update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / drop / props.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma b/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma
index 74c409bddcc77a9e4f0400ff8ef65757f5252e28..1d662e8d45b075b64650974e49b51cb9f3021f9d 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/drop/props.ma
@@ -16,131 +16,81 @@
 
 include "basic_1/drop/fwd.ma".
 
-theorem drop_skip_bind:
+lemma drop_skip_bind:
  \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h 
 d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b) 
 (lift h d u)) (CHead e (Bind b) u))))))))
 \def
  \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda 
-(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(let TMP_1 \def (Bind b) 
-in (let TMP_2 \def (r TMP_1 d) in (let TMP_9 \def (\lambda (n: nat).(let 
-TMP_3 \def (S d) in (let TMP_4 \def (Bind b) in (let TMP_5 \def (lift h n u) 
-in (let TMP_6 \def (CHead c TMP_4 TMP_5) in (let TMP_7 \def (Bind b) in (let 
-TMP_8 \def (CHead e TMP_7 u) in (drop h TMP_3 TMP_6 TMP_8)))))))) in (let 
-TMP_10 \def (Bind b) in (let TMP_11 \def (drop_skip TMP_10 h d c e H u) in 
-(let TMP_12 \def (refl_equal nat d) in (eq_ind nat TMP_2 TMP_9 TMP_11 d 
-TMP_12))))))))))))).
+(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b) 
+d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e 
+(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))).
 
-theorem drop_skip_flat:
+lemma drop_skip_flat:
  \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h 
 (S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat 
 f) (lift h (S d) u)) (CHead e (Flat f) u))))))))
 \def
  \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda 
-(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(let TMP_1 \def (Flat 
-f) in (let TMP_2 \def (r TMP_1 d) in (let TMP_9 \def (\lambda (n: nat).(let 
-TMP_3 \def (S d) in (let TMP_4 \def (Flat f) in (let TMP_5 \def (lift h n u) 
-in (let TMP_6 \def (CHead c TMP_4 TMP_5) in (let TMP_7 \def (Flat f) in (let 
-TMP_8 \def (CHead e TMP_7 u) in (drop h TMP_3 TMP_6 TMP_8)))))))) in (let 
-TMP_10 \def (Flat f) in (let TMP_11 \def (drop_skip TMP_10 h d c e H u) in 
-(let TMP_12 \def (S d) in (let TMP_13 \def (S d) in (let TMP_14 \def 
-(refl_equal nat TMP_13) in (eq_ind nat TMP_2 TMP_9 TMP_11 TMP_12 
-TMP_14))))))))))))))).
+(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat 
+f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead 
+e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S 
+d))))))))).
 
-theorem drop_ctail:
+lemma drop_ctail:
  \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop 
 h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1) 
 (CTail k u c2))))))))
 \def
- \lambda (c1: C).(let TMP_3 \def (\lambda (c: C).(\forall (c2: C).(\forall 
-(d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u: 
-T).(let TMP_1 \def (CTail k u c) in (let TMP_2 \def (CTail k u c2) in (drop h 
-d TMP_1 TMP_2)))))))))) in (let TMP_32 \def (\lambda (n: nat).(\lambda (c2: 
-C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) 
-c2)).(\lambda (k: K).(\lambda (u: T).(let TMP_4 \def (CSort n) in (let TMP_5 
-\def (eq C c2 TMP_4) in (let TMP_6 \def (eq nat h O) in (let TMP_7 \def (eq 
-nat d O) in (let TMP_8 \def (CSort n) in (let TMP_9 \def (CTail k u TMP_8) in 
-(let TMP_10 \def (CTail k u c2) in (let TMP_11 \def (drop h d TMP_9 TMP_10) 
-in (let TMP_30 \def (\lambda (H0: (eq C c2 (CSort n))).(\lambda (H1: (eq nat 
-h O)).(\lambda (H2: (eq nat d O)).(let TMP_15 \def (\lambda (n0: nat).(let 
-TMP_12 \def (CSort n) in (let TMP_13 \def (CTail k u TMP_12) in (let TMP_14 
-\def (CTail k u c2) in (drop n0 d TMP_13 TMP_14))))) in (let TMP_19 \def 
-(\lambda (n0: nat).(let TMP_16 \def (CSort n) in (let TMP_17 \def (CTail k u 
-TMP_16) in (let TMP_18 \def (CTail k u c2) in (drop O n0 TMP_17 TMP_18))))) 
-in (let TMP_20 \def (CSort n) in (let TMP_24 \def (\lambda (c: C).(let TMP_21 
-\def (CSort n) in (let TMP_22 \def (CTail k u TMP_21) in (let TMP_23 \def 
-(CTail k u c) in (drop O O TMP_22 TMP_23))))) in (let TMP_25 \def (CSort n) 
-in (let TMP_26 \def (CTail k u TMP_25) in (let TMP_27 \def (drop_refl TMP_26) 
-in (let TMP_28 \def (eq_ind_r C TMP_20 TMP_24 TMP_27 c2 H0) in (let TMP_29 
-\def (eq_ind_r nat O TMP_19 TMP_28 d H2) in (eq_ind_r nat O TMP_15 TMP_29 h 
-H1))))))))))))) in (let TMP_31 \def (drop_gen_sort n h d c2 H) in (and3_ind 
-TMP_5 TMP_6 TMP_7 TMP_11 TMP_30 TMP_31)))))))))))))))))) in (let TMP_106 \def 
-(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall 
-(h: nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d 
-(CTail k u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: 
-T).(\lambda (c3: C).(\lambda (d: nat).(let TMP_36 \def (\lambda (n: 
-nat).(\forall (h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (k0: 
-K).(\forall (u: T).(let TMP_33 \def (CHead c2 k t) in (let TMP_34 \def (CTail 
-k0 u TMP_33) in (let TMP_35 \def (CTail k0 u c3) in (drop h n TMP_34 
-TMP_35))))))))) in (let TMP_58 \def (\lambda (h: nat).(let TMP_40 \def 
-(\lambda (n: nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall 
-(u: T).(let TMP_37 \def (CHead c2 k t) in (let TMP_38 \def (CTail k0 u 
-TMP_37) in (let TMP_39 \def (CTail k0 u c3) in (drop n O TMP_38 
-TMP_39)))))))) in (let TMP_51 \def (\lambda (H: (drop O O (CHead c2 k t) 
-c3)).(\lambda (k0: K).(\lambda (u: T).(let TMP_41 \def (CHead c2 k t) in (let 
-TMP_45 \def (\lambda (c: C).(let TMP_42 \def (CHead c2 k t) in (let TMP_43 
-\def (CTail k0 u TMP_42) in (let TMP_44 \def (CTail k0 u c) in (drop O O 
-TMP_43 TMP_44))))) in (let TMP_46 \def (CHead c2 k t) in (let TMP_47 \def 
-(CTail k0 u TMP_46) in (let TMP_48 \def (drop_refl TMP_47) in (let TMP_49 
-\def (CHead c2 k t) in (let TMP_50 \def (drop_gen_refl TMP_49 c3 H) in 
-(eq_ind C TMP_41 TMP_45 TMP_48 c3 TMP_50))))))))))) in (let TMP_57 \def 
-(\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall 
-(k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail k0 u 
-c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0: 
-K).(\lambda (u: T).(let TMP_52 \def (CTail k0 u c2) in (let TMP_53 \def 
-(CTail k0 u c3) in (let TMP_54 \def (r k n) in (let TMP_55 \def 
-(drop_gen_drop k c2 c3 t n H0) in (let TMP_56 \def (IHc c3 O TMP_54 TMP_55 k0 
-u) in (drop_drop k n TMP_52 TMP_53 TMP_56 t))))))))))) in (nat_ind TMP_40 
-TMP_51 TMP_57 h))))) in (let TMP_105 \def (\lambda (n: nat).(\lambda (H: 
-((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (k0: 
-K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail k0 u 
-c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t) 
-c3)).(\lambda (k0: K).(\lambda (u: T).(let TMP_60 \def (\lambda (e: 
-C).(\lambda (v: T).(let TMP_59 \def (CHead e k v) in (eq C c3 TMP_59)))) in 
-(let TMP_63 \def (\lambda (_: C).(\lambda (v: T).(let TMP_61 \def (r k n) in 
-(let TMP_62 \def (lift h TMP_61 v) in (eq T t TMP_62))))) in (let TMP_65 \def 
-(\lambda (e: C).(\lambda (_: T).(let TMP_64 \def (r k n) in (drop h TMP_64 c2 
-e)))) in (let TMP_66 \def (S n) in (let TMP_67 \def (CHead c2 k t) in (let 
-TMP_68 \def (CTail k0 u TMP_67) in (let TMP_69 \def (CTail k0 u c3) in (let 
-TMP_70 \def (drop h TMP_66 TMP_68 TMP_69) in (let TMP_103 \def (\lambda (x0: 
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d: 
+nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u: 
+T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda 
+(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) 
+c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h 
+O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0: 
+(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d 
+O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n)) 
+(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u 
+(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O 
+(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2 
+H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2: 
+C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: 
+nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k 
+u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: 
+C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n 
+(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u 
+(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n: 
+nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop 
+n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O 
+(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t) 
+(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c))) 
+(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3 
+H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to 
+(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail 
+k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0: 
+K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O 
+(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n: 
+nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to 
+(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail 
+k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t) 
+c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda 
+(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t 
+(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e))) 
+(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0: 
 C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2: 
-(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let 
-TMP_74 \def (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k t) c) 
-\to (\forall (k1: K).(\forall (u0: T).(let TMP_71 \def (CHead c2 k t) in (let 
-TMP_72 \def (CTail k1 u0 TMP_71) in (let TMP_73 \def (CTail k1 u0 c) in (drop 
-h0 n TMP_72 TMP_73))))))))) in (let TMP_75 \def (CHead x0 k x1) in (let H4 
-\def (eq_ind C c3 TMP_74 H TMP_75 H1) in (let TMP_76 \def (CHead x0 k x1) in 
-(let TMP_81 \def (\lambda (c: C).(let TMP_77 \def (S n) in (let TMP_78 \def 
-(CHead c2 k t) in (let TMP_79 \def (CTail k0 u TMP_78) in (let TMP_80 \def 
-(CTail k0 u c) in (drop h TMP_77 TMP_79 TMP_80)))))) in (let TMP_86 \def 
-(\lambda (t0: T).(\forall (h0: nat).((drop h0 n (CHead c2 k t0) (CHead x0 k 
-x1)) \to (\forall (k1: K).(\forall (u0: T).(let TMP_82 \def (CHead c2 k t0) 
-in (let TMP_83 \def (CTail k1 u0 TMP_82) in (let TMP_84 \def (CHead x0 k x1) 
-in (let TMP_85 \def (CTail k1 u0 TMP_84) in (drop h0 n TMP_83 
-TMP_85)))))))))) in (let TMP_87 \def (r k n) in (let TMP_88 \def (lift h 
-TMP_87 x1) in (let H5 \def (eq_ind T t TMP_86 H4 TMP_88 H2) in (let TMP_89 
-\def (r k n) in (let TMP_90 \def (lift h TMP_89 x1) in (let TMP_96 \def 
-(\lambda (t0: T).(let TMP_91 \def (S n) in (let TMP_92 \def (CHead c2 k t0) 
-in (let TMP_93 \def (CTail k0 u TMP_92) in (let TMP_94 \def (CHead x0 k x1) 
-in (let TMP_95 \def (CTail k0 u TMP_94) in (drop h TMP_91 TMP_93 
-TMP_95))))))) in (let TMP_97 \def (CTail k0 u c2) in (let TMP_98 \def (CTail 
-k0 u x0) in (let TMP_99 \def (r k n) in (let TMP_100 \def (IHc x0 TMP_99 h H3 
-k0 u) in (let TMP_101 \def (drop_skip k h n TMP_97 TMP_98 TMP_100 x1) in (let 
-TMP_102 \def (eq_ind_r T TMP_90 TMP_96 TMP_101 t H2) in (eq_ind_r C TMP_76 
-TMP_81 TMP_102 c3 H1)))))))))))))))))))))))) in (let TMP_104 \def 
-(drop_gen_skip_l c2 c3 t h n k H0) in (ex3_2_ind C T TMP_60 TMP_63 TMP_65 
-TMP_70 TMP_103 TMP_104))))))))))))))))) in (nat_ind TMP_36 TMP_58 TMP_105 
-d)))))))))) in (C_ind TMP_3 TMP_32 TMP_106 c1)))).
+(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4 
+\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k 
+t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2 
+k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k 
+x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u 
+c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n 
+(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop 
+h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4 
+(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: 
+T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) 
+(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) 
+x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) 
+c1).
