theorem a:
\forall A:Set.
- \forall x,y : A.
- not (x=y) \to not (x=y).
+ \forall x: A.
+ not (x=x) \to not (x=x).
intros.
apply H.
qed.
theorem stupid :
- let x \def 0 + 1 in x + 1= x + 1.
+ let x \def 0 + 1 in x + 2 = x + 2.
intros.
clearbody x.
simplify.
alias symbol "eq" (instance 0) = "leibnitz's equality".
-theorem stupid: 0 = 0.
+theorem stupid: 1 = 1.
constructor 1.
qed.
-theorem stupid: \forall a:Prop. a \to not a \to 0 = 1.
+theorem stupid: \forall a:Prop. a \to not a \to 0 = 2.
intros.
letin H \def (H1 H).
contradiction.
alias num (instance 0) = "natural number".
alias symbol "eq" (instance 0) = "leibnitz's equality".
-theorem stupid: 0 = 0.
- cut 0 = 0.
+theorem stupid: 3 = 3.
+ cut 3 = 3.
assumption.
reflexivity.
qed.