alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
alias id "plus_comm" = "cic:/Coq/Arith/Plus/plus_comm.con".
-theorem a :
- \forall A:Set.
- \forall x,y : A.
- not (x = y) \to not(y = x).
-intros.
-unfold not. (* simplify. *)
-intro. apply H.
-symmetry.
-exact H1.
-qed.
-
theorem t: let f \def \lambda x,y. x y in f (\lambda x.S x) O = S O.
- intros. simplify. change in \vdash (? ? (? %) ?) with O.
+ intros. simplify. change in \vdash (? ? (? ? %) ?) with O.
reflexivity. qed.
-
theorem X: \forall x:nat. let myplus \def plus x in myplus (S O) = S x.
intros. simplify. change in \vdash (? ? (% ?) ?) with (plus x).
rewrite > plus_comm. reflexivity. qed.
-
+
theorem R: \forall x:nat. let uno \def x + O in S O + uno = 1 + x.
intros. simplify.
change in \vdash (? ? (? %) ?) with (x + O).