 
 theorem drop_conf_lt:
  \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall 
@@ -150,537 +100,298 @@ nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
 (e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
 h (r k d) c0 e0)))))))))))))
 \def
- \lambda (k: K).(\lambda (i: nat).(let TMP_8 \def (\lambda (n: nat).(\forall 
-(u: T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to 
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c 
-e) \to (let TMP_3 \def (\lambda (v: T).(\lambda (_: C).(let TMP_1 \def (r k 
-d) in (let TMP_2 \def (lift h TMP_1 v) in (eq T u TMP_2))))) in (let TMP_5 
-\def (\lambda (v: T).(\lambda (e0: C).(let TMP_4 \def (CHead e0 k v) in (drop 
-n O e TMP_4)))) in (let TMP_7 \def (\lambda (_: T).(\lambda (e0: C).(let 
-TMP_6 \def (r k d) in (drop h TMP_6 c0 e0)))) in (ex3_2 T C TMP_3 TMP_5 
-TMP_7))))))))))))) in (let TMP_74 \def (\lambda (u: T).(\lambda (c0: 
-C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k u))).(\lambda (e: 
-C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h (S (plus O d)) c 
-e)).(let TMP_11 \def (\lambda (c1: C).(let TMP_9 \def (plus O d) in (let 
-TMP_10 \def (S TMP_9) in (drop h TMP_10 c1 e)))) in (let TMP_12 \def (CHead 
-c0 k u) in (let TMP_13 \def (CHead c0 k u) in (let TMP_14 \def (drop_gen_refl 
-c TMP_13 H) in (let H1 \def (eq_ind C c TMP_11 H0 TMP_12 TMP_14) in (let 
-TMP_16 \def (\lambda (e0: C).(\lambda (v: T).(let TMP_15 \def (CHead e0 k v) 
-in (eq C e TMP_15)))) in (let TMP_20 \def (\lambda (_: C).(\lambda (v: 
-T).(let TMP_17 \def (plus O d) in (let TMP_18 \def (r k TMP_17) in (let 
-TMP_19 \def (lift h TMP_18 v) in (eq T u TMP_19)))))) in (let TMP_23 \def 
-(\lambda (e0: C).(\lambda (_: T).(let TMP_21 \def (plus O d) in (let TMP_22 
-\def (r k TMP_21) in (drop h TMP_22 c0 e0))))) in (let TMP_26 \def (\lambda 
-(v: T).(\lambda (_: C).(let TMP_24 \def (r k d) in (let TMP_25 \def (lift h 
-TMP_24 v) in (eq T u TMP_25))))) in (let TMP_28 \def (\lambda (v: T).(\lambda 
-(e0: C).(let TMP_27 \def (CHead e0 k v) in (drop O O e TMP_27)))) in (let 
-TMP_30 \def (\lambda (_: T).(\lambda (e0: C).(let TMP_29 \def (r k d) in 
-(drop h TMP_29 c0 e0)))) in (let TMP_31 \def (ex3_2 T C TMP_26 TMP_28 TMP_30) 
-in (let TMP_71 \def (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C e 
-(CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d)) 
-x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(let TMP_32 \def (CHead 
-x0 k x1) in (let TMP_40 \def (\lambda (c1: C).(let TMP_35 \def (\lambda (v: 
-T).(\lambda (_: C).(let TMP_33 \def (r k d) in (let TMP_34 \def (lift h 
-TMP_33 v) in (eq T u TMP_34))))) in (let TMP_37 \def (\lambda (v: T).(\lambda 
-(e0: C).(let TMP_36 \def (CHead e0 k v) in (drop O O c1 TMP_36)))) in (let 
-TMP_39 \def (\lambda (_: T).(\lambda (e0: C).(let TMP_38 \def (r k d) in 
-(drop h TMP_38 c0 e0)))) in (ex3_2 T C TMP_35 TMP_37 TMP_39))))) in (let 
-TMP_41 \def (plus O d) in (let TMP_42 \def (r k TMP_41) in (let TMP_43 \def 
-(lift h TMP_42 x1) in (let TMP_52 \def (\lambda (t: T).(let TMP_46 \def 
-(\lambda (v: T).(\lambda (_: C).(let TMP_44 \def (r k d) in (let TMP_45 \def 
-(lift h TMP_44 v) in (eq T t TMP_45))))) in (let TMP_49 \def (\lambda (v: 
-T).(\lambda (e0: C).(let TMP_47 \def (CHead x0 k x1) in (let TMP_48 \def 
-(CHead e0 k v) in (drop O O TMP_47 TMP_48))))) in (let TMP_51 \def (\lambda 
-(_: T).(\lambda (e0: C).(let TMP_50 \def (r k d) in (drop h TMP_50 c0 e0)))) 
-in (ex3_2 T C TMP_46 TMP_49 TMP_51))))) in (let TMP_58 \def (\lambda (v: 
-T).(\lambda (_: C).(let TMP_53 \def (plus O d) in (let TMP_54 \def (r k 
-TMP_53) in (let TMP_55 \def (lift h TMP_54 x1) in (let TMP_56 \def (r k d) in 
-(let TMP_57 \def (lift h TMP_56 v) in (eq T TMP_55 TMP_57)))))))) in (let 
-TMP_61 \def (\lambda (v: T).(\lambda (e0: C).(let TMP_59 \def (CHead x0 k x1) 
-in (let TMP_60 \def (CHead e0 k v) in (drop O O TMP_59 TMP_60))))) in (let 
-TMP_63 \def (\lambda (_: T).(\lambda (e0: C).(let TMP_62 \def (r k d) in 
-(drop h TMP_62 c0 e0)))) in (let TMP_64 \def (r k d) in (let TMP_65 \def 
-(lift h TMP_64 x1) in (let TMP_66 \def (refl_equal T TMP_65) in (let TMP_67 
-\def (CHead x0 k x1) in (let TMP_68 \def (drop_refl TMP_67) in (let TMP_69 
-\def (ex3_2_intro T C TMP_58 TMP_61 TMP_63 x1 x0 TMP_66 TMP_68 H4) in (let 
-TMP_70 \def (eq_ind_r T TMP_43 TMP_52 TMP_69 u H3) in (eq_ind_r C TMP_32 
-TMP_40 TMP_70 e H2)))))))))))))))))))))) in (let TMP_72 \def (plus O d) in 
-(let TMP_73 \def (drop_gen_skip_l c0 e u h TMP_72 k H1) in (ex3_2_ind C T 
-TMP_16 TMP_20 TMP_23 TMP_31 TMP_71 TMP_73)))))))))))))))))))))))) in (let 
-TMP_283 \def (\lambda (i0: nat).(\lambda (H: ((\forall (u: T).(\forall (c0: 
-C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to (\forall (e: C).(\forall 
-(h: nat).(\forall (d: nat).((drop h (S (plus i0 d)) c e) \to (ex3_2 T C 
+ \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u: 
+T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall 
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) 
+(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u: 
+T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k 
+u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop 
+h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S 
+(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in 
+(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) 
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v)))) 
+(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C 
 (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: 
-T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v)))) (\lambda (_: T).(\lambda 
-(e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda (u: T).(\lambda (c0: 
-C).(\lambda (c: C).(let TMP_83 \def (\lambda (c1: C).((drop (S i0) O c1 
-(CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h (S (plus (S i0) d)) c1 e) \to (let TMP_77 \def (\lambda (v: 
-T).(\lambda (_: C).(let TMP_75 \def (r k d) in (let TMP_76 \def (lift h 
-TMP_75 v) in (eq T u TMP_76))))) in (let TMP_80 \def (\lambda (v: T).(\lambda 
-(e0: C).(let TMP_78 \def (S i0) in (let TMP_79 \def (CHead e0 k v) in (drop 
-TMP_78 O e TMP_79))))) in (let TMP_82 \def (\lambda (_: T).(\lambda (e0: 
-C).(let TMP_81 \def (r k d) in (drop h TMP_81 c0 e0)))) in (ex3_2 T C TMP_77 
-TMP_80 TMP_82)))))))))) in (let TMP_118 \def (\lambda (n: nat).(\lambda (_: 
-(drop (S i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) 
-e)).(let TMP_84 \def (CSort n) in (let TMP_85 \def (eq C e TMP_84) in (let 
-TMP_86 \def (eq nat h O) in (let TMP_87 \def (S i0) in (let TMP_88 \def (plus 
-TMP_87 d) in (let TMP_89 \def (S TMP_88) in (let TMP_90 \def (eq nat TMP_89 
-O) in (let TMP_93 \def (\lambda (v: T).(\lambda (_: C).(let TMP_91 \def (r k 
-d) in (let TMP_92 \def (lift h TMP_91 v) in (eq T u TMP_92))))) in (let 
-TMP_96 \def (\lambda (v: T).(\lambda (e0: C).(let TMP_94 \def (S i0) in (let 
-TMP_95 \def (CHead e0 k v) in (drop TMP_94 O e TMP_95))))) in (let TMP_98 
-\def (\lambda (_: T).(\lambda (e0: C).(let TMP_97 \def (r k d) in (drop h 
-TMP_97 c0 e0)))) in (let TMP_99 \def (ex3_2 T C TMP_93 TMP_96 TMP_98) in (let 
-TMP_113 \def (\lambda (_: (eq C e (CSort n))).(\lambda (_: (eq nat h 
-O)).(\lambda (H4: (eq nat (S (plus (S i0) d)) O)).(let TMP_100 \def (S i0) in 
-(let TMP_101 \def (plus TMP_100 d) in (let TMP_102 \def (S TMP_101) in (let 
-TMP_103 \def (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) 
-\Rightarrow True])) in (let H5 \def (eq_ind nat TMP_102 TMP_103 I O H4) in 
-(let TMP_106 \def (\lambda (v: T).(\lambda (_: C).(let TMP_104 \def (r k d) 
-in (let TMP_105 \def (lift h TMP_104 v) in (eq T u TMP_105))))) in (let 
-TMP_109 \def (\lambda (v: T).(\lambda (e0: C).(let TMP_107 \def (S i0) in 
-(let TMP_108 \def (CHead e0 k v) in (drop TMP_107 O e TMP_108))))) in (let 
-TMP_111 \def (\lambda (_: T).(\lambda (e0: C).(let TMP_110 \def (r k d) in 
-(drop h TMP_110 c0 e0)))) in (let TMP_112 \def (ex3_2 T C TMP_106 TMP_109 
-TMP_111) in (False_ind TMP_112 H5))))))))))))) in (let TMP_114 \def (S i0) in 
-(let TMP_115 \def (plus TMP_114 d) in (let TMP_116 \def (S TMP_115) in (let 
-TMP_117 \def (drop_gen_sort n h TMP_116 e H1) in (and3_ind TMP_85 TMP_86 
-TMP_90 TMP_99 TMP_113 TMP_117))))))))))))))))))))))) in (let TMP_282 \def 
-(\lambda (c1: C).(\lambda (H0: (((drop (S i0) O c1 (CHead c0 k u)) \to 
+T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda 
+(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda 
+(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d)) 
+x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k 
+x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift 
+h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T 
+(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v: 
+T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
+(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda 
+(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda 
+(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v: 
+T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k 
+d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0 
+e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall 
+(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to 
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d)) 
+c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) 
+v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda 
+(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0) 
+O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v: 
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S 
+i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind 
+(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: 
+T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort 
+n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d)) 
+O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee 
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) 
+(\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) H5))))) (drop_gen_sort n h 
+(S (plus (S i0) d)) e H1)))))))) (\lambda (c1: C).(\lambda (H0: (((drop (S 
+i0) O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v: 
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda 
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop h (r k d) c0 e0))))))))))).(\lambda (k0: K).(K_ind (\lambda (k1: 
+K).(\forall (t: T).((drop (S i0) O (CHead c1 k1 t) (CHead c0 k u)) \to 
 (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0) 
-d)) c1 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k 
-d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) 
-(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))).(\lambda 
-(k0: K).(let TMP_127 \def (\lambda (k1: K).(\forall (t: T).((drop (S i0) O 
-(CHead c1 k1 t) (CHead c0 k u)) \to (\forall (e: C).(\forall (h: 
-nat).(\forall (d: nat).((drop h (S (plus (S i0) d)) (CHead c1 k1 t) e) \to 
-(let TMP_121 \def (\lambda (v: T).(\lambda (_: C).(let TMP_119 \def (r k d) 
-in (let TMP_120 \def (lift h TMP_119 v) in (eq T u TMP_120))))) in (let 
-TMP_124 \def (\lambda (v: T).(\lambda (e0: C).(let TMP_122 \def (S i0) in 
-(let TMP_123 \def (CHead e0 k v) in (drop TMP_122 O e TMP_123))))) in (let 
-TMP_126 \def (\lambda (_: T).(\lambda (e0: C).(let TMP_125 \def (r k d) in 
-(drop h TMP_125 c0 e0)))) in (ex3_2 T C TMP_121 TMP_124 TMP_126))))))))))) in 
-(let TMP_203 \def (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) 
-O (CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: 
+d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e 
+(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 
+e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O 
+(CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: 
 nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) d)) (CHead c1 
-(Bind b) t) e)).(let TMP_130 \def (\lambda (e0: C).(\lambda (v: T).(let 
-TMP_128 \def (Bind b) in (let TMP_129 \def (CHead e0 TMP_128 v) in (eq C e 
-TMP_129))))) in (let TMP_136 \def (\lambda (_: C).(\lambda (v: T).(let 
-TMP_131 \def (Bind b) in (let TMP_132 \def (S i0) in (let TMP_133 \def (plus 
-TMP_132 d) in (let TMP_134 \def (r TMP_131 TMP_133) in (let TMP_135 \def 
