--- /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/tlist/defs.ma".
+
+theorem tslt_wf__q_ind:
+ \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList
+\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0)
+\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
+\def
+ let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
+TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen
+ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat
+(tslen ts)))))).
+
+theorem tslt_wf_ind:
+ \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1:
+TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts:
+TList).(P ts)))
+\def
+ let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
+TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
+Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt
+(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts:
+TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n:
+nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda
+(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t))
+m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2
+\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to
+(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0)
+H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen
+ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
+
+theorem theads_tapp:
+ \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T
+(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads
+k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda
+(t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k
+v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0
+(THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads
+k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))).
+
+theorem tcons_tapp_ex:
+ \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda
+(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
+\def
+ \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2
+TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp
+ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen
+ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda
+(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal
+TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t:
+T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T
+(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2
+t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen
+ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in
+(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t
+t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0)
+(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq
+TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda
+(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda
+(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq
+nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2:
+TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons
+t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S
+(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n:
+nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons
+t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq
+nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2))))
+(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2))))
+(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat
+(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
+
+theorem tlist_ind_rev:
+ \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts:
+TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts:
+TList).(P ts))))
+\def
+ \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0:
+((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts
+t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t))
+(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1:
+TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1:
+TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1))))
+\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0))
+\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in
+(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t
+t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0)
+(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1:
+T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat
+(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P
+t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen
+(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0)
+H4))))) H3))))))) ts2)) ts)))).
+
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