1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "ground_1/blt/defs.ma".
20 \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
22 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
23 (eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0
24 \def (match H in le with [le_n \Rightarrow (\lambda (H0: (eq nat (S y)
25 O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in nat with [O
26 \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq bool
27 (blt y O) true) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m)
28 O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat with [O
29 \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S
30 y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal nat O)))))
31 (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq bool (blt
32 y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n))
33 \to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S n))).(refl_equal
34 bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool
35 (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)])
36 true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1)))))
40 \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
42 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
43 (eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O
44 y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y:
45 nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind
46 (\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false)))
47 (\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le with [le_n
48 \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n)
49 (\lambda (e: nat).(match e in nat with [O \Rightarrow False | (S _)
50 \Rightarrow True])) I O H1) in (False_ind (eq bool (blt O (S n)) false) H2)))
51 | (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def
52 (eq_ind nat (S m) (\lambda (e: nat).(match e in nat with [O \Rightarrow False
53 | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool
54 (blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0:
55 nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n))
56 false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y))))
60 \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
62 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
63 y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O)
64 true)).(let H0 \def (match H in eq with [refl_equal \Rightarrow (\lambda (H0:
65 (eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) (\lambda (e:
66 bool).(match e in bool with [true \Rightarrow False | false \Rightarrow
67 True])) I true H0) in (False_ind (lt y O) H1)))]) in (H0 (refl_equal bool
68 true))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y
69 n) true) \to (lt y n))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq
70 bool (blt n0 (S n)) true) \to (lt n0 (S n)))) (\lambda (_: (eq bool true
71 true)).(le_S_n (S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n)))))
72 (\lambda (n0: nat).(\lambda (_: (((eq bool (match n0 with [O \Rightarrow true
73 | (S m) \Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq
74 bool (blt n0 n) true)).(lt_n_S n0 n (H n0 H1))))) y)))) x).
77 \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
79 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
80 y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O)
81 false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq
82 bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda
83 (n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0:
84 (eq bool (blt O (S n)) false)).(let H1 \def (match H0 in eq with [refl_equal
85 \Rightarrow (\lambda (H1: (eq bool (blt O (S n)) false)).(let H2 \def (eq_ind
86 bool (blt O (S n)) (\lambda (e: bool).(match e in bool with [true \Rightarrow
87 True | false \Rightarrow False])) I false H1) in (False_ind (le (S n) O)
88 H2)))]) in (H1 (refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_:
89 (((eq bool (blt n0 (S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool
90 (blt (S n0) (S n)) false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S
91 n n0 (H n0 H1))))))) y)))) x).