+(\lambda n,m:nat.if_then_else (leb n m) (le n m) (Not (le n m))) ? ? ? n m).
+simplify.intros.apply le_O_n.
+simplify.exact le_Sn_O.
+intros 2.simplify.elim (leb n1 m1).
+simplify.apply le_n_S.apply H.
+simplify.intros.apply H.apply le_S_n.assumption.
+qed.
+
+inductive Z : Set \def
+ OZ : Z
+| pos : nat \to Z
+| neg : nat \to Z.
+
+definition Z_of_nat \def
+\lambda n. match n with
+[ O \Rightarrow OZ
+| (S n)\Rightarrow pos n].
+
+coercion Z_of_nat.
+
+definition neg_Z_of_nat \def
+\lambda n. match n with
+[ O \Rightarrow OZ
+| (S n)\Rightarrow neg n].
+
+definition absZ \def
+\lambda z.
+ match z with
+[ OZ \Rightarrow O
+| (pos n) \Rightarrow n
+| (neg n) \Rightarrow n].
+
+definition OZ_testb \def
+\lambda z.
+match z with
+[ OZ \Rightarrow true
+| (pos n) \Rightarrow false
+| (neg n) \Rightarrow false].
+
+theorem OZ_discr :
+\forall z. if_then_else (OZ_testb z) (eq Z z OZ) (Not (eq Z z OZ)).
+intros.elim z.simplify.apply refl_equal.
+simplify.intros.
+cut match neg e with
+[ OZ \Rightarrow True
+| (pos n) \Rightarrow False
+| (neg n) \Rightarrow False].
+apply Hcut. elim (sym_eq ? ? ? H).simplify.exact I.
+simplify.intros.
+cut match pos e with
+[ OZ \Rightarrow True
+| (pos n) \Rightarrow False
+| (neg n) \Rightarrow False].
+apply Hcut. elim (sym_eq ? ? ? H).simplify.exact I.
+qed.
+
+definition Zsucc \def
+\lambda z. match z with
+[ OZ \Rightarrow pos O
+| (pos n) \Rightarrow pos (S n)
+| (neg n) \Rightarrow
+ match n with
+ [ O \Rightarrow OZ
+ | (S p) \Rightarrow neg p]].
+
+definition Zpred \def
+\lambda z. match z with
+[ OZ \Rightarrow neg O
+| (pos n) \Rightarrow
+ match n with
+ [ O \Rightarrow OZ
+ | (S p) \Rightarrow pos p]
+| (neg n) \Rightarrow neg (S n)].
+
+theorem Zpred_succ: \forall z:Z. eq Z (Zpred (Zsucc z)) z.
+intros.elim z.apply refl_equal.
+elim e.apply refl_equal.
+apply refl_equal.
+apply refl_equal.
+qed.
+
+theorem Zsucc_pred: \forall z:Z. eq Z (Zsucc (Zpred z)) z.
+intros.elim z.apply refl_equal.
+apply refl_equal.
+elim e.apply refl_equal.
+apply refl_equal.
+qed.
+
+inductive compare :Set \def
+| LT : compare
+| EQ : compare
+| GT : compare.
+
+let rec nat_compare n m: compare \def
+match n with
+[ O \Rightarrow
+ match m with
+ [ O \Rightarrow EQ
+ | (S q) \Rightarrow LT ]
+| (S p) \Rightarrow
+ match m with
+ [ O \Rightarrow GT
+ | (S q) \Rightarrow nat_compare p q]].
+
+definition compare_invert: compare \to compare \def
+ \lambda c.
+ match c with
+ [ LT \Rightarrow GT
+ | EQ \Rightarrow EQ
+ | GT \Rightarrow LT ].
+
+theorem nat_compare_invert: \forall n,m:nat.
+eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
+intros.
+apply nat_double_ind (\lambda n,m.eq compare (nat_compare n m) (compare_invert (nat_compare m n))).
+intros.elim n1.simplify.apply refl_equal.
+simplify.apply refl_equal.
+intro.elim n1.simplify.apply refl_equal.
+simplify.apply refl_equal.
+intros.simplify.elim H.apply refl_equal.
+qed.
+
+let rec Zplus x y : Z \def
+ match x with
+ [ OZ \Rightarrow y
+ | (pos m) \Rightarrow
+ match y with
+ [ OZ \Rightarrow x
+ | (pos n) \Rightarrow (pos (S (plus m n)))
+ | (neg n) \Rightarrow
+ match nat_compare m n with
+ [ LT \Rightarrow (neg (pred (minus n m)))
+ | EQ \Rightarrow OZ
+ | GT \Rightarrow (pos (pred (minus m n)))]]
+ | (neg m) \Rightarrow
+ match y with
+ [ OZ \Rightarrow x
+ | (pos n) \Rightarrow
+ match nat_compare m n with
+ [ LT \Rightarrow (pos (pred (minus n m)))
+ | EQ \Rightarrow OZ
+ | GT \Rightarrow (neg (pred (minus m n)))]
+ | (neg n) \Rightarrow (neg (S (plus m n)))]].
+
+theorem Zplus_z_O: \forall z:Z. eq Z (Zplus z OZ) z.
+intro.elim z.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+simplify.apply refl_equal.
+qed.