--- /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 "Base-1/blt/defs.ma".
+
+theorem 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 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat
+n O) \to (eq bool (blt y O) true)))) with [le_n \Rightarrow (\lambda (H0: (eq
+nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in
+nat return (\lambda (_: nat).Prop) 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 in nat return (\lambda (_: nat).Prop)
+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).
+
+theorem 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 in le return (\lambda (n0:
+nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to (eq bool (blt O (S n))
+false)))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def
+(eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_:
+nat).Prop) 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 in nat return (\lambda (_: nat).Prop) 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).
+
+theorem 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 in eq return (\lambda (b: bool).(\lambda (_: (eq
+? ? b)).((eq bool b true) \to (lt y O)))) 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 in bool return (\lambda (_: bool).Prop) 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).
+
+theorem 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 in eq return (\lambda
+(b: bool).(\lambda (_: (eq ? ? b)).((eq bool b false) \to (le (S n) O))))
+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
+in bool return (\lambda (_: bool).Prop) 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).
+
projects / helm.git / blobdiff