X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fgetl%2Fdrop.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fgetl%2Fdrop.ma;h=507560f8857fa0c8fc1929d3db7915a9611e2ef1;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/getl/props.ma".
+
+include "LambdaDelta-1/clear/drop.ma".
+
+theorem getl_drop:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: 
+nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: 
+C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to 
+(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: 
+T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b) 
+u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n) 
+e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: 
+T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0 
+e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: 
+T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t) 
+(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: 
+(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear 
+(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e))) 
+(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind 
+b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) \Rightarrow 
+c1])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 
+(CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: 
+C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | 
+(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with 
+[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) 
+(CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in 
+((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda 
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) 
+(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e 
+(Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e 
+c0)).(eq_ind_r C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 (Bind b0) t) 
+c1)) (eq_ind B b (\lambda (b1: B).(drop (S O) O (CHead c0 (Bind b1) t) c0)) 
+(drop_drop (Bind b) O c0 c0 (drop_refl c0) t) b0 H5) e H6)))) H3)) H2)))) 
+(\lambda (f: F).(\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) 
+u))).(drop_clear_O b (CHead c0 (Flat f) t) e u (clear_flat c0 (CHead e (Bind 
+b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1) f t) e O (drop_refl 
+e)))) k (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) (\lambda (n: 
+nat).(\lambda (_: (((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S 
+n) O (CHead c0 k t) e)))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead e 
+(Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: 
+nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t 
+n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
+
+theorem getl_drop_conf_lt:
+ \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda 
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
+h d c0 e0)))))))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: 
+C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to 
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) 
+c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n: 
+nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i 
+(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i 
+(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda 
+(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: 
+nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda 
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
+h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t) 
+(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def 
+(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C 
+(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0 
+(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x: 
+C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead 
+c1 (Bind b) u))).(C_ind (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to 
+((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead 
+e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) 
+(\lambda (n: nat).(\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda 
+(H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind 
+b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) (\lambda (x0: C).(\lambda 
+(IHx: (((drop i O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) 
+\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(\lambda (k0: K).(\lambda 
+(t0: T).(\lambda (H5: (drop i O (CHead c0 k t) (CHead x0 k0 t0))).(\lambda 
+(H6: (clear (CHead x0 k0 t0) (CHead c1 (Bind b) u))).(K_ind (\lambda (k1: 
+K).((drop i O (CHead c0 k t) (CHead x0 k1 t0)) \to ((clear (CHead x0 k1 t0) 
+(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) (\lambda (b0: 
+B).(\lambda (H7: (drop i O (CHead c0 k t) (CHead x0 (Bind b0) t0))).(\lambda 
+(H8: (clear (CHead x0 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def 
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow c1 | (CHead c2 _ _) \Rightarrow c2])) (CHead c1 (Bind 
+b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 
+H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k1 _) \Rightarrow 
+(match k1 in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | 
+(Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead x0 (Bind b0) t0) 
+(clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def 
+(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with 
+[(CSort _) \Rightarrow u | (CHead _ _ t1) \Rightarrow t1])) (CHead c1 (Bind 
+b) u) (CHead x0 (Bind b0) t0) (clear_gen_bind b0 x0 (CHead c1 (Bind b) u) t0 
+H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 x0)).(let H14 
+\def (eq_ind_r T t0 (\lambda (t1: T).(drop i O (CHead c0 k t) (CHead x0 (Bind 
+b0) t1))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b1: B).(drop i 
+O (CHead c0 k t) (CHead x0 (Bind b1) u))) H14 b H12) in (let H16 \def 
+(eq_ind_r C x0 (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to ((clear c2 
+(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx c1 H13) in 
+(let H17 \def (eq_ind_r C x0 (\lambda (c2: C).(drop i O (CHead c0 k t) (CHead 
+c2 (Bind b) u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: 
+C).(drop i O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: 
