--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "ground_1A/blt/defs.ma".
+
+lemma lt_blt:
+ \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
+(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0
+\def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1
+\def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O \Rightarrow False |
+(S _) \Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true)
+H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def
+(eq_ind nat (S m) (\lambda (e: nat).(match e with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq bool (blt
+y O) true)) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n:
+nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq bool (blt y n)
+true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to
+(eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S n))).(refl_equal bool
+true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0
+with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda
+(H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) y)))) x).
+
+lemma le_bge:
+ \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
+(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O
+y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y:
+nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind
+(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false)))
+(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 with [le_n \Rightarrow
+(\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n) (\lambda (e:
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1)
+in (False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow
+(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e:
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2)
+in (False_ind ((le (S n) m) \to (eq bool (blt O (S n)) false)) H3)) H1))]) in
+(H1 (refl_equal nat O)))) (\lambda (n0: nat).(\lambda (_: (((le (S n) n0) \to
+(eq bool (blt n0 (S n)) false)))).(\lambda (H1: (le (S n) (S n0))).(H n0
+(le_S_n n n0 H1))))) y)))) x).
+
+lemma blt_lt:
+ \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
+y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O)
+true)).(let H0 \def (match H with [refl_equal \Rightarrow (\lambda (H0: (eq
+bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) (\lambda (e:
+bool).(match e with [true \Rightarrow False | false \Rightarrow True])) I
+true H0) in (False_ind (lt y O) H1)))]) in (H0 (refl_equal bool true)))))
+(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n) true)
+\to (lt y n))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt
+n0 (S n)) true) \to (lt n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n
+(S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0:
+nat).(\lambda (_: (((eq bool (match n0 with [O \Rightarrow true | (S m)
+\Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq bool
+(blt n0 n) true)).(lt_n_S n0 n (H n0 H1))))) y)))) x).
+
+lemma bge_le:
+ \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
+\def
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
+y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O)
+false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq
+bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda
+(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0:
+(eq bool (blt O (S n)) false)).(let H1 \def (match H0 with [refl_equal
+\Rightarrow (\lambda (H1: (eq bool (blt O (S n)) false)).(let H2 \def (eq_ind
+bool (blt O (S n)) (\lambda (e: bool).(match e with [true \Rightarrow True |
+false \Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in
+(H1 (refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool
+(blt n0 (S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0)
+(S n)) false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0
+H1))))))) y)))) x).
+