interpretation "leibnitz's non-equality" 'neq t x y = (Not (eq t x y)).
+(* this is refl
ntheorem reflexive_eq : ∀A:Type. reflexive A (eq A).
-//; nqed.
-
+//; nqed. *)
+
+(* this is sym_eq
ntheorem symmetric_eq: ∀A:Type. symmetric A (eq A).
-//; nqed.
+//; nqed. *)
ntheorem transitive_eq : ∀A:Type. transitive A (eq A).
#A; #x; #y; #z; #H1; #H2; nrewrite > H1; //; nqed.
+(*
ntheorem symmetric_not_eq: ∀A:Type. symmetric A (λx,y.x ≠ y).
+/3/; nqed.
+*)
+
+ntheorem symmetric_not_eq: ∀A:Type. ∀x,y:A. x ≠ y → y ≠ x.
+/3/; nqed.
+
+(*
#A; #x; #y; #H; #K; napply H; napply symmetric_eq; //; nqed.
+*)
ntheorem eq_f: ∀A,B:Type.∀f:A→B.∀x,y:A. x=y → f x = f y.
#A; #B; #f; #x; #y; #H; nrewrite > H; //; nqed.
intros.elim H.apply refl_eq.
qed. *)
+(* deleterio per auto*)
ntheorem eq_f2: ∀A,B,C:Type.∀f:A→B→C.
∀x1,x2:A.∀y1,y2:B. x1=x2 → y1=y2 → f x1 y1 = f x2 y2.
#A; #B; #C; #f; #x1; #x2; #y1; #y2; #E1; #E2; nrewrite > E1; nrewrite > E2;//.
-nqed.
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+nqed.
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