record relation_pair (BP1,BP2: basic_pair): Type ≝
{ concr_rel: arrows1 ? (concr BP1) (concr BP2);
form_rel: arrows1 ? (form BP1) (form BP2);
- commute: concr_rel ∘ ⊩ = ⊩ ∘ form_rel
+ commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
}.
notation "hvbox (r \sub \c)" with precedence 90 for @{'concr_rel $r}.
∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
intros;
constructor 1;
- [ apply (λr,r'. r \sub\c ∘ ⊩ = r' \sub\c ∘ ⊩);
+ [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
| simplify;
intros;
apply refl1;
]
qed.
-lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → ⊩ \circ r \sub\f = ⊩ \circ r'\sub\f.
+lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
intros 7 (o1 o2 r r' H c1 f2);
split; intro H1;
[ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
intro;
constructor 1;
[1,2: apply id1;
- | lapply (id_neutral_left1 ? (concr o) ? (⊩)) as H;
- lapply (id_neutral_right1 ?? (form o) (⊩)) as H1;
+ | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
+ lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
apply (.= H);
apply (H1 \sup -1);]
qed.
constructor 1;
[ intros (r r1);
constructor 1;
- [ apply (r \sub\c ∘ r1 \sub\c)
- | apply (r \sub\f ∘ r1 \sub\f)
+ [ apply (r1 \sub\c ∘ r \sub\c)
+ | apply (r1 \sub\f ∘ r \sub\f)
| lapply (commute ?? r) as H;
lapply (commute ?? r1) as H1;
apply (.= ASSOC1);
apply (.= H‡#);
apply ASSOC1]
| intros;
- change with (a\sub\c ∘ b\sub\c ∘ ⊩ = a'\sub\c ∘ b'\sub\c ∘ ⊩);
- change in H with (a \sub\c ∘ ⊩ = a' \sub\c ∘ ⊩);
- change in H1 with (b \sub\c ∘ ⊩ = b' \sub\c ∘ ⊩);
+ change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
+ change in H with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
+ change in H1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
apply (.= ASSOC1);
apply (.= #‡H1);
apply (.= #‡(commute ?? b'));
| apply id_relation_pair
| apply relation_pair_composition
| intros;
- change with (a12\sub\c ∘ a23\sub\c ∘ a34\sub\c ∘ ⊩ =
- (a12\sub\c ∘ (a23\sub\c ∘ a34\sub\c) ∘ ⊩));
+ change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
+ ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
apply (ASSOC1‡#);
| intros;
- change with ((id_relation_pair o1)\sub\c ∘ a\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
- apply ((id_neutral_left1 ????)‡#);
- | intros;
- change with (a\sub\c ∘ (id_relation_pair o2)\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
+ change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
apply ((id_neutral_right1 ????)‡#);
- ]
+ | intros;
+ change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
+ apply ((id_neutral_left1 ????)‡#);]
qed.
-definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o) ≝ λo.ext ? ? (rel o).
+definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
+ intros; constructor 1;
+ [ apply (ext ? ? (rel o));
+ | intros;
+ apply (.= #‡H);
+ apply refl1]
+qed.
definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
λo.extS ?? (rel o).