apply (H1 \sup -1);]
qed.
-definition Orelation_pair_composition:
- ∀o1,o2,o3. binary_morphism2 (Orelation_pair_setoid o1 o2) (Orelation_pair_setoid o2 o3) (Orelation_pair_setoid o1 o3).
- intros;
- constructor 1;
- [ intros (r r1);
+lemma Orelation_pair_composition:
+ ∀o1,o2,o3:Obasic_pair.
+ Orelation_pair_setoid o1 o2 → Orelation_pair_setoid o2 o3→Orelation_pair_setoid o1 o3.
+intros 3 (o1 o2 o3);
+ intros (r r1);
constructor 1;
[ apply (r1 \sub\c ∘ r \sub\c)
| apply (r1 \sub\f ∘ r \sub\f)
apply rule (.= ASSOC ^ -1);
apply (.= H‡#);
apply rule ASSOC]
- | intros;
+qed.
+
+
+lemma Orelation_pair_composition_is_morphism:
+ ∀o1,o2,o3:Obasic_pair.
+ Πa,a':Orelation_pair_setoid o1 o2.Πb,b':Orelation_pair_setoid o2 o3.
+ a=a' →b=b' →
+ Orelation_pair_composition o1 o2 o3 a b
+ = Orelation_pair_composition o1 o2 o3 a' b'.
+intros;
change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
apply (.= e‡#);
apply rule (.= ASSOC);
apply (.= #‡(Ocommute ?? b')\sup -1);
- apply rule (ASSOC \sup -1)]
+ apply rule (ASSOC \sup -1);
qed.
-
-definition OBP: category2.
- constructor 1;
- [ apply Obasic_pair
- | apply Orelation_pair_setoid
- | apply Oid_relation_pair
- | apply Orelation_pair_composition
- | intros;
+
+definition Orelation_pair_composition_morphism:
+ ∀o1,o2,o3. binary_morphism2 (Orelation_pair_setoid o1 o2) (Orelation_pair_setoid o2 o3) (Orelation_pair_setoid o1 o3).
+intros; constructor 1;
+[ apply Orelation_pair_composition;
+| apply Orelation_pair_composition_is_morphism;]
+qed.
+
+lemma Orelation_pair_composition_morphism_assoc:
+∀o1,o2,o3,o4:Obasic_pair
+ .Πa12:Orelation_pair_setoid o1 o2
+ .Πa23:Orelation_pair_setoid o2 o3
+ .Πa34:Orelation_pair_setoid o3 o4
+ .Orelation_pair_composition_morphism o1 o3 o4
+ (Orelation_pair_composition_morphism o1 o2 o3 a12 a23) a34
+ =Orelation_pair_composition_morphism o1 o2 o4 a12
+ (Orelation_pair_composition_morphism o2 o3 o4 a23 a34).
+ intros;
change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
apply rule (ASSOC‡#);
- | intros;
+qed.
+
+lemma Orelation_pair_composition_morphism_respects_id:
+Πo1:Obasic_pair
+.Πo2:Obasic_pair
+ .Πa:Orelation_pair_setoid o1 o2
+ .Orelation_pair_composition_morphism o1 o1 o2 (Oid_relation_pair o1) a=a.
+ intros;
change with (⊩ ∘ (a\sub\c ∘ (Oid_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
apply ((id_neutral_right2 ????)‡#);
- | intros;
+qed.
+
+lemma Orelation_pair_composition_morphism_respects_id_r:
+Πo1:Obasic_pair
+.Πo2:Obasic_pair
+ .Πa:Orelation_pair_setoid o1 o2
+ .Orelation_pair_composition_morphism o1 o2 o2 a (Oid_relation_pair o2)=a.
+intros;
change with (⊩ ∘ ((Oid_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
- apply ((id_neutral_left2 ????)‡#);]
+ apply ((id_neutral_left2 ????)‡#);
+qed.
+
+definition OBP: category2.
+ constructor 1;
+ [ apply Obasic_pair
+ | apply Orelation_pair_setoid
+ | apply Oid_relation_pair
+ | apply Orelation_pair_composition_morphism
+ | apply Orelation_pair_composition_morphism_assoc;
+ | apply Orelation_pair_composition_morphism_respects_id;
+ | apply Orelation_pair_composition_morphism_respects_id_r;]
qed.
definition Obasic_pair_of_objs2_OBP: objs2 OBP → Obasic_pair ≝ λx.x.