+++ /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-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).
-(* COMMENTS
-Initial nodes: 291
-END *)
-
-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).
-(* COMMENTS
-Initial nodes: 293
-END *)
-
-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).
-(* COMMENTS
-Initial nodes: 252
-END *)
-
-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).
-(* COMMENTS
-Initial nodes: 262
-END *)
-
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