[ true \Rightarrow n = m
| false \Rightarrow n \neq m].
intros.
-apply nat_elim2
+apply (nat_elim2
(\lambda n,m:nat.match (eqb n m) with
[ true \Rightarrow n = m
-| false \Rightarrow n \neq m]).
+| false \Rightarrow n \neq m])).
intro.elim n1.
simplify.reflexivity.
simplify.apply not_eq_O_S.
intro.
simplify.
-intro. apply not_eq_O_S n1.apply sym_eq.assumption.
+intro. apply (not_eq_O_S n1).apply sym_eq.assumption.
intros.simplify.
generalize in match H.
-elim (eqb n1 m1).
+elim ((eqb n1 m1)).
simplify.apply eq_f.apply H1.
simplify.intro.apply H1.apply inj_S.assumption.
qed.
(n=m \to (P true)) \to (n \neq m \to (P false)) \to (P (eqb n m)).
intros.
cut
-match (eqb n m) with
+(match (eqb n m) with
[ true \Rightarrow n = m
-| false \Rightarrow n \neq m] \to (P (eqb n m)).
+| false \Rightarrow n \neq m] \to (P (eqb n m))).
apply Hcut.apply eqb_to_Prop.
-elim eqb n m.
-apply (H H2).
-apply (H1 H2).
+elim (eqb n m).
+apply ((H H2)).
+apply ((H1 H2)).
qed.
theorem eqb_n_n: \forall n. eqb n n = true.
theorem eq_to_eqb_true: \forall n,m:nat.
n = m \to eqb n m = true.
-intros.apply eqb_elim n m.
+intros.apply (eqb_elim n m).
intros. reflexivity.
-intros.apply False_ind.apply H1 H.
+intros.apply False_ind.apply (H1 H).
qed.
theorem not_eq_to_eqb_false: \forall n,m:nat.
\lnot (n = m) \to eqb n m = false.
-intros.apply eqb_elim n m.
-intros. apply False_ind.apply H H1.
+intros.apply (eqb_elim n m).
+intros. apply False_ind.apply (H H1).
intros.reflexivity.
qed.
[ true \Rightarrow n \leq m
| false \Rightarrow n \nleq m].
intros.
-apply nat_elim2
+apply (nat_elim2
(\lambda n,m:nat.match (leb n m) with
[ true \Rightarrow n \leq m
-| false \Rightarrow n \nleq m]).
+| false \Rightarrow n \nleq m])).
simplify.exact le_O_n.
simplify.exact not_le_Sn_O.
-intros 2.simplify.elim (leb n1 m1).
+intros 2.simplify.elim ((leb n1 m1)).
simplify.apply le_S_S.apply H.
simplify.intros.apply H.apply le_S_S_to_le.assumption.
qed.
P (leb n m).
intros.
cut
-match (leb n m) with
+(match (leb n m) with
[ true \Rightarrow n \leq m
-| false \Rightarrow n \nleq m] \to (P (leb n m)).
+| false \Rightarrow n \nleq m] \to (P (leb n m))).
apply Hcut.apply leb_to_Prop.
-elim leb n m.
-apply (H H2).
-apply (H1 H2).
+elim (leb n m).
+apply ((H H2)).
+apply ((H1 H2)).
qed.
let rec nat_compare n m: compare \def
qed.
theorem S_pred: \forall n:nat.lt O n \to eq nat n (S (pred n)).
-intro.elim n.apply False_ind.exact not_le_Sn_O O H.
+intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
apply eq_f.apply pred_Sn.
qed.
\forall n,m:nat.lt O n \to lt O m \to
eq compare (nat_compare n m) (nat_compare (pred n) (pred m)).
intros.
-apply lt_O_n_elim n H.
-apply lt_O_n_elim m H1.
+apply (lt_O_n_elim n H).
+apply (lt_O_n_elim m H1).
intros.
simplify.reflexivity.
qed.
| EQ \Rightarrow n=m
| GT \Rightarrow m < n ].
intros.
-apply nat_elim2 (\lambda n,m.match (nat_compare n m) with
+apply (nat_elim2 (\lambda n,m.match (nat_compare n m) with
[ LT \Rightarrow n < m
| EQ \Rightarrow n=m
- | GT \Rightarrow m < n ]).
+ | GT \Rightarrow m < n ])).
intro.elim n1.simplify.reflexivity.
simplify.apply le_S_S.apply le_O_n.
intro.simplify.apply le_S_S. apply le_O_n.
-intros 2.simplify.elim (nat_compare n1 m1).
+intros 2.simplify.elim ((nat_compare n1 m1)).
simplify. apply le_S_S.apply H.
simplify. apply eq_f. apply H.
simplify. apply le_S_S.apply H.
theorem nat_compare_n_m_m_n: \forall n,m:nat.
nat_compare n m = compare_invert (nat_compare m n).
intros.
-apply nat_elim2 (\lambda n,m. nat_compare n m = compare_invert (nat_compare m n)).
+apply (nat_elim2 (\lambda n,m. nat_compare n m = compare_invert (nat_compare m n))).
intros.elim n1.simplify.reflexivity.
simplify.reflexivity.
intro.elim n1.simplify.reflexivity.
(n < m \to P LT) \to (n=m \to P EQ) \to (m < n \to P GT) \to
(P (nat_compare n m)).
intros.
-cut match (nat_compare n m) with
+cut (match (nat_compare n m) with
[ LT \Rightarrow n < m
| EQ \Rightarrow n=m
| GT \Rightarrow m < n] \to
-(P (nat_compare n m)).
+(P (nat_compare n m))).
apply Hcut.apply nat_compare_to_Prop.
-elim (nat_compare n m).
-apply (H H3).
-apply (H1 H3).
-apply (H2 H3).
+elim ((nat_compare n m)).
+apply ((H H3)).
+apply ((H1 H3)).
+apply ((H2 H3)).
qed.