+C).(drop h (r (Bind b) d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead 
+e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) 
+(\lambda (x1: T).(\lambda (x2: C).(\lambda (H18: (eq T u (lift h (r (Bind b) 
+d) x1))).(\lambda (H19: (drop i O e (CHead x2 (Bind b) x1))).(\lambda (H20: 
+(drop h (r (Bind b) d) c1 x2)).(let H21 \def (eq_ind T u (\lambda (t1: 
+T).((drop i O (CHead c0 k t) c1) \to ((clear c1 (CHead c1 (Bind b) t1)) \to 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))))) H16 (lift h (r (Bind b) d) x1) 
+H18) in (eq_ind_r T (lift h (r (Bind b) d) x1) (\lambda (t1: T).(ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: 
+T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x1) (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))) x1 x2 (refl_equal T (lift h d x1)) 
+(getl_intro i e (CHead x2 (Bind b) x1) (CHead x2 (Bind b) x1) H19 (clear_bind 
+b x2 x1)) H20) u H18))))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H17 
+e h d H1))))))))) H10)) H9))))) (\lambda (f: F).(\lambda (H7: (drop i O 
+(CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda (H8: (clear (CHead x0 (Flat 
+f) t0) (CHead c1 (Bind b) u))).(nat_ind (\lambda (n: nat).((drop h (S (plus n 
+d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead x0 (Flat f) t0)) 
+\to ((((drop n O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C (\lambda (v: 
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
+h d c1 e0)))))))) (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) 
+e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead x0 (Flat f) t0))).(\lambda 
+(IHx0: (((drop O O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 (Bind b) u)) 
+\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))))))).(let H11 \def (f_equal C C 
+(\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) 
+\Rightarrow c0 | (CHead c2 _ _) \Rightarrow c2])) (CHead c0 k t) (CHead x0 
+(Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 (Flat f) t0) H10)) in 
+((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda 
+(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1 _) \Rightarrow k1])) 
+(CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead x0 
+(Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0 
+in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t1) 
+\Rightarrow t1])) (CHead c0 k t) (CHead x0 (Flat f) t0) (drop_gen_refl (CHead 
+c0 k t) (CHead x0 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat 
+f))).(\lambda (H15: (eq C c0 x0)).(let H16 \def (eq_ind_r C x0 (\lambda (c2: 
+C).(clear c2 (CHead c1 (Bind b) u))) (clear_gen_flat f x0 (CHead c1 (Bind b) 
+u) t0 H8) c0 H15) in (let H17 \def (eq_ind_r C x0 (\lambda (c2: C).((drop O O 
+(CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))))) IHx0 c0 H15) in (let H18 \def 
+(eq_ind K k (\lambda (k1: K).((drop O O (CHead c0 k1 t) c0) \to ((clear c0 
+(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (Flat f) 
+H14) in (let H19 \def (eq_ind K k (\lambda (k1: K).(drop h (S (plus O d)) 
+(CHead c0 k1 t) e)) H9 (Flat f) H14) in (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 O d)) v)))) (\lambda (e0: 
+C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (H20: (eq C e (CHead x1 (Flat f) x2))).(\lambda (H21: (eq T t 
+(lift h (r (Flat f) (plus O d)) x2))).(\lambda (H22: (drop h (r (Flat f) 
+(plus O d)) c0 x1)).(let H23 \def (f_equal T T (\lambda (e0: T).e0) t (lift h 
+(r (Flat f) (plus O d)) x2) H21) in (let H24 \def (eq_ind C e (\lambda (c2: 
+C).((drop O O (CHead c0 (Flat f) t) c0) \to ((clear c0 (CHead c1 (Bind b) u)) 
+\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda 
+(_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H18 (CHead x1 (Flat f) x2) 
+H20) in (eq_ind_r C (CHead x1 (Flat f) x2) (\lambda (c2: C).(ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl O c2 (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))) (let H25 \def (eq_ind T t (\lambda 
+(t1: T).((drop O O (CHead c0 (Flat f) t1) c0) \to ((clear c0 (CHead c1 (Bind 
+b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 
+(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H24 
+(lift h (S d) x2) H23) in (let H26 \def (H c1 u O (getl_intro O c0 (CHead c1 
+(Bind b) u) c0 (drop_refl c0) H16) x1 h d H22) in (ex3_2_ind T C (\lambda (v: 
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl O x1 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop 
+h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d 
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead 
+e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) 
+(\lambda (x3: T).(\lambda (x4: C).(\lambda (H27: (eq T u (lift h d 
+x3))).(\lambda (H28: (getl O x1 (CHead x4 (Bind b) x3))).(\lambda (H29: (drop 
+h d c1 x4)).(let H30 \def (eq_ind T u (\lambda (t1: T).((drop O O (CHead c0 
+(Flat f) (lift h (S d) x2)) c0) \to ((clear c0 (CHead c1 (Bind b) t1)) \to 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H25 (lift h d 
+x3) H27) in (let H31 \def (eq_ind T u (\lambda (t1: T).(clear c0 (CHead c1 
+(Bind b) t1))) H16 (lift h d x3) H27) in (eq_ind_r T (lift h d x3) (\lambda 
+(t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) (CHead e0 
+(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) 
+(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x3) (lift h 
+d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x1 (Flat f) x2) 
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) 
+x3 x4 (refl_equal T (lift h d x3)) (getl_flat x1 (CHead x4 (Bind b) x3) O H28 
+f x2) H29) u H27)))))))) H26))) e H20)))))))) (drop_gen_skip_l c0 e t h (plus 
+O d) (Flat f) H19))))))))) H12)) H11))))) (\lambda (i0: nat).(\lambda (IHi: 
+(((drop h (S (plus i0 d)) (CHead c0 k t) e) \to ((drop i0 O (CHead c0 k t) 
+(CHead x0 (Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 
+(CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u 
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind 
+b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T 
+C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl i0 e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))))))).(\lambda (H9: (drop h (S (plus 
