theorem leb_to_Prop: \forall n,m:nat.
match (leb n m) with
-[ true \Rightarrow (le n m)
-| false \Rightarrow (Not (le n m))].
+[ true \Rightarrow n \leq m
+| false \Rightarrow \lnot (n \leq m)].
intros.
apply nat_elim2
(\lambda n,m:nat.match (leb n m) with
-[ true \Rightarrow (le n m)
-| false \Rightarrow (Not (le n m))]).
+[ true \Rightarrow n \leq m
+| false \Rightarrow \lnot (n \leq m)]).
simplify.exact le_O_n.
simplify.exact not_le_Sn_O.
intros 2.simplify.elim (leb n1 m1).
qed.
theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop.
-((le n m) \to (P true)) \to ((Not (le n m)) \to (P false)) \to
+(n \leq m \to (P true)) \to (\not (n \leq m) \to (P false)) \to
P (leb n m).
intros.
cut
match (leb n m) with
-[ true \Rightarrow (le n m)
-| false \Rightarrow (Not (le n m))] \to (P (leb n m)).
+[ true \Rightarrow n \leq m
+| false \Rightarrow \lnot (n \leq m)] \to (P (leb n m)).
apply Hcut.apply leb_to_Prop.
elim leb n m.
apply (H H2).
theorem nat_compare_to_Prop: \forall n,m:nat.
match (nat_compare n m) with
- [ LT \Rightarrow (lt n m)
+ [ LT \Rightarrow n < m
| EQ \Rightarrow n=m
- | GT \Rightarrow (lt m n) ].
+ | GT \Rightarrow m < n ].
intros.
apply nat_elim2 (\lambda n,m.match (nat_compare n m) with
- [ LT \Rightarrow (lt n m)
+ [ LT \Rightarrow n < m
| EQ \Rightarrow n=m
- | GT \Rightarrow (lt 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.
qed.
theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
-((lt n m) \to (P LT)) \to (n=m \to (P EQ)) \to ((lt m n) \to (P GT)) \to
+(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
-[ LT \Rightarrow (lt n m)
+[ LT \Rightarrow n < m
| EQ \Rightarrow n=m
-| GT \Rightarrow (lt m n)] \to
+| GT \Rightarrow m < n] \to
(P (nat_compare n m)).
apply Hcut.apply nat_compare_to_Prop.
elim (nat_compare n m).