_bin C) _)).
+ eapply transitivity; [ idtac |
+ apply (fmor_preserves_id (bin_second(BinoidalCat:=enr_v_bin C) _)) ].
+ apply fmor_respects.
+ apply iso_comp2.
+ Qed.
+
Lemma cancell_lemma `(F:PreMonoidalFunctor) b :
iso_backward (mf_i F) ⋉ (F b) >>> #(pmon_cancell (F b)) ~~
#((mf_first F b) _) >>> F \ #(pmon_cancell b).
apply (fmor_respects (-⋉ F b)).
apply iso_comp2.
Qed.
-
+*)
Lemma cancell_coherent (b:enr_v K) :
#(pmon_cancell(PreMonoidalCat:=enr_c_pm C) (garrow_functor b)) ~~
(#(iso_id (enr_c_i C)) ⋉ garrow_functor b >>>
set (@comp_respects) as qq.
unfold Proper in qq.
unfold respectful in qq.
- set (qq _ _ _ _ _ _ (iso_backward (mf_i F) ⋉ F b) (iso_backward (mf_i F) ⋉ F b) (reflexivity _) _ _ q) as q'.
+ set (qq _ _ _ _ _ _ (F b ⋊ iso_backward (mf_i F)) (F b ⋊ iso_backward (mf_i F)) (reflexivity _) _ _ q) as q'.
setoid_rewrite <- associativity in q'.
clear q qq.
- setoid_rewrite (fmor_preserves_comp (-⋉ F b)) in q'.
+ setoid_rewrite (fmor_preserves_comp (F b ⋊ -)) in q'.
eapply transitivity; [ apply q' | idtac ].
clear q'.
setoid_rewrite <- associativity.
apply comp_respects; try reflexivity.
symmetry.
apply iso_shift_left.
+ eapply transitivity.
+ apply comp_respects; [ idtac | reflexivity ].
+ apply mf_consistent.
setoid_rewrite iso_comp1.
symmetry.
- eapply transitivity; [ idtac | eapply (fmor_preserves_id (-⋉ F b))].
- apply (fmor_respects (-⋉ F b)).
+ eapply transitivity; [ idtac | eapply (fmor_preserves_id (F b ⋊ - ))].
+ apply (fmor_respects (F b ⋊ -)).
apply iso_comp2.
Qed.
#(pmon_cancelr(PreMonoidalCat:=enr_c_pm C) (garrow_functor b)) ~~
(garrow_functor b ⋊ #(iso_id (enr_c_i C)) >>>
#((garrow_second b) (enr_v_i K))) >>> garrow_functor \ #(pmon_cancelr(PreMonoidalCat:=enr_v_mon K) b).
-
Opaque Underlying.
Opaque fmor.
intros; simpl.
; mf_second := garrow_second
; mf_i := iso_id _ }.
intros; reflexivity.
- intros.
- unfold garrow_functor.
- unfold fmor.
- Opaque fmor. simpl.
- unfold step2_functor.
- admit.
- Transparent fmor.
-
+ intros; apply (reification_host_lang_pure _ _ _ reification).
apply cancell_coherent.
apply cancelr_coherent.
- admit.
+ apply assoc_coherent.
Defined.
Definition garrow_from_reification : GeneralizedArrow K CM :=
- {| ga_functor_monoidal := garrow_monoidal |}.
+ {| ga_functor_monoidal := garrow_monoidal
+ ; ga_host_lang_pure := reification_host_lang_pure _ _ _ reification
+ |}.
End GArrowFromReification.
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