+(S i0) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S i0) O (CHead c0 k t) 
+(CHead x0 (Flat f) t0))).(\lambda (IHx0: (((drop (S i0) O (CHead c0 k t) x0) 
+\to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda 
+(_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) 
+e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
+e0)))))))).(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 t (lift h (r k (plus (S i0) d)) 
+v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S i0) d)) c0 e0))) 
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda 
+(v: T).(\lambda (e0: C).(getl (S i0) e (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (H11: (eq C e (CHead x1 k x2))).(\lambda (H12: (eq T t (lift h (r 
+k (plus (S i0) d)) x2))).(\lambda (H13: (drop h (r k (plus (S i0) d)) c0 
+x1)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S 
+i0) d)) x2) H12) in (let H15 \def (eq_ind C e (\lambda (c2: C).((drop h (S 
+(plus i0 d)) (CHead c0 k t) c2) \to ((drop i0 O (CHead c0 k t) (CHead x0 
+(Flat f) t0)) \to ((((drop i0 O (CHead c0 k t) x0) \to ((clear x0 (CHead c1 
+(Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d 
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl i0 c2 (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0)))))))) IHi (CHead x1 k x2) H11) in (let 
+H16 \def (eq_ind C e (\lambda (c2: C).((drop (S i0) O (CHead c0 k t) x0) \to 
+((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: 
+C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
+e0))))))) IHx0 (CHead x1 k x2) H11) in (eq_ind_r C (CHead x1 k x2) (\lambda 
+(c2: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) 
+(\lambda (v: T).(\lambda (e0: C).(getl (S i0) c2 (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H17 \def (eq_ind T 
+t (\lambda (t1: T).((drop (S i0) O (CHead c0 k t1) x0) \to ((clear x0 (CHead 
+c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift 
+h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) 
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 
+e0))))))) H16 (lift h (r k (S (plus i0 d))) x2) H14) in (let H18 \def (eq_ind 
+T t (\lambda (t1: T).((drop h (S (plus i0 d)) (CHead c0 k t1) (CHead x1 k 
+x2)) \to ((drop i0 O (CHead c0 k t1) (CHead x0 (Flat f) t0)) \to ((((drop i0 
+O (CHead c0 k t1) x0) \to ((clear x0 (CHead c1 (Bind b) u)) \to (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) \to (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl i0 (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) H15 (lift h (r k (S 
+(plus i0 d))) x2) H14) in (let H19 \def (eq_ind nat (r k (plus (S i0) d)) 
+(\lambda (n: nat).(drop h n c0 x1)) H13 (plus (r k (S i0)) d) (r_plus k (S 
+i0) d)) in (let H20 \def (eq_ind nat (r k (S i0)) (\lambda (n: nat).(drop h 
+(plus n d) c0 x1)) H19 (S (r k i0)) (r_S k i0)) in (let H21 \def (H c1 u (r k 
+i0) (getl_intro (r k i0) c0 (CHead c1 (Bind b) u) (CHead x0 (Flat f) t0) 
+(drop_gen_drop k c0 (CHead x0 (Flat f) t0) t i0 H10) (clear_flat x0 (CHead c1 
+(Bind b) u) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) f t0)) x1 h d 
+H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d 
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k i0) x1 (CHead e0 (Bind b) 
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda 
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x3: T).(\lambda (x4: 
+C).(\lambda (H22: (eq T u (lift h d x3))).(\lambda (H23: (getl (r k i0) x1 
+(CHead x4 (Bind b) x3))).(\lambda (H24: (drop h d c1 x4)).(let H25 \def 
+(eq_ind T u (\lambda (t1: T).((drop (S i0) O (CHead c0 k (lift h (r k (S 
+(plus i0 d))) x2)) x0) \to ((clear x0 (CHead c1 (Bind b) t1)) \to (ex3_2 T C 
+(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) H17 (lift h d x3) 
+H22) in (let H26 \def (eq_ind T u (\lambda (t1: T).(clear x0 (CHead c1 (Bind 
+b) t1))) (clear_gen_flat f x0 (CHead c1 (Bind b) u) t0 H8) (lift h d x3) H22) 
+in (eq_ind_r T (lift h d x3) (\lambda (t1: T).(ex3_2 T C (\lambda (v: 
+T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: 
+C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) (\lambda (_: 
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: 
+T).(\lambda (_: C).(eq T (lift h d x3) (lift h d v)))) (\lambda (v: 
+T).(\lambda (e0: C).(getl (S i0) (CHead x1 k x2) (CHead e0 (Bind b) v)))) 
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x3 x4 (refl_equal T (lift 
+h d x3)) (getl_head k i0 x1 (CHead x4 (Bind b) x3) H23 x2) H24) u H22)))))))) 
+H21)))))) e H11))))))))) (drop_gen_skip_l c0 e t h (plus (S i0) d) k 
+H9))))))) i H1 H7 IHx)))) k0 H5 H6))))))) x H3 H4)))) H2)))))))))))))) c)).
+
+theorem getl_drop_conf_ge:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall 
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d 
+h) i) \to (getl (minus i h) e a)))))))))
+\def
+ \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c 
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h 
+d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H) 
+in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0 
+a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c 
+x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i 
+x c H3 e h d H0 H1) H4)))) H2)))))))))).
+
+theorem getl_conf_ge_drop:
+ \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i: 
+nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1 
+c2) \to (drop i O c2 e))))))))
+\def
+ \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda 
+(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O)) 
+(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e 
+u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S 
+i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in 
+H3)))))))).
+
+theorem getl_drop_conf_rev:
+ \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to 
+(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i 
+c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) 
+(\lambda (c1: C).(drop (S i) j c1 e1)))))))))))
+\def
+ \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1 
+e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i: 
+nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2 
