+
+lemma lift_inj:
+ \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T
+(lift h d x) (lift h d t)) \to (eq T x t)))))
+\def
+ \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t
+t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def
+(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H
+(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t
+H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq
+T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d
+(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt
+n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d
+d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift
+h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h))
+(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0
+t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
+T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t)
+(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1))
+\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d:
+nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t
+t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to
+(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall
+(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0
+t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
+(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T
+(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1
+(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z))))
+(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift
+h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r
+T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2))
+(sym_eq T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_eq T (THead (Bind
+b) t t0) (THead (Bind b) x0 x1) (f_equal3 K T T T THead (Bind b) (Bind b) t
+x0 t0 x1 (refl_equal K (Bind b)) (H x0 h d H4) (H0 x1 h (S d) H5)))) t1
+H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) t1 h d
+H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall (t0:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to
+(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall
+(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0
+t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
+(eq T (lift h d (THead (Flat f) t t0)) (lift h d t1))).(let H2 \def (eq_ind T
+(lift h d (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1
+(THead (Flat f) (lift h d t) (lift h d t0)) (lift_flat f t t0 h d)) in
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat f) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift h d z)))) (eq T
+(THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq
+T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d
+x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d x1))).(eq_ind_r T (THead
+(Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat f) t t0) t2)) (sym_eq T
+(THead (Flat f) x0 x1) (THead (Flat f) t t0) (sym_eq T (THead (Flat f) t t0)
+(THead (Flat f) x0 x1) (f_equal3 K T T T THead (Flat f) (Flat f) t x0 t0 x1
+(refl_equal K (Flat f)) (H x0 h d H4) (H0 x1 h d H5)))) t1 H3))))))
+(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x).
+
+lemma lift_gen_lift:
+ \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2:
+nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
+t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
+t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1:
+nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to
+((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2:
+T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2
+t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1
+d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1)
+x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t
+(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T
+(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda
+(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n)
+(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T
+(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1
+d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
+(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2
+(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda
+(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda
+(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2
+h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n
+d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t
+(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in
+(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T
+(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t:
+T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n))
+(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef
+n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2
+(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n
+(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2))))
+(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n))
+(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1))
+(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x
+(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
+(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2
+T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n)
+(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1))
+(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef
+n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1))
+t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n
+h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t))
+(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x
+(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2)))
+(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq
+T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
+(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus
+n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2)
+h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4)
+(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda
+(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2
+d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus
+n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus
+(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans
+h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T
+(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T
+t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))
+(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1
+d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n
+h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0)
+(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2)
+h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T
+(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2)))
+(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2
+n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2
+h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2)
+(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2)))))
+(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t:
+T).(eq T (TLRef (plus (minus n h2) h2)) t)) (sym_eq T (TLRef (plus (minus
+(plus (minus n h2) h2) h2) h2)) (TLRef (plus (minus n h2) h2)) (f_equal nat T
+TLRef (plus (minus (plus (minus n h2) h2) h2) h2) (plus (minus n h2) h2)
+(f_equal2 nat nat nat plus (minus (plus (minus n h2) h2) h2) (minus n h2) h2
+h2 (minus_plus_r (minus n h2) h2) (refl_equal nat h2)))) (lift h2 d2 (TLRef
+(minus (plus (minus n h2) h2) h2))) (lift_lref_ge (minus (plus (minus n h2)
+h2) h2) h2 d2 (le_minus d2 (plus (minus n h2) h2) h2 (le_plus_plus d2 (minus
