(**************************************************************************)
-(* ___ *)
+(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* ||T|| *)
[refl_eq \Rightarrow H]).
qed.
-theorem reflexive_eq : \forall A:Type. reflexive A (eq A).
-simplify.intros.apply refl_eq.
-qed.
+variant reflexive_eq : \forall A:Type. reflexive A (eq A)
+\def refl_eq.
+(* simplify.intros.apply refl_eq. *)
theorem symmetric_eq: \forall A:Type. symmetric A (eq A).
unfold symmetric.intros.elim H. apply refl_eq.
qed.
-theorem sym_eq : \forall A:Type.\forall x,y:A. x=y \to y=x
+variant sym_eq : \forall A:Type.\forall x,y:A. x=y \to y=x
\def symmetric_eq.
theorem transitive_eq : \forall A:Type. transitive A (eq A).
unfold transitive.intros.elim H1.assumption.
qed.
-theorem trans_eq : \forall A:Type.\forall x,y,z:A. x=y \to y=z \to x=z
+variant trans_eq : \forall A:Type.\forall x,y,z:A. x=y \to y=z \to x=z
\def transitive_eq.
theorem eq_elim_r:
theorem eq_f: \forall A,B:Type.\forall f:A\to B.
\forall x,y:A. x=y \to f x = f y.
-intros.elim H.reflexivity.
+intros.elim H.apply refl_eq.
+qed.
+
+theorem eq_f': \forall A,B:Type.\forall f:A\to B.
+\forall x,y:A. x=y \to f y = f x.
+intros.elim H.apply refl_eq.
qed.
+(*
coercion cic:/matita/logic/equality/sym_eq.con.
coercion cic:/matita/logic/equality/eq_f.con.
+*)
default "equality"
cic:/matita/logic/equality/eq.ind
- cic:/matita/logic/equality/sym_eq.con
- cic:/matita/logic/equality/trans_eq.con
+ cic:/matita/logic/equality/symmetric_eq.con
+ cic:/matita/logic/equality/transitive_eq.con
cic:/matita/logic/equality/eq_ind.con
cic:/matita/logic/equality/eq_elim_r.con
cic:/matita/logic/equality/eq_f.con
- cic:/matita/logic/equality/eq_OF_eq.con. (* \x.sym (eq_f x) *)
+ cic:/matita/logic/equality/eq_f'.con. (* \x.sym (eq_f x) *)
theorem eq_f2: \forall A,B,C:Type.\forall f:A\to B \to C.
\forall x1,x2:A. \forall y1,y2:B.