-(lift h TMP_134 v) in (eq T t TMP_135)))))))) in (let TMP_141 \def (\lambda 
-(e0: C).(\lambda (_: T).(let TMP_137 \def (Bind b) in (let TMP_138 \def (S 
-i0) in (let TMP_139 \def (plus TMP_138 d) in (let TMP_140 \def (r TMP_137 
-TMP_139) in (drop h TMP_140 c1 e0))))))) in (let TMP_144 \def (\lambda (v: 
-T).(\lambda (_: C).(let TMP_142 \def (r k d) in (let TMP_143 \def (lift h 
-TMP_142 v) in (eq T u TMP_143))))) in (let TMP_147 \def (\lambda (v: 
-T).(\lambda (e0: C).(let TMP_145 \def (S i0) in (let TMP_146 \def (CHead e0 k 
-v) in (drop TMP_145 O e TMP_146))))) in (let TMP_149 \def (\lambda (_: 
-T).(\lambda (e0: C).(let TMP_148 \def (r k d) in (drop h TMP_148 c0 e0)))) in 
-(let TMP_150 \def (ex3_2 T C TMP_144 TMP_147 TMP_149) in (let TMP_198 \def 
-(\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e (CHead x0 (Bind b) 
-x1))).(\lambda (_: (eq T t (lift h (r (Bind b) (plus (S i0) d)) 
-x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1 x0)).(let TMP_151 
-\def (Bind b) in (let TMP_152 \def (CHead x0 TMP_151 x1) in (let TMP_161 \def 
-(\lambda (c2: C).(let TMP_155 \def (\lambda (v: T).(\lambda (_: C).(let 
-TMP_153 \def (r k d) in (let TMP_154 \def (lift h TMP_153 v) in (eq T u 
-TMP_154))))) in (let TMP_158 \def (\lambda (v: T).(\lambda (e0: C).(let 
-TMP_156 \def (S i0) in (let TMP_157 \def (CHead e0 k v) in (drop TMP_156 O c2 
-TMP_157))))) in (let TMP_160 \def (\lambda (_: T).(\lambda (e0: C).(let 
-TMP_159 \def (r k d) in (drop h TMP_159 c0 e0)))) in (ex3_2 T C TMP_155 
-TMP_158 TMP_160))))) in (let TMP_162 \def (Bind b) in (let TMP_163 \def 
-(CHead c0 k u) in (let TMP_164 \def (drop_gen_drop TMP_162 c1 TMP_163 t i0 
-H1) in (let H6 \def (H u c0 c1 TMP_164 x0 h d H5) in (let TMP_167 \def 
-(\lambda (v: T).(\lambda (_: C).(let TMP_165 \def (r k d) in (let TMP_166 
-\def (lift h TMP_165 v) in (eq T u TMP_166))))) in (let TMP_169 \def (\lambda 
-(v: T).(\lambda (e0: C).(let TMP_168 \def (CHead e0 k v) in (drop i0 O x0 
-TMP_168)))) in (let TMP_171 \def (\lambda (_: T).(\lambda (e0: C).(let 
-TMP_170 \def (r k d) in (drop h TMP_170 c0 e0)))) in (let TMP_174 \def 
-(\lambda (v: T).(\lambda (_: C).(let TMP_172 \def (r k d) in (let TMP_173 
-\def (lift h TMP_172 v) in (eq T u TMP_173))))) in (let TMP_179 \def (\lambda 
-(v: T).(\lambda (e0: C).(let TMP_175 \def (S i0) in (let TMP_176 \def (Bind 
-b) in (let TMP_177 \def (CHead x0 TMP_176 x1) in (let TMP_178 \def (CHead e0 
-k v) in (drop TMP_175 O TMP_177 TMP_178))))))) in (let TMP_181 \def (\lambda 
-(_: T).(\lambda (e0: C).(let TMP_180 \def (r k d) in (drop h TMP_180 c0 
-e0)))) in (let TMP_182 \def (ex3_2 T C TMP_174 TMP_179 TMP_181) in (let 
-TMP_196 \def (\lambda (x2: T).(\lambda (x3: C).(\lambda (H7: (eq T u (lift h 
-(r k d) x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k x2))).(\lambda (H9: 
-(drop h (r k d) c0 x3)).(let TMP_185 \def (\lambda (v: T).(\lambda (_: 
-C).(let TMP_183 \def (r k d) in (let TMP_184 \def (lift h TMP_183 v) in (eq T 
-u TMP_184))))) in (let TMP_190 \def (\lambda (v: T).(\lambda (e0: C).(let 
-TMP_186 \def (S i0) in (let TMP_187 \def (Bind b) in (let TMP_188 \def (CHead 
-x0 TMP_187 x1) in (let TMP_189 \def (CHead e0 k v) in (drop TMP_186 O TMP_188 
-TMP_189))))))) in (let TMP_192 \def (\lambda (_: T).(\lambda (e0: C).(let 
-TMP_191 \def (r k d) in (drop h TMP_191 c0 e0)))) in (let TMP_193 \def (Bind 
-b) in (let TMP_194 \def (CHead x3 k x2) in (let TMP_195 \def (drop_drop 
-TMP_193 i0 x0 TMP_194 H8 x1) in (ex3_2_intro T C TMP_185 TMP_190 TMP_192 x2 
-x3 H7 TMP_195 H9)))))))))))) in (let TMP_197 \def (ex3_2_ind T C TMP_167 
-TMP_169 TMP_171 TMP_182 TMP_196 H6) in (eq_ind_r C TMP_152 TMP_161 TMP_197 e 
-H3)))))))))))))))))))))) in (let TMP_199 \def (S i0) in (let TMP_200 \def 
-(plus TMP_199 d) in (let TMP_201 \def (Bind b) in (let TMP_202 \def 
-(drop_gen_skip_l c1 e t h TMP_200 TMP_201 H2) in (ex3_2_ind C T TMP_130 
-TMP_136 TMP_141 TMP_150 TMP_198 TMP_202)))))))))))))))))))) in (let TMP_281 
-\def (\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c1 
-(Flat f) t) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H2: (drop h (S (plus (S i0) d)) (CHead c1 (Flat f) t) e)).(let 
-TMP_206 \def (\lambda (e0: C).(\lambda (v: T).(let TMP_204 \def (Flat f) in 
-(let TMP_205 \def (CHead e0 TMP_204 v) in (eq C e TMP_205))))) in (let 
-TMP_212 \def (\lambda (_: C).(\lambda (v: T).(let TMP_207 \def (Flat f) in 
-(let TMP_208 \def (S i0) in (let TMP_209 \def (plus TMP_208 d) in (let 
-TMP_210 \def (r TMP_207 TMP_209) in (let TMP_211 \def (lift h TMP_210 v) in 
-(eq T t TMP_211)))))))) in (let TMP_217 \def (\lambda (e0: C).(\lambda (_: 
-T).(let TMP_213 \def (Flat f) in (let TMP_214 \def (S i0) in (let TMP_215 
-\def (plus TMP_214 d) in (let TMP_216 \def (r TMP_213 TMP_215) in (drop h 
-TMP_216 c1 e0))))))) in (let TMP_220 \def (\lambda (v: T).(\lambda (_: 
-C).(let TMP_218 \def (r k d) in (let TMP_219 \def (lift h TMP_218 v) in (eq T 
-u TMP_219))))) in (let TMP_223 \def (\lambda (v: T).(\lambda (e0: C).(let 
-TMP_221 \def (S i0) in (let TMP_222 \def (CHead e0 k v) in (drop TMP_221 O e 
-TMP_222))))) in (let TMP_225 \def (\lambda (_: T).(\lambda (e0: C).(let 
-TMP_224 \def (r k d) in (drop h TMP_224 c0 e0)))) in (let TMP_226 \def (ex3_2 
-T C TMP_220 TMP_223 TMP_225) in (let TMP_276 \def (\lambda (x0: C).(\lambda 
-(x1: T).(\lambda (H3: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t 
-(lift h (r (Flat f) (plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Flat f) 
-(plus (S i0) d)) c1 x0)).(let TMP_227 \def (Flat f) in (let TMP_228 \def 
-(CHead x0 TMP_227 x1) in (let TMP_237 \def (\lambda (c2: C).(let TMP_231 \def 
-(\lambda (v: T).(\lambda (_: C).(let TMP_229 \def (r k d) in (let TMP_230 
-\def (lift h TMP_229 v) in (eq T u TMP_230))))) in (let TMP_234 \def (\lambda 
-(v: T).(\lambda (e0: C).(let TMP_232 \def (S i0) in (let TMP_233 \def (CHead 
-e0 k v) in (drop TMP_232 O c2 TMP_233))))) in (let TMP_236 \def (\lambda (_: 
-T).(\lambda (e0: C).(let TMP_235 \def (r k d) in (drop h TMP_235 c0 e0)))) in 
-(ex3_2 T C TMP_231 TMP_234 TMP_236))))) in (let TMP_240 \def (\lambda (v: 
-T).(\lambda (_: C).(let TMP_238 \def (r k d) in (let TMP_239 \def (lift h 
-TMP_238 v) in (eq T u TMP_239))))) in (let TMP_243 \def (\lambda (v: 
-T).(\lambda (e0: C).(let TMP_241 \def (S i0) in (let TMP_242 \def (CHead e0 k 
-v) in (drop TMP_241 O x0 TMP_242))))) in (let TMP_245 \def (\lambda (_: 
-T).(\lambda (e0: C).(let TMP_244 \def (r k d) in (drop h TMP_244 c0 e0)))) in 
-(let TMP_248 \def (\lambda (v: T).(\lambda (_: C).(let TMP_246 \def (r k d) 
-in (let TMP_247 \def (lift h TMP_246 v) in (eq T u TMP_247))))) in (let 
-TMP_253 \def (\lambda (v: T).(\lambda (e0: C).(let TMP_249 \def (S i0) in 
-(let TMP_250 \def (Flat f) in (let TMP_251 \def (CHead x0 TMP_250 x1) in (let 
-TMP_252 \def (CHead e0 k v) in (drop TMP_249 O TMP_251 TMP_252))))))) in (let 
-TMP_255 \def (\lambda (_: T).(\lambda (e0: C).(let TMP_254 \def (r k d) in 
-(drop h TMP_254 c0 e0)))) in (let TMP_256 \def (ex3_2 T C TMP_248 TMP_253 
-TMP_255) in (let TMP_270 \def (\lambda (x2: T).(\lambda (x3: C).(\lambda (H6: 
-(eq T u (lift h (r k d) x2))).(\lambda (H7: (drop (S i0) O x0 (CHead x3 k 
-x2))).(\lambda (H8: (drop h (r k d) c0 x3)).(let TMP_259 \def (\lambda (v: 
-T).(\lambda (_: C).(let TMP_257 \def (r k d) in (let TMP_258 \def (lift h 
-TMP_257 v) in (eq T u TMP_258))))) in (let TMP_264 \def (\lambda (v: 
-T).(\lambda (e0: C).(let TMP_260 \def (S i0) in (let TMP_261 \def (Flat f) in 
-(let TMP_262 \def (CHead x0 TMP_261 x1) in (let TMP_263 \def (CHead e0 k v) 
-in (drop TMP_260 O TMP_262 TMP_263))))))) in (let TMP_266 \def (\lambda (_: 
-T).(\lambda (e0: C).(let TMP_265 \def (r k d) in (drop h TMP_265 c0 e0)))) in 
-(let TMP_267 \def (Flat f) in (let TMP_268 \def (CHead x3 k x2) in (let 
-TMP_269 \def (drop_drop TMP_267 i0 x0 TMP_268 H7 x1) in (ex3_2_intro T C 
-TMP_259 TMP_264 TMP_266 x2 x3 H6 TMP_269 H8)))))))))))) in (let TMP_271 \def 
-(Flat f) in (let TMP_272 \def (CHead c0 k u) in (let TMP_273 \def 
-(drop_gen_drop TMP_271 c1 TMP_272 t i0 H1) in (let TMP_274 \def (H0 TMP_273 
-x0 h d H5) in (let TMP_275 \def (ex3_2_ind T C TMP_240 TMP_243 TMP_245 
-TMP_256 TMP_270 TMP_274) in (eq_ind_r C TMP_228 TMP_237 TMP_275 e 
-H3)))))))))))))))))))))) in (let TMP_277 \def (S i0) in (let TMP_278 \def 
-(plus TMP_277 d) in (let TMP_279 \def (Flat f) in (let TMP_280 \def 
-(drop_gen_skip_l c1 e t h TMP_278 TMP_279 H2) in (ex3_2_ind C T TMP_206 
-TMP_212 TMP_217 TMP_226 TMP_276 TMP_280)))))))))))))))))))) in (K_ind TMP_127 
-TMP_203 TMP_281 k0))))))) in (C_ind TMP_83 TMP_118 TMP_282 c))))))))) in 
-(nat_ind TMP_8 TMP_74 TMP_283 i))))).
+(Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e 
+(CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r 
+(Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
+(Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S 
+i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 
+e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e (CHead x0 
+(Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) (plus (S i0) d)) 
+x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1 x0)).(eq_ind_r C 
+(CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda (v: T).(\lambda 
+(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop 
+(S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k 
+d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b) c1 (CHead c0 k 
+u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq 
+T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O x0 
+(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))) 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) 
+(\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead 
+e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) 
+(\lambda (x2: T).(\lambda (x3: C).(\lambda (H7: (eq T u (lift h (r k d) 
+x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k x2))).(\lambda (H9: (drop h (r 
+k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h 
+(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Bind 
+b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 
+e0))) x2 x3 H7 (drop_drop (Bind b) i0 x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) 
+e H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Bind b) H2))))))))) 
+(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c1 (Flat 
+f) t) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H2: (drop h (S (plus (S i0) d)) (CHead c1 (Flat f) t) 
+e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat 
+f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) (plus (S 
+i0) d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) (plus (S 
+i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h 
+(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda 
+(x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e (CHead x0 (Flat f) 
+x1))).(\lambda (_: (eq T t (lift h (r (Flat f) (plus (S i0) d)) 
+x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1 x0)).(eq_ind_r C 
+(CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda (v: T).(\lambda 
+(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop 
+(S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k 
+d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h 
+(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O x0 (CHead e0 k 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))) (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: 
+T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) (CHead e0 k v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (x2: 
+T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d) x2))).(\lambda (H7: 
+(drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h (r k d) c0 
+x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) 
+v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) 
+(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))) 
+x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1) H8)))))) (H0 
+(drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e H3)))))) 
+(drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2))))))))) k0)))) c)))))) 
+i)).
