set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/getl/drop".
-include "getl/fwd.ma".
+include "getl/props.ma".
include "clear/drop.ma".
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))).((match x in C return (\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))))))) with [(CSort n) \Rightarrow (\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))))))) | (CHead c2 k0 t0) \Rightarrow (\lambda (H5: (drop i O
+(CHead c0 k t) (CHead c2 k0 t0))).(\lambda (H6: (clear (CHead c2 k0 t0)
+(CHead c1 (Bind b) u))).((match k0 in K return (\lambda (k1: K).((drop i O
+(CHead c0 k t) (CHead c2 k1 t0)) \to ((clear (CHead c2 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))))))) with [(Bind b0) \Rightarrow
+(\lambda (H7: (drop i O (CHead c0 k t) (CHead c2 (Bind b0) t0))).(\lambda
+(H8: (clear (CHead c2 (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 c _ _) \Rightarrow c])) (CHead c1 (Bind b)
+u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (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 _ k _) \Rightarrow
+(match k in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat
+_) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind b0) t0)
+(clear_gen_bind b0 c2 (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 _ _ t) \Rightarrow t])) (CHead c1 (Bind b)
+u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0
+H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 c2)).(let H14
+\def (eq_ind_r T t0 (\lambda (t0: T).(drop i O (CHead c0 k t) (CHead c2 (Bind
+b0) t0))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b: B).(drop i O
+(CHead c0 k t) (CHead c2 (Bind b) u))) H14 b H12) in (let H16 \def (eq_ind_r
+C c2 (\lambda (c: C).(drop i O (CHead c0 k t) (CHead c (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 (x0: T).(\lambda (x1:
+C).(\lambda (H17: (eq T u (lift h (r (Bind b) d) x0))).(\lambda (H18: (drop i
+O e (CHead x1 (Bind b) x0))).(\lambda (H19: (drop h (r (Bind b) d) c1
+x1)).(eq_ind_r T (lift h (r (Bind b) d) x0) (\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) x0) (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))) x0 x1 (refl_equal T (lift h d x0))
+(getl_intro i e (CHead x1 (Bind b) x0) (CHead x1 (Bind b) x0) H18 (clear_bind
+b x1 x0)) H19) u H17)))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H16 e
+h d H1)))))))) H10)) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (drop i O
+(CHead c0 k t) (CHead c2 (Flat f) t0))).(\lambda (H8: (clear (CHead c2 (Flat
+f) t0) (CHead c1 (Bind b) u))).((match i in nat return (\lambda (n:
+nat).((drop h (S (plus n d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t)
+(CHead c2 (Flat f) t0)) \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))))))) with [O
+\Rightarrow (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) e)).(\lambda
+(H10: (drop O O (CHead c0 k t) (CHead c2 (Flat f) t0))).(let H11 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k t)
+(CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (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 _ k _) \Rightarrow
+k])) (CHead c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t)
+(CHead c2 (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 _ _ t) \Rightarrow t])) (CHead c0 k t) (CHead c2 (Flat f) t0)
+(drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0) H10)) in (\lambda (H14:
+(eq K k (Flat f))).(\lambda (H15: (eq C c0 c2)).(let H16 \def (eq_ind_r C c2
+(\lambda (c: C).(clear c (CHead c1 (Bind b) u))) (clear_gen_flat f c2 (CHead
+c1 (Bind b) u) t0 H8) c0 H15) in (let H17 \def (eq_ind K k (\lambda (k:
+K).(drop h (S (plus O d)) (CHead c0 k 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 (x0: C).(\lambda (x1:
+T).(\lambda (H18: (eq C e (CHead x0 (Flat f) x1))).(\lambda (H19: (eq T t
+(lift h (r (Flat f) (plus O d)) x1))).(\lambda (H20: (drop h (r (Flat f)
+(plus O d)) c0 x0)).(let H21 \def (f_equal T T (\lambda (e0: T).e0) t (lift h
+(r (Flat f) (plus O d)) x1) H19) in (eq_ind_r C (CHead x0 (Flat f) x1)
+(\lambda (c3: 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 c3 (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H22 \def (H c1 u O
+(getl_intro O c0 (CHead c1 (Bind b) u) c0 (drop_refl c0) H16) x0 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 O x0 (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 x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2: T).(\lambda (x3:
+C).(\lambda (H23: (eq T u (lift h d x2))).(\lambda (H24: (getl O x0 (CHead x3
+(Bind b) x2))).(\lambda (H25: (drop h d c1 x3)).(let H26 \def (eq_ind T u
+(\lambda (t: T).(clear c0 (CHead c1 (Bind b) t))) H16 (lift h d x2) H23) in
+(eq_ind_r T (lift h d x2) (\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 x0 (Flat f) x1) (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 x2) (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift
+h d x2)) (getl_flat x0 (CHead x3 (Bind b) x2) O H24 f x1) H25) u H23)))))))
+H22)) e H18))))))) (drop_gen_skip_l c0 e t h (plus O d) (Flat f) H17)))))))
+H12)) H11)))) | (S n) \Rightarrow (\lambda (H9: (drop h (S (plus (S n) d))
+(CHead c0 k t) e)).(\lambda (H10: (drop (S n) O (CHead c0 k t) (CHead c2
+(Flat f) t0))).(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 n)
+d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S n) 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 n) e (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H11: (eq C e (CHead x0 k x1))).(\lambda (H12:
+(eq T t (lift h (r k (plus (S n) d)) x1))).(\lambda (H13: (drop h (r k (plus
+(S n) d)) c0 x0)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h
+(r k (plus (S n) d)) x1) H12) in (eq_ind_r C (CHead x0 k x1) (\lambda (c3:
+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 n) c3 (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H15 \def (eq_ind
+nat (r k (plus (S n) d)) (\lambda (n: nat).(drop h n c0 x0)) H13 (plus (r k
+(S n)) d) (r_plus k (S n) d)) in (let H16 \def (eq_ind nat (r k (S n))
+(\lambda (n: nat).(drop h (plus n d) c0 x0)) H15 (S (r k n)) (r_S k n)) in
+(let H17 \def (H c1 u (r k n) (getl_intro (r k n) c0 (CHead c1 (Bind b) u)
+(CHead c2 (Flat f) t0) (drop_gen_drop k c0 (CHead c2 (Flat f) t0) t n H10)
+(clear_flat c2 (CHead c1 (Bind b) u) (clear_gen_flat f c2 (CHead c1 (Bind b)
+u) t0 H8) f t0)) x0 h d H16) 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 n) x0
+(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 n) (CHead x0 k x1) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2:
+T).(\lambda (x3: C).(\lambda (H18: (eq T u (lift h d x2))).(\lambda (H19:
+(getl (r k n) x0 (CHead x3 (Bind b) x2))).(\lambda (H20: (drop h d c1
+x3)).(let H21 \def (eq_ind T u (\lambda (t: T).(clear c2 (CHead c1 (Bind b)
+t))) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) (lift h d x2) H18) in
+(eq_ind_r T (lift h d x2) (\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 n) (CHead x0 k x1) (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 x2) (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift
+h d x2)) (getl_head k n x0 (CHead x3 (Bind b) x2) H19 x1) H20) u H18)))))))
+H17)))) e H11))))))) (drop_gen_skip_l c0 e t h (plus (S n) d) k H9))))]) H1
+H7)))]) H5 H6)))]) 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_comm 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).
+
projects / helm.git / blobdiff