lemma eq_to_eq':
∀o1,o2.∀r,r':relation_pair_setoid o1 o2. r =_1 r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
- intros 7 (o1 o2 r r' H c1 f2);
- split; intro H1;
- [ apply (. (commute ?? r')^-1 ??);
- apply (. H^-1 ??);
- apply (. commute ?? r ??);
- assumption;
- | apply (. (commute ?? r)^-1 ??);
- apply (. H ??);
- apply (. commute ?? r' ??);
- assumption;
- ]
+ intros 5 (o1 o2 r r' H);
+ apply (.= (commute ?? r)^-1);
+ change in H with (⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
+ apply rule (.= H);
+ apply (commute ?? r').
qed.
definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
∀P,Q. arrows1 BP P Q → relation_pair_setoid P Q ≝ λP,Q,x.x.
coercion relation_pair_setoid_of_arrows1_BP.
+(*
definition BPext: ∀o: BP. (form o) ⇒_1 Ω^(concr o).
intros; constructor 1;
[ apply (ext ? ? (rel o));
interpretation "basic pair relation for subsets" 'Vdash2 x y c = (fun21 (concr ?) ?? (relS c) x y).
interpretation "basic pair relation for subsets (non applied)" 'Vdash c = (fun21 ??? (relS c)).
+*)