-(*
+ (*
||M|| This file is part of HELM, an Hypertextual, Electronic
||A|| Library of Mathematics, developed at the Computer Science
||T|| Department of the University of Bologna, Italy.
#A #x #P #Hx #y #Heq (cases Heq); //; qed.
theorem sym_eq: ∀A.∀x,y:A. x = y → y = x.
-#A #x #y #Heq @(rewrite_l A x (λz.z=x)); //; qed.
+#A #x #y #Heq @(rewrite_l A x (λz.z=x)) // qed-.
theorem rewrite_r: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. y = x → P y.
#A #x #P #Hx #y #Heq (cases (sym_eq ? ? ? Heq)); //; qed.
#A #B #Ha #Heq (elim Heq); //; qed.
theorem trans_eq : ∀A.∀x,y,z:A. x = y → y = z → x = z.
-#A #x #y #z #H1 #H2 >H1; //; qed.
+#A #x #y #z #H1 #H2 >H1; //; qed-.
theorem eq_f: ∀A,B.∀f:A→B.∀x,y:A. x=y → f x = f y.
#A #B #f #x #y #H >H; //; qed.
lemma injective_compose : ∀A,B,C,f,g.
injective A B f → injective B C g → injective A C (λx.g (f x)).
-/3/; qed.
+/3/; qed-.
(* extensional equality *)