notation "ℱ\sub 1 x" non associative with precedence 60 for @{'F1 $x}.
notation > "ℱ_1" non associative with precedence 90 for @{F1 ???}.
-interpretation "F1" 'F1 x = (F1 ___ x).
+interpretation "F1" 'F1 x = (F1 ??? x).
notation "ℱ\sub 2 x" non associative with precedence 60 for @{'F2 $x}.
notation > "ℱ_2" non associative with precedence 90 for @{F2 ???}.
-interpretation "F2" 'F2 x = (F2 ___ x).
+interpretation "F2" 'F2 x = (F2 ??? x).
lemma REW : ∀C1,C2: CAT2.∀F:arrows3 CAT2 C1 C2.∀X,Y:Fo ?? F.
arrows2 C2 (F (ℱ_1 X)) (F (ℱ_1 Y)) →
notation "ℳ\sub 1 x" non associative with precedence 60 for @{'Fm1 $x}.
notation > "ℳ_1" non associative with precedence 90 for @{Fm1 ?????}.
-interpretation "Fm1" 'Fm1 x = (Fm1 _____ x).
+interpretation "Fm1" 'Fm1 x = (Fm1 ????? x).
notation "ℳ\sub 2 x" non associative with precedence 60 for @{'Fm2 $x}.
notation > "ℳ_2" non associative with precedence 90 for @{Fm2 ?????}.
-interpretation "Fm2" 'Fm2 x = (Fm2 _____ x).
+interpretation "Fm2" 'Fm2 x = (Fm2 ????? x).
definition Fm :
∀C1,C2: CAT2.∀F:arrows3 CAT2 C1 C2.
| intros; apply (id_neutral_right2 C2 ?? (ℳ_2 a));
| intros; apply (id_neutral_left2 C2 ?? (ℳ_2 a));]
qed.
+
+definition faithful ≝
+ λC1,C2.λF:arrows3 CAT2 C1 C2.∀S,T.∀f,g:arrows2 C1 S T.
+ map_arrows2 ?? F ?? f = map_arrows2 ?? F ?? g → f=g.
+
+definition Ylppa : ∀C1,C2: CAT2.∀F:arrows3 CAT2 C1 C2.
+ faithful ?? F → let rC2 ≝ Apply ?? F in arrows3 CAT2 rC2 C1.
+intros; constructor 1;
+[ intro; apply (ℱ_1 o);
+| intros; constructor 1;
+ [ intros; apply (ℳ_1 c);
+ | apply hide; intros; apply f; change in e with (ℳ_2 a = ℳ_2 a');
+ lapply (FmP ????? a) as H1; lapply (FmP ????? a') as H2;
+ cut (REW ????? (map_arrows2 ?? F ?? (ℳ_1 a)) =
+ REW ????? (map_arrows2 ?? F ?? (ℳ_1 a')));[2:
+ apply (.= H1); apply (.= e); apply H2^-1;]
+ clear H1 H2 e; cases S in a a' Hcut; cases T;
+ cases H; cases H1; simplify; intros; assumption;]
+| intro; apply rule #;
+| intros; simplify; apply rule #;]
+qed.
+
+
+