+n h2) h2 h2 (le_minus d2 n h2 H4) (le_n h2))))) n (le_plus_minus_sym h2 n
+(le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2
+(plus d2 h1) (minus (plus n h1) h2) (arith0 h2 d2 n H4 h1) x
+H5)))))))))))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall
+(x: T).(\forall (h1: nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2:
+nat).((le d1 d2) \to ((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2
+T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift
+h2 d2 t2))))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (x:
+T).(\forall (h1: nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2:
+nat).((le d1 d2) \to ((eq T (lift h1 d1 t0) (lift h2 (plus d2 h1) x)) \to
+(ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t0
+(lift h2 d2 t2))))))))))))).(\lambda (x: T).(\lambda (h1: nat).(\lambda (h2:
+nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (H1: (le d1 d2)).(\lambda
+(H2: (eq T (lift h1 d1 (THead k t t0)) (lift h2 (plus d2 h1) x))).(K_ind
+(\lambda (k0: K).((eq T (lift h1 d1 (THead k0 t t0)) (lift h2 (plus d2 h1)
+x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2:
+T).(eq T (THead k0 t t0) (lift h2 d2 t2)))))) (\lambda (b: B).(\lambda (H3:
+(eq T (lift h1 d1 (THead (Bind b) t t0)) (lift h2 (plus d2 h1) x))).(let H4
+\def (eq_ind T (lift h1 d1 (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2
+(lift h2 (plus d2 h1) x))) H3 (THead (Bind b) (lift h1 d1 t) (lift h1 (S d1)
+t0)) (lift_bind b t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z:
+T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
+(lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T (lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) z)))) (ex2 T (\lambda
+(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) t
+t0) (lift h2 d2 t2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T
+x (THead (Bind b) x0 x1))).(\lambda (H6: (eq T (lift h1 d1 t) (lift h2 (plus
+d2 h1) x0))).(\lambda (H7: (eq T (lift h1 (S d1) t0) (lift h2 (S (plus d2
+h1)) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T t2 (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead
+(Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))) (ex2 T (\lambda (t2:
+T).(eq T (THead (Bind b) x0 x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x2: T).(\lambda (H8: (eq T
+x0 (lift h1 d1 x2))).(\lambda (H9: (eq T t (lift h2 d2 x2))).(eq_ind_r T
+(lift h1 d1 x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind
+b) t2 x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t t0)
+(lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2 x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift h1 d1 t3)))
+(\lambda (t3: T).(eq T (THead (Bind b) t2 t0) (lift h2 d2 t3))))) (let H10
+\def (refl_equal nat (plus (S d2) h1)) in (let H11 \def (eq_ind nat (S (plus
+d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1) t0) (lift h2 n x1))) H7 (plus
+(S d2) h1) H10) in (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 (S d1) t2)))
+(\lambda (t2: T).(eq T t0 (lift h2 (S d2) t2))) (ex2 T (\lambda (t2: T).(eq T
+(THead (Bind b) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(THead (Bind b) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3:
+T).(\lambda (H12: (eq T x1 (lift h1 (S d1) x3))).(\lambda (H13: (eq T t0
+(lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S d1) x3) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) t2) (lift h1 d1 t3)))
+(\lambda (t3: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift h2 d2
+t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda (t2: T).(ex2 T (\lambda (t3:
+T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1
+t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift h2 d2 x2) t2) (lift h2 d2
+t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2)
+(lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b)
+(lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2 t2))) (THead (Bind b) x2 x3)
+(eq_ind_r T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (\lambda
+(t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) t2))
+(refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3))) (lift h1
+d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1 d1)) (eq_ind_r T (THead
+(Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (\lambda (t2: T).(eq T (THead
+(Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) t2)) (refl_equal T (THead (Bind
+b) (lift h2 d2 x2) (lift h2 (S d2) x3))) (lift h2 d2 (THead (Bind b) x2 x3))
+(lift_bind b x2 x3 h2 d2))) t0 H13) x1 H12)))) (H0 x1 h1 h2 (S d1) (S d2)
+(le_n_S d1 d2 H1) H11)))) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x
+H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S d1) t0) x h2 (plus d2
+h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift h1 d1 (THead (Flat f) t
+t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead
+(Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3
+(THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t t0 h1 d1)) in
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2
+h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 d1 t0) (lift h2
+(plus d2 h1) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda
+(t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Flat f) x0 x1))).(\lambda
+(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
+(lift h1 d1 t0) (lift h2 (plus d2 h1) x1))).(eq_ind_r T (THead (Flat f) x0
+x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
+(\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (ex2_ind T
+(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
+d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Flat f) x0 x1) (lift h1 d1
+t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2))))
+(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
+t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead (Flat f) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
+T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
+x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1
+d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t2 t0)
+(lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq T
+(THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3:
+T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda (H11: (eq T t0 (lift h2
+d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: T).(ex2 T (\lambda (t3:
+T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3:
+T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T
+(lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat
+f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T
+(THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda
+(t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1
+t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))
+(lift h2 d2 t2))) (THead (Flat f) x2 x3) (eq_ind_r T (THead (Flat f) (lift h1
+d1 x2) (lift h1 d1 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1
+x2) (lift h1 d1 x3)) t2)) (refl_equal T (THead (Flat f) (lift h1 d1 x2) (lift
+h1 d1 x3))) (lift h1 d1 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h1 d1))
+(eq_ind_r T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) (\lambda (t2:
+T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) t2)) (refl_equal T
+(THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))) (lift h2 d2 (THead (Flat f)
+x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1
+H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_flat f
+(lift h1 d1 t) (lift h1 d1 t0) x h2 (plus d2 h1) H4))))) k H2)))))))))))))
+t1).