 
 theorem drop_conf_ge:
  \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to 
 (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le 
 (plus d h) i) \to (drop (minus i h) O e a)))))))))
 \def
- \lambda (i: nat).(let TMP_2 \def (\lambda (n: nat).(\forall (a: C).(\forall 
-(c: C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
-nat).((drop h d c e) \to ((le (plus d h) n) \to (let TMP_1 \def (minus n h) 
-in (drop TMP_1 O e a))))))))))) in (let TMP_23 \def (\lambda (a: C).(\lambda 
-(c: C).(\lambda (H: (drop O O c a)).(\lambda (e: C).(\lambda (h: 
-nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(\lambda (H1: (le (plus 
-d h) O)).(let TMP_3 \def (\lambda (c0: C).(drop h d c0 e)) in (let TMP_4 \def 
-(drop_gen_refl c a H) in (let H2 \def (eq_ind C c TMP_3 H0 a TMP_4) in (let 
-TMP_5 \def (plus d h) in (let H_y \def (le_n_O_eq TMP_5 H1) in (let TMP_6 
-\def (eq nat d O) in (let TMP_7 \def (eq nat h O) in (let TMP_8 \def (minus O 
-h) in (let TMP_9 \def (drop TMP_8 O e a) in (let TMP_19 \def (\lambda (H3: 
-(eq nat d O)).(\lambda (H4: (eq nat h O)).(let TMP_10 \def (\lambda (n: 
-nat).(drop h n a e)) in (let H5 \def (eq_ind nat d TMP_10 H2 O H3) in (let 
-TMP_11 \def (\lambda (n: nat).(drop n O a e)) in (let H6 \def (eq_ind nat h 
-TMP_11 H5 O H4) in (let TMP_13 \def (\lambda (n: nat).(let TMP_12 \def (minus 
-O n) in (drop TMP_12 O e a))) in (let TMP_15 \def (\lambda (c0: C).(let 
-TMP_14 \def (minus O O) in (drop TMP_14 O c0 a))) in (let TMP_16 \def 
-(drop_refl a) in (let TMP_17 \def (drop_gen_refl a e H6) in (let TMP_18 \def 
-(eq_ind C a TMP_15 TMP_16 e TMP_17) in (eq_ind_r nat O TMP_13 TMP_18 h 
-H4)))))))))))) in (let TMP_20 \def (plus d h) in (let TMP_21 \def (sym_eq nat 
-O TMP_20 H_y) in (let TMP_22 \def (plus_O d h TMP_21) in (land_ind TMP_6 
-TMP_7 TMP_9 TMP_19 TMP_22)))))))))))))))))))))) in (let TMP_227 \def (\lambda 
-(i0: nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to 
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c: 
+C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: 
+nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e 
+a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c 
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h 
+d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda 
+(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def 
+(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus 
+O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5 
+\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def 
+(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O 
+(\lambda (n: nat).(drop (minus O n) O e a)) (eq_ind C a (\lambda (c0: 
+C).(drop (minus O O) O c0 a)) (drop_refl a) e (drop_gen_refl a e H6)) h 
+H4))))) (plus_O d h (sym_eq nat O (plus d h) H_y))))))))))))) (\lambda (i0: 
+nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to 
 (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le 
 (plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: 
-C).(\lambda (c: C).(let TMP_26 \def (\lambda (c0: C).((drop (S i0) O c0 a) 
-\to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to 
-((le (plus d h) (S i0)) \to (let TMP_24 \def (S i0) in (let TMP_25 \def 
-(minus TMP_24 h) in (drop TMP_25 O e a)))))))))) in (let TMP_75 \def (\lambda 
-(n: nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda 
-(h: nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda 
-(H2: (le (plus d h) (S i0))).(let TMP_27 \def (CSort n) in (let TMP_28 \def 
-(eq C e TMP_27) in (let TMP_29 \def (eq nat h O) in (let TMP_30 \def (eq nat 
-d O) in (let TMP_31 \def (S i0) in (let TMP_32 \def (minus TMP_31 h) in (let 
-TMP_33 \def (drop TMP_32 O e a) in (let TMP_73 \def (\lambda (H3: (eq C e 
-(CSort n))).(\lambda (H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(let 
-TMP_34 \def (CSort n) in (let TMP_35 \def (eq C a TMP_34) in (let TMP_36 \def 
-(S i0) in (let TMP_37 \def (eq nat TMP_36 O) in (let TMP_38 \def (eq nat O O) 
-in (let TMP_39 \def (S i0) in (let TMP_40 \def (minus TMP_39 h) in (let 
-TMP_41 \def (drop TMP_40 O e a) in (let TMP_70 \def (\lambda (H6: (eq C a 
-(CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(let 
-TMP_44 \def (\lambda (n0: nat).(let TMP_42 \def (plus n0 h) in (let TMP_43 
-\def (S i0) in (le TMP_42 TMP_43)))) in (let H9 \def (eq_ind nat d TMP_44 H2 
-O H5) in (let TMP_47 \def (\lambda (n0: nat).(let TMP_45 \def (plus O n0) in 
-(let TMP_46 \def (S i0) in (le TMP_45 TMP_46)))) in (let H10 \def (eq_ind nat 
-h TMP_47 H9 O H4) in (let TMP_50 \def (\lambda (n0: nat).(let TMP_48 \def (S 
-i0) in (let TMP_49 \def (minus TMP_48 n0) in (drop TMP_49 O e a)))) in (let 
-TMP_51 \def (CSort n) in (let TMP_54 \def (\lambda (c0: C).(let TMP_52 \def 
-(S i0) in (let TMP_53 \def (minus TMP_52 O) in (drop TMP_53 O c0 a)))) in 
-(let TMP_55 \def (CSort n) in (let TMP_59 \def (\lambda (c0: C).(let TMP_56 
-\def (S i0) in (let TMP_57 \def (minus TMP_56 O) in (let TMP_58 \def (CSort 
-n) in (drop TMP_57 O TMP_58 c0))))) in (let TMP_60 \def (S i0) in (let TMP_61 
-\def (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) 
-\Rightarrow True])) in (let H11 \def (eq_ind nat TMP_60 TMP_61 I O H7) in 
-(let TMP_62 \def (S i0) in (let TMP_63 \def (minus TMP_62 O) in (let TMP_64 
-\def (CSort n) in (let TMP_65 \def (CSort n) in (let TMP_66 \def (drop TMP_63 
-O TMP_64 TMP_65) in (let TMP_67 \def (False_ind TMP_66 H11) in (let TMP_68 
-\def (eq_ind_r C TMP_55 TMP_59 TMP_67 a H6) in (let TMP_69 \def (eq_ind_r C 
-TMP_51 TMP_54 TMP_68 e H3) in (eq_ind_r nat O TMP_50 TMP_69 h 
-H4)))))))))))))))))))))))) in (let TMP_71 \def (S i0) in (let TMP_72 \def 
-(drop_gen_sort n TMP_71 O a H0) in (and3_ind TMP_35 TMP_37 TMP_38 TMP_41 
-TMP_70 TMP_72))))))))))))))) in (let TMP_74 \def (drop_gen_sort n h d e H1) 
-in (and3_ind TMP_28 TMP_29 TMP_30 TMP_33 TMP_73 TMP_74))))))))))))))))) in 
-(let TMP_226 \def (\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to 
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le 
-(plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k: 
-K).(let TMP_78 \def (\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead 
-c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
-d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (let TMP_76 \def (S i0) 
-in (let TMP_77 \def (minus TMP_76 h) in (drop TMP_77 O e a))))))))))) in (let 
-TMP_148 \def (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O 
-(CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le 
-(plus d h) (S i0))).(let TMP_81 \def (\lambda (n: nat).((drop h n (CHead c0 
-(Bind b) t) e) \to ((le (plus n h) (S i0)) \to (let TMP_79 \def (S i0) in 
-(let TMP_80 \def (minus TMP_79 h) in (drop TMP_80 O e a)))))) in (let TMP_105 
-\def (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le 
-(plus O h) (S i0))).(let TMP_84 \def (\lambda (n: nat).((drop n O (CHead c0 
-(Bind b) t) e) \to ((le (plus O n) (S i0)) \to (let TMP_82 \def (S i0) in 
-(let TMP_83 \def (minus TMP_82 n) in (drop TMP_83 O e a)))))) in (let TMP_97 
-\def (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le (plus 
-O O) (S i0))).(let TMP_85 \def (Bind b) in (let TMP_86 \def (CHead c0 TMP_85 
-t) in (let TMP_89 \def (\lambda (c1: C).(let TMP_87 \def (S i0) in (let 
-TMP_88 \def (minus TMP_87 O) in (drop TMP_88 O c1 a)))) in (let TMP_90 \def 
-(Bind b) in (let TMP_91 \def (Bind b) in (let TMP_92 \def (drop_gen_drop 
-TMP_91 c0 a t i0 H1) in (let TMP_93 \def (drop_drop TMP_90 i0 c0 a TMP_92 t) 
-in (let TMP_94 \def (Bind b) in (let TMP_95 \def (CHead c0 TMP_94 t) in (let 
-TMP_96 \def (drop_gen_refl TMP_95 e H6) in (eq_ind C TMP_86 TMP_89 TMP_93 e 
-TMP_96))))))))))))) in (let TMP_104 \def (\lambda (h0: nat).(\lambda (_: 
-(((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to (drop 
-(minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Bind b) 
-t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(let TMP_98 \def (Bind b) 
-in (let TMP_99 \def (drop_gen_drop TMP_98 c0 a t i0 H1) in (let TMP_100 \def 
-(Bind b) in (let TMP_101 \def (drop_gen_drop TMP_100 c0 e t h0 H6) in (let 
-TMP_102 \def (plus O h0) in (let TMP_103 \def (le_S_n TMP_102 i0 H7) in (H a 
-c0 TMP_99 e h0 O TMP_101 TMP_103))))))))))) in (nat_ind TMP_84 TMP_97 TMP_104 
-h H4 H5)))))) in (let TMP_147 \def (\lambda (d0: nat).(\lambda (_: (((drop h 
-d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S 
-i0) h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Bind b) t) 
-e)).(\lambda (H5: (le (plus (S d0) h) (S i0))).(let TMP_108 \def (\lambda 
-(e0: C).(\lambda (v: T).(let TMP_106 \def (Bind b) in (let TMP_107 \def 
-(CHead e0 TMP_106 v) in (eq C e TMP_107))))) in (let TMP_112 \def (\lambda 
-(_: C).(\lambda (v: T).(let TMP_109 \def (Bind b) in (let TMP_110 \def (r 
-TMP_109 d0) in (let TMP_111 \def (lift h TMP_110 v) in (eq T t TMP_111)))))) 
-in (let TMP_115 \def (\lambda (e0: C).(\lambda (_: T).(let TMP_113 \def (Bind 
-b) in (let TMP_114 \def (r TMP_113 d0) in (drop h TMP_114 c0 e0))))) in (let 
-TMP_116 \def (S i0) in (let TMP_117 \def (minus TMP_116 h) in (let TMP_118 
-\def (drop TMP_117 O e a) in (let TMP_144 \def (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H6: (eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift 
-h (r (Bind b) d0) x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(let 
-TMP_119 \def (Bind b) in (let TMP_120 \def (CHead x0 TMP_119 x1) in (let 
-TMP_123 \def (\lambda (c1: C).(let TMP_121 \def (S i0) in (let TMP_122 \def 
-(minus TMP_121 h) in (drop TMP_122 O c1 a)))) in (let TMP_124 \def (minus i0 
-h) in (let TMP_125 \def (S TMP_124) in (let TMP_128 \def (\lambda (n: 
-nat).(let TMP_126 \def (Bind b) in (let TMP_127 \def (CHead x0 TMP_126 x1) in 
-(drop n O TMP_127 a)))) in (let TMP_129 \def (Bind b) in (let TMP_130 \def 
-(minus i0 h) in (let TMP_131 \def (Bind b) in (let TMP_132 \def 
-(drop_gen_drop TMP_131 c0 a t i0 H1) in (let TMP_133 \def (plus d0 h) in (let 
-TMP_134 \def (le_S_n TMP_133 i0 H5) in (let TMP_135 \def (H a c0 TMP_132 x0 h 
-d0 H8 TMP_134) in (let TMP_136 \def (drop_drop TMP_129 TMP_130 x0 a TMP_135 
-x1) in (let TMP_137 \def (S i0) in (let TMP_138 \def (minus TMP_137 h) in 
-(let TMP_139 \def (plus d0 h) in (let TMP_140 \def (le_S_n TMP_139 i0 H5) in 
-(let TMP_141 \def (le_trans_plus_r d0 h i0 TMP_140) in (let TMP_142 \def 
-(minus_Sn_m i0 h TMP_141) in (let TMP_143 \def (eq_ind nat TMP_125 TMP_128 
-TMP_136 TMP_138 TMP_142) in (eq_ind_r C TMP_120 TMP_123 TMP_143 e 
-H6))))))))))))))))))))))))))) in (let TMP_145 \def (Bind b) in (let TMP_146 
-\def (drop_gen_skip_l c0 e t h d0 TMP_145 H4) in (ex3_2_ind C T TMP_108 
-TMP_112 TMP_115 TMP_118 TMP_144 TMP_146)))))))))))))) in (nat_ind TMP_81 
-TMP_105 TMP_147 d H2 H3)))))))))))) in (let TMP_225 \def (\lambda (f: 
-F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat f) t) 
-a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h 
-d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S i0))).(let 
-TMP_151 \def (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le 
-(plus n h) (S i0)) \to (let TMP_149 \def (S i0) in (let TMP_150 \def (minus 
-TMP_149 h) in (drop TMP_150 O e a)))))) in (let TMP_174 \def (\lambda (H4: 
-(drop h O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S 
-i0))).(let TMP_154 \def (\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) 
-\to ((le (plus O n) (S i0)) \to (let TMP_152 \def (S i0) in (let TMP_153 \def 
-(minus TMP_152 n) in (drop TMP_153 O e a)))))) in (let TMP_167 \def (\lambda 
+C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall 
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d 
+h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: 
+nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: 
+(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d 
+O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda 
+(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) 
+(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: 
+(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O 
+O)).(let H9 \def (eq_ind nat d (\lambda (n0: nat).(le (plus n0 h) (S i0))) H2 
+O H5) in (let H10 \def (eq_ind nat h (\lambda (n0: nat).(le (plus O n0) (S 
+i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) 
+O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 
+a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) 
+c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H7) in (False_ind (drop 
+(minus (S i0) O) O (CSort n) (CSort n)) H11)) a H6) e H3) h H4)))))) 
+(drop_gen_sort n (S i0) O a H0))))) (drop_gen_sort n h d e H1))))))))) 
+(\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to (\forall (e: 
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d h) 
+(S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k: K).(K_ind 
+(\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead c0 k0 t) a) \to 
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d (CHead c0 k0 
+t) e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))) 
+(\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Bind 
+b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: 
+(drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le (plus d h) (S 
+i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Bind b) t) e) \to ((le 
+(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h 
+O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind 
+(\lambda (n: nat).((drop n O (CHead c0 (Bind b) t) e) \to ((le (plus O n) (S 
+i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 
+(Bind b) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 
+(Bind b) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Bind 
+b) i0 c0 a (drop_gen_drop (Bind b) c0 a t i0 H1) t) e (drop_gen_refl (CHead 
+c0 (Bind b) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead 
+c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O 
+e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Bind b) t) e)).(\lambda (H7: 
+(le (plus O (S h0)) (S i0))).(H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e 
+h0 O (drop_gen_drop (Bind b) c0 e t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h 
+H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) 
+e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e 
+a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Bind b) t) e)).(\lambda (H5: 
+(le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: 
+T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t 
+(lift h (r (Bind b) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
+(Bind b) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda 
+(x1: T).(\lambda (H6: (eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t 
+(lift h (r (Bind b) d0) x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 
+x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) 
+h) O c1 a)) (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead 
+x0 (Bind b) x1) a)) (drop_drop (Bind b) (minus i0 h) x0 a (H a c0 
+(drop_gen_drop (Bind b) c0 a t i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) 
+x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus 
+d0 h) i0 H5)))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d 
+H2 H3))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O 
+(CHead c0 (Flat f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H2: (drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le 
+(plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) 
+t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) 
+(\lambda (H4: (drop h O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O 
+h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) 
+\to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda 
 (H6: (drop O O (CHead c0 (Flat f) t) e)).(\lambda (_: (le (plus O O) (S 
-i0))).(let TMP_155 \def (Flat f) in (let TMP_156 \def (CHead c0 TMP_155 t) in 
-(let TMP_159 \def (\lambda (c1: C).(let TMP_157 \def (S i0) in (let TMP_158 
-\def (minus TMP_157 O) in (drop TMP_158 O c1 a)))) in (let TMP_160 \def (Flat 
-f) in (let TMP_161 \def (Flat f) in (let TMP_162 \def (drop_gen_drop TMP_161 
-c0 a t i0 H1) in (let TMP_163 \def (drop_drop TMP_160 i0 c0 a TMP_162 t) in 
-(let TMP_164 \def (Flat f) in (let TMP_165 \def (CHead c0 TMP_164 t) in (let 
-TMP_166 \def (drop_gen_refl TMP_165 e H6) in (eq_ind C TMP_156 TMP_159 
-TMP_163 e TMP_166))))))))))))) in (let TMP_173 \def (\lambda (h0: 
-nat).(\lambda (_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O h0) 
-(S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O 
-(CHead c0 (Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(let 
-TMP_168 \def (Flat f) in (let TMP_169 \def (drop_gen_drop TMP_168 c0 a t i0 
-H1) in (let TMP_170 \def (S h0) in (let TMP_171 \def (Flat f) in (let TMP_172 
-\def (drop_gen_drop TMP_171 c0 e t h0 H6) in (H0 TMP_169 e TMP_170 O TMP_172 
-H7)))))))))) in (nat_ind TMP_154 TMP_167 TMP_173 h H4 H5)))))) in (let 
-TMP_224 \def (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) 
-t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e 
+i0))).(eq_ind C (CHead c0 (Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) 
+O c1 a)) (drop_drop (Flat f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) 
+e (drop_gen_refl (CHead c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda 
+(_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to 
+(drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 
+(Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop 
+(Flat f) c0 a t i0 H1) e (S h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) 
+H7))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 
+(Flat f) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e 
 a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Flat f) t) e)).(\lambda (H5: 
-(le (plus (S d0) h) (S i0))).(let TMP_177 \def (\lambda (e0: C).(\lambda (v: 
-T).(let TMP_175 \def (Flat f) in (let TMP_176 \def (CHead e0 TMP_175 v) in 
-(eq C e TMP_176))))) in (let TMP_181 \def (\lambda (_: C).(\lambda (v: 
-T).(let TMP_178 \def (Flat f) in (let TMP_179 \def (r TMP_178 d0) in (let 
-TMP_180 \def (lift h TMP_179 v) in (eq T t TMP_180)))))) in (let TMP_184 \def 
-(\lambda (e0: C).(\lambda (_: T).(let TMP_182 \def (Flat f) in (let TMP_183 
-\def (r TMP_182 d0) in (drop h TMP_183 c0 e0))))) in (let TMP_185 \def (S i0) 
-in (let TMP_186 \def (minus TMP_185 h) in (let TMP_187 \def (drop TMP_186 O e 
-a) in (let TMP_221 \def (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C 
-e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0) 
-x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(let TMP_188 \def (Flat 
-f) in (let TMP_189 \def (CHead x0 TMP_188 x1) in (let TMP_192 \def (\lambda 
-(c1: C).(let TMP_190 \def (S i0) in (let TMP_191 \def (minus TMP_190 h) in 
-(drop TMP_191 O c1 a)))) in (let TMP_193 \def (S i0) in (let TMP_194 \def 
-(minus TMP_193 h) in (let TMP_195 \def (\lambda (n: nat).(drop n O x0 a)) in 
-(let TMP_196 \def (Flat f) in (let TMP_197 \def (drop_gen_drop TMP_196 c0 a t 
-i0 H1) in (let TMP_198 \def (S d0) in (let TMP_199 \def (H0 TMP_197 x0 h 
-TMP_198 H8 H5) in (let TMP_200 \def (minus i0 h) in (let TMP_201 \def (S 
-TMP_200) in (let TMP_202 \def (plus d0 h) in (let TMP_203 \def (le_S_n 
-TMP_202 i0 H5) in (let TMP_204 \def (le_trans_plus_r d0 h i0 TMP_203) in (let 
-TMP_205 \def (minus_Sn_m i0 h TMP_204) in (let H9 \def (eq_ind_r nat TMP_194 
-TMP_195 TMP_199 TMP_201 TMP_205) in (let TMP_206 \def (minus i0 h) in (let 
-TMP_207 \def (S TMP_206) in (let TMP_210 \def (\lambda (n: nat).(let TMP_208 
-\def (Flat f) in (let TMP_209 \def (CHead x0 TMP_208 x1) in (drop n O TMP_209 
-a)))) in (let TMP_211 \def (Flat f) in (let TMP_212 \def (minus i0 h) in (let 
-TMP_213 \def (drop_drop TMP_211 TMP_212 x0 a H9 x1) in (let TMP_214 \def (S 
-i0) in (let TMP_215 \def (minus TMP_214 h) in (let TMP_216 \def (plus d0 h) 
-in (let TMP_217 \def (le_S_n TMP_216 i0 H5) in (let TMP_218 \def 
-(le_trans_plus_r d0 h i0 TMP_217) in (let TMP_219 \def (minus_Sn_m i0 h 
-TMP_218) in (let TMP_220 \def (eq_ind nat TMP_207 TMP_210 TMP_213 TMP_215 
-TMP_219) in (eq_ind_r C TMP_189 TMP_192 TMP_220 e 
-H6)))))))))))))))))))))))))))))))))))) in (let TMP_222 \def (Flat f) in (let 
-TMP_223 \def (drop_gen_skip_l c0 e t h d0 TMP_222 H4) in (ex3_2_ind C T 
-TMP_177 TMP_181 TMP_184 TMP_187 TMP_221 TMP_223)))))))))))))) in (nat_ind 
-TMP_151 TMP_174 TMP_224 d H2 H3)))))))))))) in (K_ind TMP_78 TMP_148 TMP_225 
-k))))))) in (C_ind TMP_26 TMP_75 TMP_226 c)))))))) in (nat_ind TMP_2 TMP_23 
-TMP_227 i)))).
