--- /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 "LambdaDelta-1/C/defs.ma".
+
+include "LambdaDelta-1/T/props.ma".
+
+theorem clt_cong:
+ \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t:
+T).(clt (CHead c k t) (CHead d k t))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight
+d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d)
+(tweight t) H))))).
+
+theorem clt_head:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight
+c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u))))
+(le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c))
+(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
+
+theorem clt_wf__q_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to
+Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0
+c))))) P n))) \to (\forall (c: C).(P c)))
+\def
+ let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
+C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c)
+n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight
+c)))))).
+
+theorem clt_wf_ind:
+ \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c)
+\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
+\def
+ let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
+C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
+Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d)
+(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind
+(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0:
+C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
+\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat
+(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
+(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P
+c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt
+(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight
+d))))))))))))) c)))).
+
+theorem chead_ctail:
+ \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h:
+K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d))))))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3
+K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t)
+(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k:
+K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead
+(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall
+(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def
+H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d:
+C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d))))))
+(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead
+c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1:
+C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead
+c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d:
+C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0
+(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0
+k t) H1))))) H0))))))))) c).
+
+theorem clt_thead:
+ \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt
+c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0:
+C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t:
+T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))).
+
+theorem c_tail_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to
+(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t
+c))))))) \to (\forall (c: C).(P c))))
+\def
+ \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort
+n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t:
+T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P
+c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1)
+\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d:
+C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_:
+((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to
+(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind
+K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t)
+(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2
+x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def
+(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P
+d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2))
+(CHead c1 k t) H4))))) H3)))))))) c0)) c)))).
+
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