K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0)
t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1)
t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d)
-t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq T (THead k t0 t1)
-(THead k (lift O d t0) (lift O (s k d) t1)) (sym_eq T (THead k (lift O d t0)
-(lift O (s k d) t1)) (THead k t0 t1) (sym_eq T (THead k t0 t1) (THead k (lift
-O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d t0) t1
-(lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d))
-(sym_eq T (lift O (s k d) t1) t1 (H0 (s k d))))))) (lift O d (THead k t0 t1))
-(lift_head k t0 t1 O d)))))))) t).
+t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (f_equal3 K T T T THead k k
+(lift O d t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d)))
+(lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t).
theorem lift_lref_gt:
\forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef
(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) (sym_eq T (THead (Bind b) x0 x1) (THead (Bind
-b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0 (refl_equal K
-(Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (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) (sym_eq T (THead (Flat f) x0 x1) (THead (Flat f) t t0)
-(f_equal3 K T T T THead (Flat f) (Flat f) x0 t x1 t0 (refl_equal K (Flat f))
-(sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h d H5)))))) t1 H3))))))
+(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)) (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).
theorem lift_gen_lift:
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_S_n (plus d2 h1) (plus n h1) (lt_le_S (plus d2 h1) (S
-(plus n h1)) (le_lt_n_Sm (plus d2 h1) (plus n h1) (plus_le_compat 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_le_S (plus n h1) (plus (plus d2 h2) h1)
-(plus_lt_compat_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_plus_r d2 h2 n 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_plus_r d2 h2 n 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)) (f_equal nat T
-TLRef (plus (minus n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2)
-(f_equal2 nat nat nat plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2
-h2 (sym_eq nat (minus (plus (minus n h2) h2) h2) (minus n 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 (plus_le_compat d2 (minus n h2) h2 h2
-(le_minus d2 n h2 H4) (le_n h2))))) n (le_plus_minus_sym h2 n
-(le_trans_plus_r d2 h2 n 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:
+h1) (plus n h1) (plus_le_compat 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)) (plus_lt_compat_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)) (f_equal nat T TLRef (plus (minus
+n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) (f_equal2 nat nat nat
+plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 h2 (sym_eq nat (minus
+(plus (minus n h2) h2) h2) (minus n 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 (plus_le_compat 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
(\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_S_n (S d1) (S d2) (lt_le_S (S d1) (S
-(S d2)) (lt_n_S d1 (S d2) (le_lt_n_Sm 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).
+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).
theorem lift_free:
\forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T
(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d
(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k)))
-(lift_lref_lt (plus n k) h (plus d k) (lt_le_S (plus n k) (plus d k)
-(plus_lt_compat_r n d k H1))))) (\lambda (H1: (le d n)).(eq_ind_r T (TLRef
-(plus (plus n k) h)) (\lambda (t0: T).(eq T t0 (lift k e (lift h d (TLRef
-n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (TLRef (plus
-(plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef (plus (plus n h) k))
-(\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) (f_equal nat T TLRef
-(plus (plus n k) h) (plus (plus n h) k) (sym_eq nat (plus (plus n h) k) (plus
-(plus n k) h) (plus_permute_2_in_3 n h k))) (lift k e (TLRef (plus n h)))
-(lift_lref_ge (plus n h) k e (le_S_n e (plus n h) (lt_le_S e (S (plus n h))
-(le_lt_n_Sm e (plus n h) (le_plus_trans e n h H0)))))) (lift h d (TLRef n))
-(lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus n k))) (lift_lref_ge
-(plus n k) h (plus d k) (le_S_n (plus d k) (plus n k) (lt_le_S (plus d k) (S
-(plus n k)) (le_lt_n_Sm (plus d k) (plus n k) (plus_le_compat d n k k H1
-(le_n k))))))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n))
-(lift_lref_ge n k e H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda
-(H: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e:
-nat).((le e d) \to (eq T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift
-h d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall
-(k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h
-(plus k0 d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h:
+(lift_lref_lt (plus n k) h (plus d k) (plus_lt_compat_r n d k H1)))) (\lambda
+(H1: (le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T
+t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda
+(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef
+(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0))
+(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat
+(plus (plus n h) k) (plus (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k
+e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_plus_trans e n h H0)))
+(lift h d (TLRef n)) (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus
+n k))) (lift_lref_ge (plus n k) h (plus d k) (plus_le_compat d n k k H1 (le_n
+k)))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n)) (lift_lref_ge n k e
+H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h:
+nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq
+T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift h d
+t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0:
+nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0
+d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h:
nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le
e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2:
T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1)))))
projects / helm.git / blobdiff