--- /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_1A/flt/defs.ma".
+
+include "basic_1A/C/props.ma".
+
+lemma flt_thead_sx:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c
+(THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_reg_l
+(tweight u) (S (plus (tweight u) (tweight t))) (cweight c) (le_n_S (tweight
+u) (plus (tweight u) (tweight t)) (le_plus_l (tweight u) (tweight t))))))).
+
+lemma flt_thead_dx:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c
+(THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_reg_l
+(tweight t) (S (plus (tweight u) (tweight t))) (cweight c) (le_n_S (tweight
+t) (plus (tweight u) (tweight t)) (le_plus_r (tweight u) (tweight t))))))).
+
+lemma flt_shift:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c
+k u) t c (THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat
+(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt
+(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus
+(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus
+(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight
+c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight
+t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S
+(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u)
+(tweight t))))))).
+
+lemma flt_arith0:
+ \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t
+(CHead c k t) (TLRef i)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_:
+nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))).
+
+lemma flt_arith1:
+ \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle
+(CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i:
+nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i)))))))))
+\def
+ \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda
+(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_:
+K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1)
+(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H
+(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n:
+nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2)
+(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2)
+(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S
+O))))))))))).
+
+lemma flt_arith2:
+ \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1
+t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt
+c1 t1 (CHead c2 k2 t2) (TLRef j)))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda
+(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda
+(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1)
+(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight
+t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S
+O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))).
+
+lemma cle_flt_trans:
+ \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (c3: C).(\forall
+(u2: T).(\forall (u3: T).((flt c2 u2 c3 u3) \to (flt c1 u2 c3 u3)))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight
+c2))).(\lambda (c3: C).(\lambda (u2: T).(\lambda (u3: T).(\lambda (H0: (lt
+(plus (cweight c2) (tweight u2)) (plus (cweight c3) (tweight
+u3)))).(le_lt_trans (plus (cweight c1) (tweight u2)) (plus (cweight c2)
+(tweight u2)) (plus (cweight c3) (tweight u3)) (le_plus_plus (cweight c1)
+(cweight c2) (tweight u2) (tweight u2) H (le_n (tweight u2))) H0))))))).
+
+theorem flt_trans:
+ \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1
+t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt
+c1 t1 c3 t3))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3:
+T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1
+t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))).
+
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