fix miscompilation errors introduced by recent changes

[coq-categories.git] / src / PreMonoidalCenter.v

diff --git a/src/PreMonoidalCenter.v b/src/PreMonoidalCenter.v
index eb7f608..576265d 100644 (file)
--- a/src/PreMonoidalCenter.v
+++ b/src/PreMonoidalCenter.v
@@ -39,8 +39,8 @@ Lemma central_morphisms_compose `{bc:BinoidalCat}{a b c}(f:a~>b)(g:b~>c)
   Qed.
 
 (* the central morphisms of a category constitute a subcategory *)
-Definition Center `(bc:BinoidalCat) : SubCategory bc (fun _ => True) (fun _ _ _ _ f => CentralMorphism f).
-  apply Build_SubCategory; intros.
+Definition Center `(bc:BinoidalCat) : WideSubcategory bc (fun _ _ f => CentralMorphism f).
+  apply Build_WideSubcategory; intros.
   apply Build_CentralMorphism; intros.
   abstract (setoid_rewrite (fmor_preserves_id(bin_first c));
               setoid_rewrite (fmor_preserves_id(bin_first d));
@@ -51,6 +51,53 @@ Definition Center `(bc:BinoidalCat) : SubCategory bc (fun _ => True) (fun _ _ _
   apply central_morphisms_compose; auto.
   Qed.
 
+(* if one half of an iso is central, so is the other half *)
+Lemma iso_central_both `{C:BinoidalCat}{a b:C}(i:a ≅ b) : CentralMorphism #i -> CentralMorphism #i⁻¹.
+  intro cm.
+  apply Build_CentralMorphism; intros; simpl.
+    etransitivity.  
+    symmetry.
+    apply right_identity.
+    setoid_rewrite associativity.
+    setoid_replace (id (a ⊗ d)) with ((#i >>> iso_backward i) ⋉ d).
+    setoid_rewrite <- fmor_preserves_comp.
+    setoid_rewrite <- associativity.
+    setoid_rewrite juggle3.
+    setoid_rewrite <- (centralmor_first(CentralMorphism:=cm)).
+    setoid_rewrite <- associativity.
+    setoid_rewrite (fmor_preserves_comp (-⋉c)).
+    apply comp_respects; try reflexivity.
+    etransitivity; [ idtac | apply left_identity ].
+    apply comp_respects; try reflexivity.
+    etransitivity; [ idtac | apply (fmor_preserves_id (-⋉c)) ].
+    apply (fmor_respects (-⋉c)).
+    apply iso_comp2.
+    symmetry.
+    etransitivity; [ idtac | apply (fmor_preserves_id (-⋉d)) ].
+    apply (fmor_respects (-⋉d)).
+    apply iso_comp1.
+
+    etransitivity.  
+    symmetry.
+    apply left_identity.
+    setoid_replace (id (c ⊗ b)) with (c ⋊ (iso_backward i >>> #i)).
+    setoid_rewrite <- fmor_preserves_comp.
+    setoid_rewrite juggle3.
+    setoid_rewrite <- (centralmor_second(CentralMorphism:=cm)).
+    setoid_rewrite associativity.
+    apply comp_respects; try reflexivity.
+    setoid_rewrite associativity.
+    setoid_rewrite (fmor_preserves_comp (d⋊-)).
+    etransitivity; [ idtac | apply right_identity ].
+    apply comp_respects; try reflexivity.
+    etransitivity; [ idtac | apply (fmor_preserves_id (d⋊-)) ].
+    apply (fmor_respects (d⋊-)).
+    apply iso_comp1.
+    symmetry.
+    etransitivity; [ idtac | apply (fmor_preserves_id (c⋊-)) ].
+    apply (fmor_respects (c⋊-)).
+    apply iso_comp2.
+    Qed.
 
 Lemma first_preserves_centrality `{C:PreMonoidalCat}{a}{b}(f:a~~{C}~~>b){c} : CentralMorphism f -> CentralMorphism (f ⋉ c).
   intro cm.
@@ -316,21 +363,20 @@ Section Center_is_Monoidal.
 
   Context `(pm:PreMonoidalCat(I:=pmI)).
 
-  Definition Center_bobj : Center pm -> Center pm -> Center pm.
-    apply PreMonoidalSubCategory_bobj.
-    intros; auto.
-    Defined.
-
-  Definition Center_is_Binoidal : BinoidalCat (Center pm) Center_bobj.
-    apply PreMonoidalSubCategory_is_Binoidal.
-    intros.
-    apply first_preserves_centrality; auto.
-    intros.
-    apply second_preserves_centrality; auto.
+  Definition Center_is_Binoidal : BinoidalCat (Center pm) bin_obj'.
+    apply PreMonoidalWideSubcategory_is_Binoidal.
+    intros; apply first_preserves_centrality; auto.
+    intros; apply second_preserves_centrality; auto.
     Defined.
 
-  Definition Center_is_PreMonoidal : PreMonoidalCat Center_is_Binoidal (exist _ pmI I).
-    apply PreMonoidalSubCategory_PreMonoidal.
+  Definition Center_is_PreMonoidal : PreMonoidalCat Center_is_Binoidal pmI.
+    apply PreMonoidalWideSubcategory_PreMonoidal; intros.
+    apply pmon_assoc_central.
+    apply iso_central_both; apply pmon_assoc_central.
+    apply pmon_cancell_central.
+    apply iso_central_both; apply pmon_cancell_central.
+    apply pmon_cancelr_central.
+    apply iso_central_both; apply pmon_cancelr_central.
     Defined.
 
   Definition Center_is_Monoidal : MonoidalCat Center_is_PreMonoidal.
@@ -338,9 +384,9 @@ Section Center_is_Monoidal.
     apply Build_CommutativeCat.
     intros.
     apply Build_CentralMorphism; unfold hom; 
-      intros; destruct f; destruct a; destruct c; destruct d; destruct b; destruct g; simpl in *.
-    apply centralmor_second.
-    apply centralmor_second.
+      intros; destruct f; destruct g; simpl in *.
+    apply (centralmor_second(CentralMorphism:=c1)).
+    apply (centralmor_second(CentralMorphism:=c0)).
     Defined.
 
 End Center_is_Monoidal.