(**************************************************************************)
-(* ___ *)
+(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* ||T|| *)
'neq x y = (cic:/matita/logic/connectives/Not.con
(cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y)).
-theorem eq_ind':
- \forall A. \forall x:A. \forall P: \forall y:A. x=y \to Prop.
+theorem eq_rect':
+ \forall A. \forall x:A. \forall P: \forall y:A. x=y \to Type.
P ? (refl_eq ? x) \to \forall y:A. \forall p:x=y. P y p.
intros.
exact
- (match p return \lambda y. \lambda p.P y p with
- [refl_eq \Rightarrow H]).
+ (match p1 return \lambda y. \lambda p.P y p with
+ [refl_eq \Rightarrow p]).
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:
intros. elim (sym_eq ? ? ? H1).assumption.
qed.
+theorem eq_elim_r':
+ \forall A:Type.\forall x:A. \forall P: A \to Set.
+ P x \to \forall y:A. y=x \to P y.
+intros. elim (sym_eq ? ? ? H).assumption.
+qed.
+
+theorem eq_elim_r'':
+ \forall A:Type.\forall x:A. \forall P: A \to Type.
+ P x \to \forall y:A. y=x \to P y.
+intros. elim (sym_eq ? ? ? H).assumption.
+qed.
+
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.apply refl_eq.
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/transitive_eq.con
cic:/matita/logic/equality/eq_ind.con
cic:/matita/logic/equality/eq_elim_r.con
+ cic:/matita/logic/equality/eq_rec.con
+ cic:/matita/logic/equality/eq_elim_r'.con
+ cic:/matita/logic/equality/eq_rect.con
+ cic:/matita/logic/equality/eq_elim_r''.con
cic:/matita/logic/equality/eq_f.con
+(* *)
+ cic:/matita/logic/equality/eq_OF_eq.con.
+(* *)
+(*
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.
\forall u:x=y.
comp ? ? ? ? u u = refl_eq ? y.
intros.
- apply (eq_ind' ? ? ? ? ? u).
+ apply (eq_rect' ? ? ? ? ? u).
reflexivity.
qed.
\forall u:x=y.
nu_inv ? H ? ? (nu ? H ? ? u) = u.
intros.
- apply (eq_ind' ? ? ? ? ? u).
+ apply (eq_rect' ? ? ? ? ? u).
unfold nu_inv.
apply trans_sym_eq.
qed.
focus 8.
clear q; clear p.
intro.
- apply (eq_ind' ? ? ? ? ? q);
+ apply (eq_rect' ? ? ? ? ? q);
fold simplify (nu ? ? (refl_eq ? x)).
generalize in match (nu ? ? (refl_eq ? x)); intro.
apply
- (eq_ind' A x
+ (eq_rect' A x
(\lambda y. \lambda u.
eq_ind A x (\lambda a.a=y) u y u = refl_eq ? y)
? x H1).