+(le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: 
+T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t 
+(lift h (r (Flat f) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r 
+(Flat f) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda 
+(x1: T).(\lambda (H6: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t 
+(lift h (r (Flat f) d0) x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 
+x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) 
+h) O c1 a)) (let H9 \def (eq_ind_r nat (minus (S i0) h) (\lambda (n: 
+nat).(drop n O x0 a)) (H0 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) 
+H8 H5) (S (minus i0 h)) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n 
+(plus d0 h) i0 H5)))) in (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop 
+n O (CHead x0 (Flat f) x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) 
+(minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 
+h) i0 H5))))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 
+H3))))))))) k)))) c))))) i).
 
 theorem drop_conf_rev:
  \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to 
 (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: 
 C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1)))))))))
 \def
- \lambda (j: nat).(let TMP_3 \def (\lambda (n: nat).(\forall (e1: C).(\forall 
-(e2: C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O 
-c2 e2) \to (let TMP_1 \def (\lambda (c1: C).(drop n O c1 c2)) in (let TMP_2 
-\def (\lambda (c1: C).(drop i n c1 e1)) in (ex2 C TMP_1 TMP_2)))))))))) in 
-(let TMP_9 \def (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1 
+ \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2: 
+C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) 
+\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1 
+e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1 
 e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let 
-TMP_4 \def (\lambda (c: C).(drop i O c2 c)) in (let TMP_5 \def (drop_gen_refl 
-e1 e2 H) in (let H1 \def (eq_ind_r C e2 TMP_4 H0 e1 TMP_5) in (let TMP_6 \def 
-(\lambda (c1: C).(drop O O c1 c2)) in (let TMP_7 \def (\lambda (c1: C).(drop 
-i O c1 e1)) in (let TMP_8 \def (drop_refl c2) in (ex_intro2 C TMP_6 TMP_7 c2 
-TMP_8 H1))))))))))))) in (let TMP_108 \def (\lambda (j0: nat).(\lambda (IHj: 
-((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2: 
+H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl 
+e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1: 
+C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda 
+(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2: 
 C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O 
-c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(let 
-TMP_14 \def (\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to 
-(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (let TMP_11 \def 
-(\lambda (c1: C).(let TMP_10 \def (S j0) in (drop TMP_10 O c1 c2))) in (let 
-TMP_13 \def (\lambda (c1: C).(let TMP_12 \def (S j0) in (drop i TMP_12 c1 
-c))) in (ex2 C TMP_11 TMP_13))))))))) in (let TMP_39 \def (\lambda (n: 
+c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind 
+(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2: 
+C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S 
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n: 
 nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda 
-(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let TMP_15 \def 
-(CSort n) in (let TMP_16 \def (eq C e2 TMP_15) in (let TMP_17 \def (S j0) in 
-(let TMP_18 \def (eq nat TMP_17 O) in (let TMP_19 \def (eq nat O O) in (let 
-TMP_21 \def (\lambda (c1: C).(let TMP_20 \def (S j0) in (drop TMP_20 O c1 
-c2))) in (let TMP_24 \def (\lambda (c1: C).(let TMP_22 \def (S j0) in (let 
-TMP_23 \def (CSort n) in (drop i TMP_22 c1 TMP_23)))) in (let TMP_25 \def 
-(ex2 C TMP_21 TMP_24) in (let TMP_36 \def (\lambda (H1: (eq C e2 (CSort 
-n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O O)).(let TMP_26 
-\def (\lambda (c: C).(drop i O c2 c)) in (let TMP_27 \def (CSort n) in (let 
-H4 \def (eq_ind C e2 TMP_26 H0 TMP_27 H1) in (let TMP_28 \def (S j0) in (let 
-TMP_29 \def (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) 
-\Rightarrow True])) in (let H5 \def (eq_ind nat TMP_28 TMP_29 I O H2) in (let 
-TMP_31 \def (\lambda (c1: C).(let TMP_30 \def (S j0) in (drop TMP_30 O c1 
-c2))) in (let TMP_34 \def (\lambda (c1: C).(let TMP_32 \def (S j0) in (let 
-TMP_33 \def (CSort n) in (drop i TMP_32 c1 TMP_33)))) in (let TMP_35 \def 
-(ex2 C TMP_31 TMP_34) in (False_ind TMP_35 H5))))))))))))) in (let TMP_37 
-\def (S j0) in (let TMP_38 \def (drop_gen_sort n TMP_37 O e2 H) in (and3_ind 
-TMP_16 TMP_18 TMP_19 TMP_25 TMP_36 TMP_38)))))))))))))))))) in (let TMP_107 
-\def (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop (S j0) O e2 
-e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to (ex2 C 
+(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2 
+(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S 
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1: 
+(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O 
+O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n) 
+H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee with [O 
+\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (ex2 C 
+(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 
+(CSort n)))) H5)))))) (drop_gen_sort n (S j0) O e2 H)))))))) (\lambda (e2: 
+C).(\lambda (IHe1: ((\forall (e3: C).((drop (S j0) O e2 e3) \to (\forall (c2: 
+C).(\forall (i: nat).((drop i O c2 e3) \to (ex2 C (\lambda (c1: C).(drop (S 
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)))))))))).(\lambda (k: 
+K).(\lambda (t: T).(\lambda (e3: C).(\lambda (H: (drop (S j0) O (CHead e2 k 
+t) e3)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 
+e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0) O e2 e3) \to (ex2 C (\lambda 
+(c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 
+k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) j0) O e2 e3)).(let 
+H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda 
+(c1: C).(drop j0 O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e2)) (ex2 C 
 (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 
-e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda (H: 
-(drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i: 
-nat).(\lambda (H0: (drop i O c2 e3)).(let TMP_45 \def (\lambda (k0: K).((drop 
-(r k0 j0) O e2 e3) \to (let TMP_41 \def (\lambda (c1: C).(let TMP_40 \def (S 
-j0) in (drop TMP_40 O c1 c2))) in (let TMP_44 \def (\lambda (c1: C).(let 
-TMP_42 \def (S j0) in (let TMP_43 \def (CHead e2 k0 t) in (drop i TMP_42 c1 
-TMP_43)))) in (ex2 C TMP_41 TMP_44))))) in (let TMP_74 \def (\lambda (b: 
-B).(\lambda (H1: (drop (r (Bind b) j0) O e2 e3)).(let H_x \def (IHj e2 e3 H1 
-c2 i H0) in (let H2 \def H_x in (let TMP_46 \def (\lambda (c1: C).(drop j0 O 
-c1 c2)) in (let TMP_47 \def (\lambda (c1: C).(drop i j0 c1 e2)) in (let 
-TMP_49 \def (\lambda (c1: C).(let TMP_48 \def (S j0) in (drop TMP_48 O c1 
-c2))) in (let TMP_53 \def (\lambda (c1: C).(let TMP_50 \def (S j0) in (let 
-TMP_51 \def (Bind b) in (let TMP_52 \def (CHead e2 TMP_51 t) in (drop i 
-TMP_50 c1 TMP_52))))) in (let TMP_54 \def (ex2 C TMP_49 TMP_53) in (let 
-TMP_73 \def (\lambda (x: C).(\lambda (H3: (drop j0 O x c2)).(\lambda (H4: 
-(drop i j0 x e2)).(let TMP_56 \def (\lambda (c1: C).(let TMP_55 \def (S j0) 
-in (drop TMP_55 O c1 c2))) in (let TMP_60 \def (\lambda (c1: C).(let TMP_57 
-\def (S j0) in (let TMP_58 \def (Bind b) in (let TMP_59 \def (CHead e2 TMP_58 
-t) in (drop i TMP_57 c1 TMP_59))))) in (let TMP_61 \def (Bind b) in (let 
-TMP_62 \def (Bind b) in (let TMP_63 \def (r TMP_62 j0) in (let TMP_64 \def 
-(lift i TMP_63 t) in (let TMP_65 \def (CHead x TMP_61 TMP_64) in (let TMP_66 
-\def (Bind b) in (let TMP_67 \def (Bind b) in (let TMP_68 \def (r TMP_67 j0) 
-in (let TMP_69 \def (lift i TMP_68 t) in (let TMP_70 \def (drop_drop TMP_66 
-j0 x c2 H3 TMP_69) in (let TMP_71 \def (Bind b) in (let TMP_72 \def 
-(drop_skip TMP_71 i j0 x e2 H4 t) in (ex_intro2 C TMP_56 TMP_60 TMP_65 TMP_70 
-TMP_72)))))))))))))))))) in (ex2_ind C TMP_46 TMP_47 TMP_54 TMP_73 
-H2))))))))))) in (let TMP_105 \def (\lambda (f: F).(\lambda (H1: (drop (r 
-(Flat f) j0) O e2 e3)).(let H_x \def (IHe1 e3 H1 c2 i H0) in (let H2 \def H_x 
-in (let TMP_76 \def (\lambda (c1: C).(let TMP_75 \def (S j0) in (drop TMP_75 
-O c1 c2))) in (let TMP_78 \def (\lambda (c1: C).(let TMP_77 \def (S j0) in 
-(drop i TMP_77 c1 e2))) in (let TMP_80 \def (\lambda (c1: C).(let TMP_79 \def 
-(S j0) in (drop TMP_79 O c1 c2))) in (let TMP_84 \def (\lambda (c1: C).(let 
-TMP_81 \def (S j0) in (let TMP_82 \def (Flat f) in (let TMP_83 \def (CHead e2 
-TMP_82 t) in (drop i TMP_81 c1 TMP_83))))) in (let TMP_85 \def (ex2 C TMP_80 
-TMP_84) in (let TMP_104 \def (\lambda (x: C).(\lambda (H3: (drop (S j0) O x 
-c2)).(\lambda (H4: (drop i (S j0) x e2)).(let TMP_87 \def (\lambda (c1: 
-C).(let TMP_86 \def (S j0) in (drop TMP_86 O c1 c2))) in (let TMP_91 \def 
-(\lambda (c1: C).(let TMP_88 \def (S j0) in (let TMP_89 \def (Flat f) in (let 
-TMP_90 \def (CHead e2 TMP_89 t) in (drop i TMP_88 c1 TMP_90))))) in (let 
-TMP_92 \def (Flat f) in (let TMP_93 \def (Flat f) in (let TMP_94 \def (r 
-TMP_93 j0) in (let TMP_95 \def (lift i TMP_94 t) in (let TMP_96 \def (CHead x 
-TMP_92 TMP_95) in (let TMP_97 \def (Flat f) in (let TMP_98 \def (Flat f) in 
-(let TMP_99 \def (r TMP_98 j0) in (let TMP_100 \def (lift i TMP_99 t) in (let 
-TMP_101 \def (drop_drop TMP_97 j0 x c2 H3 TMP_100) in (let TMP_102 \def (Flat 
-f) in (let TMP_103 \def (drop_skip TMP_102 i j0 x e2 H4 t) in (ex_intro2 C 
-TMP_87 TMP_91 TMP_96 TMP_101 TMP_103)))))))))))))))))) in (ex2_ind C TMP_76 
-TMP_78 TMP_85 TMP_104 H2))))))))))) in (let TMP_106 \def (drop_gen_drop k e2 
-e3 t j0 H) in (K_ind TMP_45 TMP_74 TMP_105 k TMP_106)))))))))))))) in (C_ind 
-TMP_14 TMP_39 TMP_107 e1))))))) in (nat_ind TMP_3 TMP_9 TMP_108 j)))).
