--- /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/lift1/defs.ma".
+
+include "LambdaDelta-1/lift/fwd.ma".
+
+theorem lift1_sort:
+ \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
+\def
+ \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T
+(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0:
+nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p
+(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0
+n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)).
+
+theorem lift1_lref:
+ \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef
+(trans hds i))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T
+(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T
+(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
+(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p
+i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq
+T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow
+(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T
+(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false
+\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds).
+
+theorem lift1_bind:
+ \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
+(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss
+hds) t))))))
+\def
+ \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b)
+(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal
+T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead
+(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda
+(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p)
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p
+u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n
+n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0
+(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t)))))
+(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1
+(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t)))
+(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u
+t)) (H u t)))))))) hds)).
+
+theorem lift1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
+(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds
+t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f)
+(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T
+(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead
+(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u:
+T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t))
+(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u))
+(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p
+u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift
+n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f)
+(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f)
+(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1
+p (THead (Flat f) u t)) (H u t)))))))) hds)).
+
+theorem lift1_cons_tail:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq
+T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t))))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds:
+PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t)
+(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1
+(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d
+t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p
+h d) t) H))))) hds)))).
+
+theorem lifts1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts:
+TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds
+ts) (lift1 hds t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts:
+TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0
+t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1
+hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds
+(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds
+t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f)
+t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads
+(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f)
+(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1
+hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1)
+(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat
+f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H)
+(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0
+(THeads (Flat f) t1 t)))))) ts)))).
+
+theorem lifts1_nil:
+ \forall (ts: TList).(eq TList (lifts1 PNil ts) ts)
+\def
+ \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t)
+t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H:
+(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq
+TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1
+PNil t0) H)))) ts).
+
+theorem lifts1_cons:
+ \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts:
+TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts:
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t)
+(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda
+(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d
+(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1:
+TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1
+hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d
+(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0)
+H)))) ts)))).
+
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