--- /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 "Basic-1/llt/defs.ma".
+
+include "Basic-1/leq/defs.ma".
+
+theorem lweight_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat
+(lweight a1) (lweight a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight
+a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
+nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3:
+A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight
+a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda
+(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight
+a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus
+(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2
+H)))).
+(* COMMENTS
+Initial nodes: 189
+END *)
+
+theorem llt_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
+(a3: A).((llt a1 a3) \to (llt a2 a3))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1
+\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0
+(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))).
+(* COMMENTS
+Initial nodes: 61
+END *)
+
+theorem llt_trans:
+ \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2
+a3) \to (llt a1 a3)))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight
+a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans
+(lweight a1) (lweight a2) (lweight a3) H H0))))).
+(* COMMENTS
+Initial nodes: 43
+END *)
+
+theorem llt_head_sx:
+ \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1)
+(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))).
+(* COMMENTS
+Initial nodes: 29
+END *)
+
+theorem llt_head_dx:
+ \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1)
+(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))).
+(* COMMENTS
+Initial nodes: 29
+END *)
+
+theorem llt_wf__q_ind:
+ \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to
+Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0
+a))))) P n))) \to (\forall (a: A).(P a)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a)
+n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight
+a)))))).
+(* COMMENTS
+Initial nodes: 61
+END *)
+
+theorem llt_wf_ind:
+ \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1
+a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
+Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1)
+(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind
+(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0:
+A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
+\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat
+(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
+(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P
+a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt
+(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight
+a1))))))))))))) a)))).
+(* COMMENTS
+Initial nodes: 179
+END *)
+