+(CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda (H3: (drop j0 O x 
+c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S 
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t))) 
+(CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind b) j0 x c2 H3 
+(lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4 t))))) H2))))) 
+(\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2 e3)).(let H_x \def 
+(IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop 
+(S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)) (ex2 C (\lambda (c1: 
+C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat 
+f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0) O x c2)).(\lambda (H4: 
+(drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) 
+(\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t))) (CHead x (Flat f) 
+(lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0 x c2 H3 (lift i (r (Flat 
+f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t))))) H2))))) k (drop_gen_drop k 
+e2 e3 t j0 H))))))))))) e1)))) j).
 
 theorem drop_trans_le:
  \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall 
@@ -688,365 +399,196 @@ theorem drop_trans_le:
 c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h 
 (minus d i) e1 e2)))))))))))
 \def
- \lambda (i: nat).(let TMP_4 \def (\lambda (n: nat).(\forall (d: nat).((le n 
-d) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) 
-\to (\forall (e2: C).((drop n O c2 e2) \to (let TMP_1 \def (\lambda (e1: 
-C).(drop n O c1 e1)) in (let TMP_3 \def (\lambda (e1: C).(let TMP_2 \def 
-(minus d n) in (drop h TMP_2 e1 e2))) in (ex2 C TMP_1 TMP_3)))))))))))) in 
-(let TMP_16 \def (\lambda (d: nat).(\lambda (_: (le O d)).(\lambda (c1: 
-C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 
-c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(let TMP_5 \def 
-(\lambda (c: C).(drop h d c1 c)) in (let TMP_6 \def (drop_gen_refl c2 e2 H1) 
-in (let H2 \def (eq_ind C c2 TMP_5 H0 e2 TMP_6) in (let TMP_9 \def (\lambda 
-(n: nat).(let TMP_7 \def (\lambda (e1: C).(drop O O c1 e1)) in (let TMP_8 
-\def (\lambda (e1: C).(drop h n e1 e2)) in (ex2 C TMP_7 TMP_8)))) in (let 
-TMP_10 \def (\lambda (e1: C).(drop O O c1 e1)) in (let TMP_11 \def (\lambda 
-(e1: C).(drop h d e1 e2)) in (let TMP_12 \def (drop_refl c1) in (let TMP_13 
-\def (ex_intro2 C TMP_10 TMP_11 c1 TMP_12 H2) in (let TMP_14 \def (minus d O) 
-in (let TMP_15 \def (minus_n_O d) in (eq_ind nat d TMP_9 TMP_13 TMP_14 
-TMP_15))))))))))))))))))) in (let TMP_271 \def (\lambda (i0: nat).(\lambda 
-(IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1: C).(\forall (c2: 
-C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i0 O c2 
-e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda (e1: C).(drop h 
-(minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(let TMP_22 \def (\lambda 
-(n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: 
-nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to (let 
-TMP_18 \def (\lambda (e1: C).(let TMP_17 \def (S i0) in (drop TMP_17 O c1 
-e1))) in (let TMP_21 \def (\lambda (e1: C).(let TMP_19 \def (S i0) in (let 
-TMP_20 \def (minus n TMP_19) in (drop h TMP_20 e1 e2)))) in (ex2 C TMP_18 
-TMP_21))))))))))) in (let TMP_42 \def (\lambda (H: (le (S i0) O)).(\lambda 
-(c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h O c1 
-c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(let TMP_24 \def 
-(\lambda (n: nat).(let TMP_23 \def (S n) in (eq nat O TMP_23))) in (let 
-TMP_25 \def (\lambda (n: nat).(le i0 n)) in (let TMP_27 \def (\lambda (e1: 
-C).(let TMP_26 \def (S i0) in (drop TMP_26 O c1 e1))) in (let TMP_30 \def 
-(\lambda (e1: C).(let TMP_28 \def (S i0) in (let TMP_29 \def (minus O TMP_28) 
-in (drop h TMP_29 e1 e2)))) in (let TMP_31 \def (ex2 C TMP_27 TMP_30) in (let 
-TMP_40 \def (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_: 
-(le i0 x)).(let TMP_32 \def (\lambda (ee: nat).(match ee with [O \Rightarrow 
-True | (S _) \Rightarrow False])) in (let TMP_33 \def (S x) in (let H4 \def 
-(eq_ind nat O TMP_32 I TMP_33 H2) in (let TMP_35 \def (\lambda (e1: C).(let 
-TMP_34 \def (S i0) in (drop TMP_34 O c1 e1))) in (let TMP_38 \def (\lambda 
-(e1: C).(let TMP_36 \def (S i0) in (let TMP_37 \def (minus O TMP_36) in (drop 
-h TMP_37 e1 e2)))) in (let TMP_39 \def (ex2 C TMP_35 TMP_38) in (False_ind 
-TMP_39 H4)))))))))) in (let TMP_41 \def (le_gen_S i0 O H) in (ex2_ind nat 
-TMP_24 TMP_25 TMP_31 TMP_40 TMP_41))))))))))))))) in (let TMP_270 \def 
-(\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall (c1: 
-C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall (e2: 
-C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) 
-(\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda (H: (le 
-(S i0) (S d0))).(\lambda (c1: C).(let TMP_49 \def (\lambda (c: C).(\forall 
-(c2: C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop 
-(S i0) O c2 e2) \to (let TMP_44 \def (\lambda (e1: C).(let TMP_43 \def (S i0) 
-in (drop TMP_43 O c e1))) in (let TMP_48 \def (\lambda (e1: C).(let TMP_45 
-\def (S d0) in (let TMP_46 \def (S i0) in (let TMP_47 \def (minus TMP_45 
-TMP_46) in (drop h TMP_47 e1 e2))))) in (ex2 C TMP_44 TMP_48))))))))) in (let 
-TMP_106 \def (\lambda (n: nat).(\lambda (c2: C).(\lambda (h: nat).(\lambda 
-(H0: (drop h (S d0) (CSort n) c2)).(\lambda (e2: C).(\lambda (H1: (drop (S 
-i0) O c2 e2)).(let TMP_50 \def (CSort n) in (let TMP_51 \def (eq C c2 TMP_50) 
-in (let TMP_52 \def (eq nat h O) in (let TMP_53 \def (S d0) in (let TMP_54 
-\def (eq nat TMP_53 O) in (let TMP_57 \def (\lambda (e1: C).(let TMP_55 \def 
-(S i0) in (let TMP_56 \def (CSort n) in (drop TMP_55 O TMP_56 e1)))) in (let 
-TMP_61 \def (\lambda (e1: C).(let TMP_58 \def (S d0) in (let TMP_59 \def (S 
-i0) in (let TMP_60 \def (minus TMP_58 TMP_59) in (drop h TMP_60 e1 e2))))) in 
-(let TMP_62 \def (ex2 C TMP_57 TMP_61) in (let TMP_103 \def (\lambda (H2: (eq 
-C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: (eq nat (S d0) 
-O)).(let TMP_64 \def (\lambda (c: C).(let TMP_63 \def (S i0) in (drop TMP_63 
-O c e2))) in (let TMP_65 \def (CSort n) in (let H5 \def (eq_ind C c2 TMP_64 
-H1 TMP_65 H2) in (let TMP_66 \def (CSort n) in (let TMP_67 \def (eq C e2 
-TMP_66) in (let TMP_68 \def (S i0) in (let TMP_69 \def (eq nat TMP_68 O) in 
-(let TMP_70 \def (eq nat O O) in (let TMP_73 \def (\lambda (e1: C).(let 
-TMP_71 \def (S i0) in (let TMP_72 \def (CSort n) in (drop TMP_71 O TMP_72 
-e1)))) in (let TMP_77 \def (\lambda (e1: C).(let TMP_74 \def (S d0) in (let 
-TMP_75 \def (S i0) in (let TMP_76 \def (minus TMP_74 TMP_75) in (drop h 
-TMP_76 e1 e2))))) in (let TMP_78 \def (ex2 C TMP_73 TMP_77) in (let TMP_100 
-\def (\lambda (H6: (eq C e2 (CSort n))).(\lambda (H7: (eq nat (S i0) 
-O)).(\lambda (_: (eq nat O O)).(let TMP_79 \def (CSort n) in (let TMP_87 \def 
-(\lambda (c: C).(let TMP_82 \def (\lambda (e1: C).(let TMP_80 \def (S i0) in 
-(let TMP_81 \def (CSort n) in (drop TMP_80 O TMP_81 e1)))) in (let TMP_86 
-\def (\lambda (e1: C).(let TMP_83 \def (S d0) in (let TMP_84 \def (S i0) in 
-(let TMP_85 \def (minus TMP_83 TMP_84) in (drop h TMP_85 e1 c))))) in (ex2 C 
-TMP_82 TMP_86)))) in (let TMP_88 \def (S i0) in (let TMP_89 \def (\lambda 
-(ee: nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow True])) in 
-(let H9 \def (eq_ind nat TMP_88 TMP_89 I O H7) in (let TMP_92 \def (\lambda 
-(e1: C).(let TMP_90 \def (S i0) in (let TMP_91 \def (CSort n) in (drop TMP_90 
-O TMP_91 e1)))) in (let TMP_97 \def (\lambda (e1: C).(let TMP_93 \def (S d0) 
-in (let TMP_94 \def (S i0) in (let TMP_95 \def (minus TMP_93 TMP_94) in (let 
-TMP_96 \def (CSort n) in (drop h TMP_95 e1 TMP_96)))))) in (let TMP_98 \def 
-(ex2 C TMP_92 TMP_97) in (let TMP_99 \def (False_ind TMP_98 H9) in (eq_ind_r 
-C TMP_79 TMP_87 TMP_99 e2 H6))))))))))))) in (let TMP_101 \def (S i0) in (let 
-TMP_102 \def (drop_gen_sort n TMP_101 O e2 H5) in (and3_ind TMP_67 TMP_69 
-TMP_70 TMP_78 TMP_100 TMP_102)))))))))))))))))) in (let TMP_104 \def (S d0) 
-in (let TMP_105 \def (drop_gen_sort n h TMP_104 c2 H0) in (and3_ind TMP_51 
-TMP_52 TMP_54 TMP_62 TMP_103 TMP_105)))))))))))))))))) in (let TMP_269 \def 
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to 
+(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to 
+(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1 
+e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d: 
+nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: 
+nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O 
+c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2 
+(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda 
+(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C 
+(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1 
+(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0: 
+nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1: 
+C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: 
+C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda 
+(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind 
+(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2: 
+C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O 
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: 
+C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0) 
+O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h 
+O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(ex2_ind nat 
+(\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le i0 n)) (ex2 C 
+(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S 
+i0)) e1 e2))) (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_: 
+(le i0 x)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O 
+\Rightarrow True | (S _) \Rightarrow False])) I (S x) H2) in (False_ind (ex2 
+C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O 
+(S i0)) e1 e2))) H4))))) (le_gen_S i0 O H))))))))) (\lambda (d0: 
+nat).(\lambda (_: (((le (S i0) d0) \to (\forall (c1: C).(\forall (c2: 
+C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall (e2: C).((drop (S i0) O 
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: 
+C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda (H: (le (S i0) (S 
+d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: 
+nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to 
