rel: arrows2 ? concr form
}.
-notation > "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y ?}.
-notation < "x (⊩ \below c) y" with precedence 45 for @{'Vdash2 $x $y $c}.
-notation < "⊩ \sub c" with precedence 60 for @{'Vdash $c}.
-notation > "⊩ " with precedence 60 for @{'Vdash ?}.
-
interpretation "basic pair relation indexed" 'Vdash2 x y c = (rel c x y).
interpretation "basic pair relation (non applied)" 'Vdash c = (rel c).
alias symbol "eq" = "setoid1 eq".
alias symbol "compose" = "category1 composition".
+(*DIFFER*)
+
+alias symbol "eq" = "setoid2 eq".
+alias symbol "compose" = "category2 composition".
record relation_pair (BP1,BP2: basic_pair): Type2 ≝
{ concr_rel: arrows2 ? (concr BP1) (concr BP2);
form_rel: arrows2 ? (form BP1) (form BP2);
]
qed.
+definition relation_pair_of_relation_pair_setoid:
+ ∀P,Q. relation_pair_setoid P Q → relation_pair P Q ≝ λP,Q,x.x.
+coercion relation_pair_of_relation_pair_setoid.
+
lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
intros 5 (o1 o2 r r' H); change in H with (⊩ ∘ r\sub\c = ⊩ ∘ r'\sub\c);
apply (.= ((commute ?? r) \sup -1));
apply ((id_neutral_left2 ????)‡#);]
qed.
+definition basic_pair_of_objs2_BP: objs2 BP → basic_pair ≝ λx.x.
+coercion basic_pair_of_objs2_BP.
+
+definition relation_pair_setoid_of_arrows2_BP:
+ ∀P,Q.arrows2 BP P Q → relation_pair_setoid P Q ≝ λP,Q,c.c.
+coercion relation_pair_setoid_of_arrows2_BP.
(*
definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr _) __ (relS _) x y).
interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ___ (relS _)).
*)
+
+notation "□ \sub b" non associative with precedence 90 for @{'box $b}.
+notation > "□_term 90 b" non associative with precedence 90 for @{'box $b}.
+interpretation "Universal image ⊩⎻*" 'box x = (fun12 _ _ (or_f_minus_star _ _) (rel x)).
+
+notation "◊ \sub b" non associative with precedence 90 for @{'diamond $b}.
+notation > "◊_term 90 b" non associative with precedence 90 for @{'diamond $b}.
+interpretation "Existential image ⊩" 'diamond x = (fun12 _ _ (or_f _ _) (rel x)).
+
+notation "'Rest' \sub b" non associative with precedence 90 for @{'rest $b}.
+notation > "'Rest'⎽term 90 b" non associative with precedence 90 for @{'rest $b}.
+interpretation "Universal pre-image ⊩*" 'rest x = (fun12 _ _ (or_f_star _ _) (rel x)).
+
+notation "'Ext' \sub b" non associative with precedence 90 for @{'ext $b}.
+notation > "'Ext'⎽term 90 b" non associative with precedence 90 for @{'ext $b}.
+interpretation "Existential pre-image ⊩⎻" 'ext x = (fun12 _ _ (or_f_minus _ _) (rel x)).