'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.
variant reflexive_eq : \forall A:Type. reflexive A (eq A)
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/symmetric_eq.con
+ cic:/matita/logic/equality/sym_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).