+(ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: C).(drop h (minus 
+(S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (h: 
+nat).(\lambda (H0: (drop h (S d0) (CSort n) c2)).(\lambda (e2: C).(\lambda 
+(H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 (CSort n)) (eq nat h O) (eq 
+nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda 
+(e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H2: (eq C c2 (CSort 
+n))).(\lambda (_: (eq nat h O)).(\lambda (_: (eq nat (S d0) O)).(let H5 \def 
+(eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CSort n) H2) in 
+(and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (ex2 C (\lambda 
+(e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) 
+(S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort n))).(\lambda (H7: (eq nat (S 
+i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 
+C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h 
+(minus (S d0) (S i0)) e1 c)))) (let H9 \def (eq_ind nat (S i0) (\lambda (ee: 
+nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) 
+in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda 
+(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6)))) 
+(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0)))))))) 
 (\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h 
 (S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda 
 (e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) 
-e1 e2)))))))))).(\lambda (k: K).(let TMP_114 \def (\lambda (k0: K).(\forall 
-(t: T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) 
-\to (\forall (e2: C).((drop (S i0) O c3 e2) \to (let TMP_109 \def (\lambda 
-(e1: C).(let TMP_107 \def (S i0) in (let TMP_108 \def (CHead c2 k0 t) in 
-(drop TMP_107 O TMP_108 e1)))) in (let TMP_113 \def (\lambda (e1: C).(let 
-TMP_110 \def (S d0) in (let TMP_111 \def (S i0) in (let TMP_112 \def (minus 
-TMP_110 TMP_111) in (drop h TMP_112 e1 e2))))) in (ex2 C TMP_109 
-TMP_113)))))))))) in (let TMP_190 \def (\lambda (b: B).(\lambda (t: 
-T).(\lambda (c3: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 
-(Bind b) t) c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(let 
-TMP_117 \def (\lambda (e: C).(\lambda (v: T).(let TMP_115 \def (Bind b) in 
-(let TMP_116 \def (CHead e TMP_115 v) in (eq C c3 TMP_116))))) in (let 
-TMP_121 \def (\lambda (_: C).(\lambda (v: T).(let TMP_118 \def (Bind b) in 
-(let TMP_119 \def (r TMP_118 d0) in (let TMP_120 \def (lift h TMP_119 v) in 
-(eq T t TMP_120)))))) in (let TMP_124 \def (\lambda (e: C).(\lambda (_: 
-T).(let TMP_122 \def (Bind b) in (let TMP_123 \def (r TMP_122 d0) in (drop h 
-TMP_123 c2 e))))) in (let TMP_128 \def (\lambda (e1: C).(let TMP_125 \def (S 
-i0) in (let TMP_126 \def (Bind b) in (let TMP_127 \def (CHead c2 TMP_126 t) 
-in (drop TMP_125 O TMP_127 e1))))) in (let TMP_132 \def (\lambda (e1: C).(let 
-TMP_129 \def (S d0) in (let TMP_130 \def (S i0) in (let TMP_131 \def (minus 
-TMP_129 TMP_130) in (drop h TMP_131 e1 e2))))) in (let TMP_133 \def (ex2 C 
-TMP_128 TMP_132) in (let TMP_187 \def (\lambda (x0: C).(\lambda (x1: 
-T).(\lambda (H2: (eq C c3 (CHead x0 (Bind b) x1))).(\lambda (H3: (eq T t 
-(lift h (r (Bind b) d0) x1))).(\lambda (H4: (drop h (r (Bind b) d0) c2 
-x0)).(let TMP_135 \def (\lambda (c: C).(let TMP_134 \def (S i0) in (drop 
-TMP_134 O c e2))) in (let TMP_136 \def (Bind b) in (let TMP_137 \def (CHead 
-x0 TMP_136 x1) in (let H5 \def (eq_ind C c3 TMP_135 H1 TMP_137 H2) in (let 
-TMP_138 \def (Bind b) in (let TMP_139 \def (r TMP_138 d0) in (let TMP_140 
-\def (lift h TMP_139 x1) in (let TMP_149 \def (\lambda (t0: T).(let TMP_144 
-\def (\lambda (e1: C).(let TMP_141 \def (S i0) in (let TMP_142 \def (Bind b) 
-in (let TMP_143 \def (CHead c2 TMP_142 t0) in (drop TMP_141 O TMP_143 e1))))) 
-in (let TMP_148 \def (\lambda (e1: C).(let TMP_145 \def (S d0) in (let 
-TMP_146 \def (S i0) in (let TMP_147 \def (minus TMP_145 TMP_146) in (drop h 
-TMP_147 e1 e2))))) in (ex2 C TMP_144 TMP_148)))) in (let TMP_150 \def 
-(\lambda (e1: C).(drop i0 O c2 e1)) in (let TMP_152 \def (\lambda (e1: 
-C).(let TMP_151 \def (minus d0 i0) in (drop h TMP_151 e1 e2))) in (let 
-TMP_159 \def (\lambda (e1: C).(let TMP_153 \def (S i0) in (let TMP_154 \def 
-(Bind b) in (let TMP_155 \def (Bind b) in (let TMP_156 \def (r TMP_155 d0) in 
-(let TMP_157 \def (lift h TMP_156 x1) in (let TMP_158 \def (CHead c2 TMP_154 
-TMP_157) in (drop TMP_153 O TMP_158 e1)))))))) in (let TMP_163 \def (\lambda 
-(e1: C).(let TMP_160 \def (S d0) in (let TMP_161 \def (S i0) in (let TMP_162 
-\def (minus TMP_160 TMP_161) in (drop h TMP_162 e1 e2))))) in (let TMP_164 
-\def (ex2 C TMP_159 TMP_163) in (let TMP_181 \def (\lambda (x: C).(\lambda 
-(H6: (drop i0 O c2 x)).(\lambda (H7: (drop h (minus d0 i0) x e2)).(let 
-TMP_171 \def (\lambda (e1: C).(let TMP_165 \def (S i0) in (let TMP_166 \def 
-(Bind b) in (let TMP_167 \def (Bind b) in (let TMP_168 \def (r TMP_167 d0) in 
-(let TMP_169 \def (lift h TMP_168 x1) in (let TMP_170 \def (CHead c2 TMP_166 
-TMP_169) in (drop TMP_165 O TMP_170 e1)))))))) in (let TMP_175 \def (\lambda 
-(e1: C).(let TMP_172 \def (S d0) in (let TMP_173 \def (S i0) in (let TMP_174 
-\def (minus TMP_172 TMP_173) in (drop h TMP_174 e1 e2))))) in (let TMP_176 
-\def (Bind b) in (let TMP_177 \def (Bind b) in (let TMP_178 \def (r TMP_177 
-d0) in (let TMP_179 \def (lift h TMP_178 x1) in (let TMP_180 \def (drop_drop 
-TMP_176 i0 c2 x H6 TMP_179) in (ex_intro2 C TMP_171 TMP_175 x TMP_180 
-H7))))))))))) in (let TMP_182 \def (le_S_n i0 d0 H) in (let TMP_183 \def 
-(Bind b) in (let TMP_184 \def (drop_gen_drop TMP_183 x0 e2 x1 i0 H5) in (let 
-TMP_185 \def (IHi d0 TMP_182 c2 x0 h H4 e2 TMP_184) in (let TMP_186 \def 
-(ex2_ind C TMP_150 TMP_152 TMP_164 TMP_181 TMP_185) in (eq_ind_r T TMP_140 
-TMP_149 TMP_186 t H3))))))))))))))))))))))))) in (let TMP_188 \def (Bind b) 
-in (let TMP_189 \def (drop_gen_skip_l c2 c3 t h d0 TMP_188 H0) in (ex3_2_ind 
-C T TMP_117 TMP_121 TMP_124 TMP_133 TMP_187 TMP_189))))))))))))))))) in (let 
-TMP_268 \def (\lambda (f: F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: 
-nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t) c3)).(\lambda (e2: 
-C).(\lambda (H1: (drop (S i0) O c3 e2)).(let TMP_193 \def (\lambda (e: 
-C).(\lambda (v: T).(let TMP_191 \def (Flat f) in (let TMP_192 \def (CHead e 
-TMP_191 v) in (eq C c3 TMP_192))))) in (let TMP_197 \def (\lambda (_: 
-C).(\lambda (v: T).(let TMP_194 \def (Flat f) in (let TMP_195 \def (r TMP_194 
-d0) in (let TMP_196 \def (lift h TMP_195 v) in (eq T t TMP_196)))))) in (let 
-TMP_200 \def (\lambda (e: C).(\lambda (_: T).(let TMP_198 \def (Flat f) in 
-(let TMP_199 \def (r TMP_198 d0) in (drop h TMP_199 c2 e))))) in (let TMP_204 
-\def (\lambda (e1: C).(let TMP_201 \def (S i0) in (let TMP_202 \def (Flat f) 
-in (let TMP_203 \def (CHead c2 TMP_202 t) in (drop TMP_201 O TMP_203 e1))))) 
-in (let TMP_208 \def (\lambda (e1: C).(let TMP_205 \def (S d0) in (let 
-TMP_206 \def (S i0) in (let TMP_207 \def (minus TMP_205 TMP_206) in (drop h 
-TMP_207 e1 e2))))) in (let TMP_209 \def (ex2 C TMP_204 TMP_208) in (let 
-TMP_265 \def (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead 
-x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0) 
-x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let TMP_211 \def 
-(\lambda (c: C).(let TMP_210 \def (S i0) in (drop TMP_210 O c e2))) in (let 
-TMP_212 \def (Flat f) in (let TMP_213 \def (CHead x0 TMP_212 x1) in (let H5 
-\def (eq_ind C c3 TMP_211 H1 TMP_213 H2) in (let TMP_214 \def (Flat f) in 
-(let TMP_215 \def (r TMP_214 d0) in (let TMP_216 \def (lift h TMP_215 x1) in 
-(let TMP_225 \def (\lambda (t0: T).(let TMP_220 \def (\lambda (e1: C).(let 
-TMP_217 \def (S i0) in (let TMP_218 \def (Flat f) in (let TMP_219 \def (CHead 
-c2 TMP_218 t0) in (drop TMP_217 O TMP_219 e1))))) in (let TMP_224 \def 
-(\lambda (e1: C).(let TMP_221 \def (S d0) in (let TMP_222 \def (S i0) in (let 
-TMP_223 \def (minus TMP_221 TMP_222) in (drop h TMP_223 e1 e2))))) in (ex2 C 
-TMP_220 TMP_224)))) in (let TMP_227 \def (\lambda (e1: C).(let TMP_226 \def 
-(S i0) in (drop TMP_226 O c2 e1))) in (let TMP_231 \def (\lambda (e1: C).(let 
-TMP_228 \def (S d0) in (let TMP_229 \def (S i0) in (let TMP_230 \def (minus 
-TMP_228 TMP_229) in (drop h TMP_230 e1 e2))))) in (let TMP_238 \def (\lambda 
-(e1: C).(let TMP_232 \def (S i0) in (let TMP_233 \def (Flat f) in (let 
-TMP_234 \def (Flat f) in (let TMP_235 \def (r TMP_234 d0) in (let TMP_236 
-\def (lift h TMP_235 x1) in (let TMP_237 \def (CHead c2 TMP_233 TMP_236) in 
-(drop TMP_232 O TMP_237 e1)))))))) in (let TMP_242 \def (\lambda (e1: C).(let 
-TMP_239 \def (S d0) in (let TMP_240 \def (S i0) in (let TMP_241 \def (minus 
-TMP_239 TMP_240) in (drop h TMP_241 e1 e2))))) in (let TMP_243 \def (ex2 C 
-TMP_238 TMP_242) in (let TMP_260 \def (\lambda (x: C).(\lambda (H6: (drop (S 
-i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S i0)) x e2)).(let TMP_250 
-\def (\lambda (e1: C).(let TMP_244 \def (S i0) in (let TMP_245 \def (Flat f) 
-in (let TMP_246 \def (Flat f) in (let TMP_247 \def (r TMP_246 d0) in (let 
-TMP_248 \def (lift h TMP_247 x1) in (let TMP_249 \def (CHead c2 TMP_245 
-TMP_248) in (drop TMP_244 O TMP_249 e1)))))))) in (let TMP_254 \def (\lambda 
-(e1: C).(let TMP_251 \def (S d0) in (let TMP_252 \def (S i0) in (let TMP_253 
-\def (minus TMP_251 TMP_252) in (drop h TMP_253 e1 e2))))) in (let TMP_255 
-\def (Flat f) in (let TMP_256 \def (Flat f) in (let TMP_257 \def (r TMP_256 
-d0) in (let TMP_258 \def (lift h TMP_257 x1) in (let TMP_259 \def (drop_drop 
-TMP_255 i0 c2 x H6 TMP_258) in (ex_intro2 C TMP_250 TMP_254 x TMP_259 
-H7))))))))))) in (let TMP_261 \def (Flat f) in (let TMP_262 \def 
-(drop_gen_drop TMP_261 x0 e2 x1 i0 H5) in (let TMP_263 \def (IHc x0 h H4 e2 
-TMP_262) in (let TMP_264 \def (ex2_ind C TMP_227 TMP_231 TMP_243 TMP_260 
-TMP_263) in (eq_ind_r T TMP_216 TMP_225 TMP_264 t H3)))))))))))))))))))))))) 
-in (let TMP_266 \def (Flat f) in (let TMP_267 \def (drop_gen_skip_l c2 c3 t h 
-d0 TMP_266 H0) in (ex3_2_ind C T TMP_193 TMP_197 TMP_200 TMP_209 TMP_265 
-TMP_267))))))))))))))))) in (K_ind TMP_114 TMP_190 TMP_268 k))))))) in (C_ind 
-TMP_49 TMP_106 TMP_269 c1)))))))) in (nat_ind TMP_22 TMP_42 TMP_270 d))))))) 
-in (nat_ind TMP_4 TMP_16 TMP_271 i)))).