+
+lemma lifts_inj:
+ \forall (xs: TList).(\forall (ts: TList).(\forall (h: nat).(\forall (d:
+nat).((eq TList (lifts h d xs) (lifts h d ts)) \to (eq TList xs ts)))))
+\def
+ \lambda (xs: TList).(TList_ind (\lambda (t: TList).(\forall (ts:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t) (lifts h
+d ts)) \to (eq TList t ts)))))) (\lambda (ts: TList).(TList_ind (\lambda (t:
+TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d TNil) (lifts
+h d t)) \to (eq TList TNil t))))) (\lambda (_: nat).(\lambda (_:
+nat).(\lambda (_: (eq TList TNil TNil)).(refl_equal TList TNil)))) (\lambda
+(t: T).(\lambda (t0: TList).(\lambda (_: ((\forall (h: nat).(\forall (d:
+nat).((eq TList TNil (lifts h d t0)) \to (eq TList TNil t0)))))).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H0: (eq TList TNil (TCons (lift h d t)
+(lifts h d t0)))).(let H1 \def (eq_ind TList TNil (\lambda (ee: TList).(match
+ee with [TNil \Rightarrow True | (TCons _ _) \Rightarrow False])) I (TCons
+(lift h d t) (lifts h d t0)) H0) in (False_ind (eq TList TNil (TCons t t0))
+H1)))))))) ts)) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall
+(ts: TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h d t0)
+(lifts h d ts)) \to (eq TList t0 ts))))))).(\lambda (ts: TList).(TList_ind
+(\lambda (t1: TList).(\forall (h: nat).(\forall (d: nat).((eq TList (lifts h
+d (TCons t t0)) (lifts h d t1)) \to (eq TList (TCons t t0) t1))))) (\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H0: (eq TList (TCons (lift h d t) (lifts
+h d t0)) TNil)).(let H1 \def (eq_ind TList (TCons (lift h d t) (lifts h d
+t0)) (\lambda (ee: TList).(match ee with [TNil \Rightarrow False | (TCons _
+_) \Rightarrow True])) I TNil H0) in (False_ind (eq TList (TCons t t0) TNil)
+H1))))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_: ((\forall (h:
+nat).(\forall (d: nat).((eq TList (TCons (lift h d t) (lifts h d t0)) (lifts
+h d t2)) \to (eq TList (TCons t t0) t2)))))).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (eq TList (TCons (lift h d t) (lifts h d t0)) (TCons (lift
+h d t1) (lifts h d t2)))).(let H2 \def (f_equal TList T (\lambda (e:
+TList).(match e with [TNil \Rightarrow (lref_map (\lambda (x: nat).(plus x
+h)) d t) | (TCons t3 _) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0))
+(TCons (lift h d t1) (lifts h d t2)) H1) in ((let H3 \def (f_equal TList
+TList (\lambda (e: TList).(match e with [TNil \Rightarrow (lifts h d t0) |
+(TCons _ t3) \Rightarrow t3])) (TCons (lift h d t) (lifts h d t0)) (TCons
+(lift h d t1) (lifts h d t2)) H1) in (\lambda (H4: (eq T (lift h d t) (lift h
+d t1))).(eq_ind T t (\lambda (t3: T).(eq TList (TCons t t0) (TCons t3 t2)))
+(f_equal2 T TList TList TCons t t t0 t2 (refl_equal T t) (H t2 h d H3)) t1
+(lift_inj t t1 h d H4)))) H2)))))))) ts))))) xs).