-interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
-interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
-interpretation "setoid1 symmetry" 'invert r = (sym1 ____ r).
-interpretation "setoid symmetry" 'invert r = (sym ____ r).
+interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
+interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
+interpretation "setoid symmetry" 'invert r = (sym ???? r).
-interpretation "trans1" 'trans r = (trans1 _____ r).
-interpretation "trans" 'trans r = (trans _____ r).
+interpretation "trans1" 'trans r = (trans1 ????? r).
+interpretation "trans" 'trans r = (trans ????? r).
notation "† c" with precedence 90 for @{'prop1 $c }.
notation "l ‡ r" with precedence 90 for @{'prop $l $r }.
notation "#" with precedence 90 for @{'refl}.
notation "† c" with precedence 90 for @{'prop1 $c }.
notation "l ‡ r" with precedence 90 for @{'prop $l $r }.
notation "#" with precedence 90 for @{'refl}.
-interpretation "prop_1" 'prop1 c = (prop_1 _____ c).
-interpretation "prop1" 'prop l r = (prop1 ________ l r).
-interpretation "prop" 'prop l r = (prop ________ l r).
-interpretation "refl1" 'refl = (refl1 ___).
-interpretation "refl" 'refl = (refl ___).
+interpretation "prop_1" 'prop1 c = (prop_1 ????? c).
+interpretation "prop1" 'prop l r = (prop1 ???????? l r).
+interpretation "prop" 'prop l r = (prop ???????? l r).
+interpretation "refl1" 'refl = (refl1 ???).
+interpretation "refl" 'refl = (refl ???).
notation "'ASSOC'" with precedence 90 for @{'assoc}.
notation "'ASSOC1'" with precedence 90 for @{'assoc1}.
notation "'ASSOC'" with precedence 90 for @{'assoc}.
notation "'ASSOC1'" with precedence 90 for @{'assoc1}.
-interpretation "category1 composition" 'compose x y = (fun1 ___ (comp1 ____) y x).
-interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ________).
-interpretation "category composition" 'compose x y = (fun ___ (comp ____) y x).
-interpretation "category assoc" 'assoc = (comp_assoc ________).
+interpretation "category1 composition" 'compose x y = (fun1 ??? (comp1 ????) y x).
+interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ????????).
+interpretation "category composition" 'compose x y = (fun ??? (comp ????) y x).
+interpretation "category assoc" 'assoc = (comp_assoc ????????).
| intros; simplify; whd; intros; simplify; apply refl;
| intros; simplify; whd; intros; simplify; apply refl;
]
| intros; simplify; whd; intros; simplify; apply refl;
| intros; simplify; whd; intros; simplify; apply refl;
]
-qed.
\ No newline at end of file
+qed.
+
+definition setoid_OF_SET: objs1 SET → setoid.
+ intros; apply o; qed.
+
+coercion setoid_OF_SET.
+
+
+definition prop_1_SET :
+ ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:A.eq1 ? a b→eq1 ? (w a) (w b).
+intros; apply (prop_1 A B w a b H);
+qed.
+
+interpretation "SET dagger" 'prop1 h = (prop_1_SET ? ? ? ? ? h).