+H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))).
+
+theorem drop_getl_trans_lt:
+ \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2: 
+C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda 
+(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda 
+(e1: C).(drop h (minus d (S i)) e1 e2)))))))))))))
+\def
+ \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 
+c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i 
+c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b) 
+v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e: 
+C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead 
+e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d 
+(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: 
+(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1 
+e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1: 
+C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: 
+C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O 
+c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat 
+(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i))) 
+(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b 
+e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h 
+(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C 
+(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) 
+(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda 
+(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda 
+(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i 
+c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h 
+(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus 
+d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d (le_S 
+(S i) d H)) c1 c2 h H0 x H3))))) H2)))))))))))).
+
+theorem drop_getl_trans_le:
+ \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall 
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 
+e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) 
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: 
+C).(\lambda (e1: C).(clear e1 e2))))))))))))
+\def
+ \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1: 
+C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 
+c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def 
+(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) 
+(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_: 
+C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) 
+e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x: 
+C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def 
+(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i 
+O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda 
+(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: 
+C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 
+e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h 
+(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i 
+O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) 
+(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5))))) 
+H2)))))))))).
+
+theorem drop_getl_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).((getl i c2 e2) 
+\to ((le d i) \to (getl (plus i h) c1 e2)))))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: 
+nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: 
+C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def 
+(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) 
+(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x: 
+C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro 
+(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))).
+
+theorem getl_drop_trans:
+ \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to 
+(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i 
+h)) O c1 e2)))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: 
+nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 
+e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2: 
+C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2: 
+C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2 
+H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda 
+(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2: 
+C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2 
+e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall 
+(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2: 
+C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead 
+c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: 
+C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b) 
+t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop 
+(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead 
+c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S 
+i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2)) 
+H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2 
+(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead 
+c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2 
+t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_: 
+(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i: 
+nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t) 
+e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2: 
+C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S 
+i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop 
+(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2 
+i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f: 
+F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n: 
+nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i: 
+nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t) 
+e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2: 
+C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f) 
+(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2) 
+(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0) 
+t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to 
+(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i 
+n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2 
+(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i) 
+O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S 
+(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1).
+