+e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: 
+T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to 
+(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S 
+i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 
+e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: 
+nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2: 
+C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: 
+C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda 
+(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_: 
+T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O 
+(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 
+e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 
+(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda 
+(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: 
+C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h 
+(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O 
+(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 
+e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop 
+h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 
+(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S 
+d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda 
+(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) 
+O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop 
+h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r 
+(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2 
+(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 
+t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3: 
+C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t) 
+c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T 
+(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_: 
+C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e: 
+C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1: 
+C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus 
+(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C 
+c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0) 
+x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3 
+(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in 
+(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: 
+C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus 
+(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1)) 
+(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1: 
+C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) 
+(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: 
+C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S 
+i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) 
+(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S 
+i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1)) 
+H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3))))))) 
+(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
 
 theorem drop_trans_ge:
  \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: 
 nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 
 e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
 \def
- \lambda (i: nat).(let TMP_2 \def (\lambda (n: nat).(\forall (c1: C).(\forall 
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: 
+C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2)))))))))) 
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: 
+nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O 
+c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) 
+O c1 c)) (let H_y \def (le_n_O_eq d H1) in (let H2 \def (eq_ind_r nat d 
+(\lambda (n: nat).(drop h n c1 c2)) H O H_y) in H2)) e2 (drop_gen_refl c2 e2 
+H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall 
 (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall 
-(e2: C).((drop n O c2 e2) \to ((le d n) \to (let TMP_1 \def (plus n h) in 
-(drop TMP_1 O c1 e2))))))))))) in (let TMP_7 \def (\lambda (c1: C).(\lambda 
-(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 
-c2)).(\lambda (e2: C).(\lambda (H0: (drop O O c2 e2)).(\lambda (H1: (le d 
-O)).(let TMP_4 \def (\lambda (c: C).(let TMP_3 \def (plus O h) in (drop TMP_3 
-O c1 c))) in (let H_y \def (le_n_O_eq d H1) in (let TMP_5 \def (\lambda (n: 
-nat).(drop h n c1 c2)) in (let H2 \def (eq_ind_r nat d TMP_5 H O H_y) in (let 
-TMP_6 \def (drop_gen_refl c2 e2 H0) in (eq_ind C c2 TMP_4 H2 e2 
-TMP_6)))))))))))))) in (let TMP_165 \def (\lambda (i0: nat).(\lambda (IHi: 
-((\forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop 
-h d c1 c2) \to (\forall (e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop 
-(plus i0 h) O c1 e2))))))))))).(\lambda (c1: C).(let TMP_10 \def (\lambda (c: 
-C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to 
-(\forall (e2: C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (let TMP_8 
-\def (S i0) in (let TMP_9 \def (plus TMP_8 h) in (drop TMP_9 O c 
-e2))))))))))) in (let TMP_56 \def (\lambda (n: nat).(\lambda (c2: C).(\lambda 
-(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda 
-(e2: C).(\lambda (H0: (drop (S i0) O c2 e2)).(\lambda (H1: (le d (S 
-i0))).(let TMP_11 \def (CSort n) in (let TMP_12 \def (eq C c2 TMP_11) in (let 
-TMP_13 \def (eq nat h O) in (let TMP_14 \def (eq nat d O) in (let TMP_15 \def 
-(plus i0 h) in (let TMP_16 \def (S TMP_15) in (let TMP_17 \def (CSort n) in 
-(let TMP_18 \def (drop TMP_16 O TMP_17 e2) in (let TMP_54 \def (\lambda (H2: 
-(eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d 
-O)).(let TMP_22 \def (\lambda (n0: nat).(let TMP_19 \def (plus i0 n0) in (let 
-TMP_20 \def (S TMP_19) in (let TMP_21 \def (CSort n) in (drop TMP_20 O TMP_21 
-e2))))) in (let TMP_24 \def (\lambda (n0: nat).(let TMP_23 \def (S i0) in (le 
-n0 TMP_23))) in (let H5 \def (eq_ind nat d TMP_24 H1 O H4) in (let TMP_26 
-\def (\lambda (c: C).(let TMP_25 \def (S i0) in (drop TMP_25 O c e2))) in 
-(let TMP_27 \def (CSort n) in (let H6 \def (eq_ind C c2 TMP_26 H0 TMP_27 H2) 
-in (let TMP_28 \def (CSort n) in (let TMP_29 \def (eq C e2 TMP_28) in (let 
-TMP_30 \def (S i0) in (let TMP_31 \def (eq nat TMP_30 O) in (let TMP_32 \def 
-(eq nat O O) in (let TMP_33 \def (plus i0 O) in (let TMP_34 \def (S TMP_33) 
-in (let TMP_35 \def (CSort n) in (let TMP_36 \def (drop TMP_34 O TMP_35 e2) 
-in (let TMP_50 \def (\lambda (H7: (eq C e2 (CSort n))).(\lambda (H8: (eq nat 
-(S i0) O)).(\lambda (_: (eq nat O O)).(let TMP_37 \def (CSort n) in (let 
-TMP_41 \def (\lambda (c: C).(let TMP_38 \def (plus i0 O) in (let TMP_39 \def 
-(S TMP_38) in (let TMP_40 \def (CSort n) in (drop TMP_39 O TMP_40 c))))) in 
-(let TMP_42 \def (S i0) in (let TMP_43 \def (\lambda (ee: nat).(match ee with 
-[O \Rightarrow False | (S _) \Rightarrow True])) in (let H10 \def (eq_ind nat 
-TMP_42 TMP_43 I O H8) in (let TMP_44 \def (plus i0 O) in (let TMP_45 \def (S 
-TMP_44) in (let TMP_46 \def (CSort n) in (let TMP_47 \def (CSort n) in (let 
-TMP_48 \def (drop TMP_45 O TMP_46 TMP_47) in (let TMP_49 \def (False_ind 
-TMP_48 H10) in (eq_ind_r C TMP_37 TMP_41 TMP_49 e2 H7))))))))))))))) in (let 
-TMP_51 \def (S i0) in (let TMP_52 \def (drop_gen_sort n TMP_51 O e2 H6) in 
-(let TMP_53 \def (and3_ind TMP_29 TMP_31 TMP_32 TMP_36 TMP_50 TMP_52) in 
-(eq_ind_r nat O TMP_22 TMP_53 h H3))))))))))))))))))))))) in (let TMP_55 \def 
-(drop_gen_sort n h d c2 H) in (and3_ind TMP_12 TMP_13 TMP_14 TMP_18 TMP_54 
-TMP_55))))))))))))))))))) in (let TMP_164 \def (\lambda (c2: C).(\lambda 
-(IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 c3) 
-\to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d (S i0)) \to (drop (S 
-(plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: 
-C).(\lambda (d: nat).(let TMP_60 \def (\lambda (n: nat).(\forall (h: 
-nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 
-e2) \to ((le n (S i0)) \to (let TMP_57 \def (plus i0 h) in (let TMP_58 \def 
-(S TMP_57) in (let TMP_59 \def (CHead c2 k t) in (drop TMP_58 O TMP_59 
-e2)))))))))) in (let TMP_111 \def (\lambda (h: nat).(let TMP_64 \def (\lambda 
-(n: nat).((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O 
-c3 e2) \to ((le O (S i0)) \to (let TMP_61 \def (plus i0 n) in (let TMP_62 
-\def (S TMP_61) in (let TMP_63 \def (CHead c2 k t) in (drop TMP_62 O TMP_63 
-e2))))))))) in (let TMP_77 \def (\lambda (H: (drop O O (CHead c2 k t) 
-c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (_: (le O 
-(S i0))).(let TMP_66 \def (\lambda (c: C).(let TMP_65 \def (S i0) in (drop 
-TMP_65 O c e2))) in (let TMP_67 \def (CHead c2 k t) in (let TMP_68 \def 
-(CHead c2 k t) in (let TMP_69 \def (drop_gen_refl TMP_68 c3 H) in (let H2 
-\def (eq_ind_r C c3 TMP_66 H0 TMP_67 TMP_69) in (let TMP_72 \def (\lambda (n: 
-nat).(let TMP_70 \def (S n) in (let TMP_71 \def (CHead c2 k t) in (drop 
-TMP_70 O TMP_71 e2)))) in (let TMP_73 \def (drop_gen_drop k c2 e2 t i0 H2) in 
-(let TMP_74 \def (drop_drop k i0 c2 e2 TMP_73 t) in (let TMP_75 \def (plus i0 
-O) in (let TMP_76 \def (plus_n_O i0) in (eq_ind nat i0 TMP_72 TMP_74 TMP_75 
-TMP_76))))))))))))))) in (let TMP_110 \def (\lambda (n: nat).(\lambda (_: 
-(((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) 
-\to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) 
+(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 
+e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: 
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2: 
+C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c 
+e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: 
+nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0: 
+(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort 
+n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda 
+(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d 
+O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) 
+e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4) 
+in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort 
+n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop 
+(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda 
+(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) 
+(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind 
+nat (S i0) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _) 
+\Rightarrow True])) I O H8) in (False_ind (drop (S (plus i0 O)) O (CSort n) 
+(CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O e2 H6)))) h H3)))) 
+(drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall 
+(c3: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 c3) \to (\forall 
+(e2: C).((drop (S i0) O c3 e2) \to ((le d (S i0)) \to (drop (S (plus i0 h)) O 
+c2 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (d: 
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c2 k t) 
+c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le n (S i0)) \to (drop 
+(S (plus i0 h)) O (CHead c2 k t) e2))))))) (\lambda (h: nat).(nat_ind 
+(\lambda (n: nat).((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop 
+(S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) 
+e2)))))) (\lambda (H: (drop O O (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda 
+(H0: (drop (S i0) O c3 e2)).(\lambda (_: (le O (S i0))).(let H2 \def 
+(eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead c2 k t) 
+(drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat i0 (\lambda (n: 
+nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 (drop_gen_drop k 
+c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda (n: nat).(\lambda 
+(_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 
+e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) 
 e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (e2: 
-C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le O (S i0))).(let 
-TMP_78 \def (plus i0 n) in (let TMP_79 \def (S TMP_78) in (let TMP_82 \def 
-(\lambda (n0: nat).(let TMP_80 \def (S n0) in (let TMP_81 \def (CHead c2 k t) 
-in (drop TMP_80 O TMP_81 e2)))) in (let TMP_83 \def (plus i0 n) in (let 
-TMP_84 \def (S TMP_83) in (let TMP_85 \def (plus i0 n) in (let TMP_86 \def (r 
-k TMP_85) in (let TMP_87 \def (S TMP_86) in (let TMP_88 \def (\lambda (n0: 
-nat).(drop n0 O c2 e2)) in (let TMP_89 \def (r k n) in (let TMP_90 \def (plus 
-i0 TMP_89) in (let TMP_92 \def (\lambda (n0: nat).(let TMP_91 \def (S n0) in 
-(drop TMP_91 O c2 e2))) in (let TMP_93 \def (r k n) in (let TMP_94 \def 
-(drop_gen_drop k c2 c3 t n H0) in (let TMP_95 \def (IHc c3 O TMP_93 TMP_94 e2 
-H1 H2) in (let TMP_96 \def (plus i0 n) in (let TMP_97 \def (r k TMP_96) in 
-(let TMP_98 \def (r_plus_sym k i0 n) in (let TMP_99 \def (eq_ind_r nat TMP_90 
-TMP_92 TMP_95 TMP_97 TMP_98) in (let TMP_100 \def (plus i0 n) in (let TMP_101 
-\def (S TMP_100) in (let TMP_102 \def (r k TMP_101) in (let TMP_103 \def 
-(plus i0 n) in (let TMP_104 \def (r_S k TMP_103) in (let TMP_105 \def 
-(eq_ind_r nat TMP_87 TMP_88 TMP_99 TMP_102 TMP_104) in (let TMP_106 \def 
-(drop_drop k TMP_84 c2 e2 TMP_105 t) in (let TMP_107 \def (S n) in (let 
-TMP_108 \def (plus i0 TMP_107) in (let TMP_109 \def (plus_n_Sm i0 n) in 
-(eq_ind nat TMP_79 TMP_82 TMP_106 TMP_108 
-TMP_109)))))))))))))))))))))))))))))))))))) in (nat_ind TMP_64 TMP_77 TMP_110 
-h))))) in (let TMP_163 \def (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: 
-nat).((drop h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 
-e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) 
-e2)))))))).(\lambda (h: nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) 
-c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le 
-(S d0) (S i0))).(let TMP_113 \def (\lambda (e: C).(\lambda (v: T).(let 
-TMP_112 \def (CHead e k v) in (eq C c3 TMP_112)))) in (let TMP_116 \def 
-(\lambda (_: C).(\lambda (v: T).(let TMP_114 \def (r k d0) in (let TMP_115 
-\def (lift h TMP_114 v) in (eq T t TMP_115))))) in (let TMP_118 \def (\lambda 
-(e: C).(\lambda (_: T).(let TMP_117 \def (r k d0) in (drop h TMP_117 c2 e)))) 
-in (let TMP_119 \def (plus i0 h) in (let TMP_120 \def (S TMP_119) in (let 
-TMP_121 \def (CHead c2 k t) in (let TMP_122 \def (drop TMP_120 O TMP_121 e2) 
-in (let TMP_161 \def (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 
-(CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4: 
-(drop h (r k d0) c2 x0)).(let TMP_126 \def (\lambda (c: C).(\forall (h0: 
-nat).((drop h0 d0 (CHead c2 k t) c) \to (\forall (e3: C).((drop (S i0) O c 
-e3) \to ((le d0 (S i0)) \to (let TMP_123 \def (plus i0 h0) in (let TMP_124 
-\def (S TMP_123) in (let TMP_125 \def (CHead c2 k t) in (drop TMP_124 O 
-TMP_125 e3)))))))))) in (let TMP_127 \def (CHead x0 k x1) in (let H5 \def 
-(eq_ind C c3 TMP_126 IHd TMP_127 H2) in (let TMP_129 \def (\lambda (c: 
-C).(let TMP_128 \def (S i0) in (drop TMP_128 O c e2))) in (let TMP_130 \def 
-(CHead x0 k x1) in (let H6 \def (eq_ind C c3 TMP_129 H0 TMP_130 H2) in (let 
-TMP_134 \def (\lambda (t0: T).(\forall (h0: nat).((drop h0 d0 (CHead c2 k t0) 
-(CHead x0 k x1)) \to (\forall (e3: C).((drop (S i0) O (CHead x0 k x1) e3) \to 
-((le d0 (S i0)) \to (let TMP_131 \def (plus i0 h0) in (let TMP_132 \def (S 
-TMP_131) in (let TMP_133 \def (CHead c2 k t0) in (drop TMP_132 O TMP_133 
-e3)))))))))) in (let TMP_135 \def (r k d0) in (let TMP_136 \def (lift h 
-TMP_135 x1) in (let H7 \def (eq_ind T t TMP_134 H5 TMP_136 H3) in (let 
-TMP_137 \def (r k d0) in (let TMP_138 \def (lift h TMP_137 x1) in (let 
-TMP_142 \def (\lambda (t0: T).(let TMP_139 \def (plus i0 h) in (let TMP_140 
-\def (S TMP_139) in (let TMP_141 \def (CHead c2 k t0) in (drop TMP_140 O 
-TMP_141 e2))))) in (let TMP_143 \def (plus i0 h) in (let TMP_146 \def 
-(\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to 
-(let TMP_144 \def (plus i0 h) in (let TMP_145 \def (r k0 TMP_144) in (drop 
-TMP_145 O c2 e2)))))) in (let TMP_152 \def (\lambda (b: B).(\lambda (H8: 
-(drop h (r (Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 
-e2)).(let TMP_147 \def (Bind b) in (let TMP_148 \def (r TMP_147 d0) in (let 
-TMP_149 \def (Bind b) in (let TMP_150 \def (r TMP_149 d0) in (let TMP_151 
-\def (le_S_n TMP_150 i0 H1) in (IHi c2 x0 TMP_148 h H8 e2 H9 TMP_151))))))))) 
-in (let TMP_155 \def (\lambda (f: F).(\lambda (H8: (drop h (r (Flat f) d0) c2 
-x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0 e2)).(let TMP_153 \def (Flat f) 
-in (let TMP_154 \def (r TMP_153 d0) in (IHc x0 TMP_154 h H8 e2 H9 H1)))))) in 
-(let TMP_156 \def (drop_gen_drop k x0 e2 x1 i0 H6) in (let TMP_157 \def 
-(K_ind TMP_146 TMP_152 TMP_155 k H4 TMP_156) in (let TMP_158 \def (r k d0) in 
-(let TMP_159 \def (lift h TMP_158 x1) in (let TMP_160 \def (drop_drop k 
-TMP_143 c2 e2 TMP_157 TMP_159) in (eq_ind_r T TMP_138 TMP_142 TMP_160 t 
-H3)))))))))))))))))))))))))))) in (let TMP_162 \def (drop_gen_skip_l c2 c3 t 
-h d0 k H) in (ex3_2_ind C T TMP_113 TMP_116 TMP_118 TMP_122 TMP_161 
-TMP_162))))))))))))))))) in (nat_ind TMP_60 TMP_111 TMP_163 d)))))))))) in 
-(C_ind TMP_10 TMP_56 TMP_164 c1))))))) in (nat_ind TMP_2 TMP_7 TMP_165 i)))).
+C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le O (S i0))).(eq_ind 
+nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O (CHead c2 k t) e2)) 
+(drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k (plus i0 n))) 
+(\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r k n)) 
+(\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) (drop_gen_drop k 
+c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 n)) (r k (S (plus 
+i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm i0 n)))))))) h)) 
+(\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop h d0 (CHead c2 k 
+t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d0 (S i0)) \to 
+(drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: nat).(\lambda 
+(H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop 
+(S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S i0))).(ex3_2_ind C T (\lambda 
+(e: C).(\lambda (v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: 
+T).(eq T t (lift h (r k d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r 
+k d0) c2 e))) (drop (S (plus i0 h)) O (CHead c2 k t) e2) (\lambda (x0: 
+C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 k x1))).(\lambda (H3: 
+(eq T t (lift h (r k d0) x1))).(\lambda (H4: (drop h (r k d0) c2 x0)).(let H5 
+\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 d0 (CHead c2 k 
+t) c) \to (\forall (e3: C).((drop (S i0) O c e3) \to ((le d0 (S i0)) \to 
+(drop (S (plus i0 h0)) O (CHead c2 k t) e3))))))) IHd (CHead x0 k x1) H2) in 
+(let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead x0 
+k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: 
+nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k x1)) \to (\forall (e3: 
+C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S i0)) \to (drop (S (plus 
+i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k d0) x1) H3) in (eq_ind_r 
+T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O (CHead c2 k 
+t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0: K).((drop h (r 
+k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0 (plus i0 h)) O c2 
+e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r (Bind b) d0) c2 x0)).(\lambda 
+(H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind b) d0) h H8 e2 H9 
+(le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda (H8: (drop h (r 
+(Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0 e2)).(IHc x0 (r 
+(Flat f) d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1 i0 H6)) (lift h 
+(r k d0) x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k H))))))))) 
+d))))))) c1)))) i).