(* This file was automatically generated: do not edit *********************)
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/drop1/getl".
+include "LambdaDelta-1/drop1/fwd.ma".
-include "drop1/defs.ma".
-
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
theorem drop1_getl_trans:
\forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
(trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v))))))))))))))
(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda
(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl
-i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H in drop1 return (\lambda
-(p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p c c0)).((eq
-PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex2 C (\lambda (e2:
-C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b)
-v))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil
-PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2
-(\lambda (c0: C).((eq C c0 c1) \to (ex2 C (\lambda (e2: C).(drop1 PNil e2
-e1)) (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))) (\lambda (H4: (eq
-C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop1 PNil
-e2 e1)) (\lambda (e2: C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro2 C
-(\lambda (e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2
-(Bind b) v))) e1 (drop1_nil e1) H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2
-H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds0 H2) \Rightarrow (\lambda (H3:
-(eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5:
-(eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e:
-PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
-\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
-(False_ind ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1
-hds0 c3 c4) \to (ex2 C (\lambda (e2: C).(drop1 PNil e2 e1)) (\lambda (e2:
-C).(getl i c2 (CHead e2 (Bind b) v)))))))) H6)) H4 H5 H1 H2))))]) in (H1
-(refl_equal PList PNil) (refl_equal C c2) (refl_equal C c1)))))))))))
-(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: PList).(\lambda (H:
-((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2 c1) \to (\forall (b:
-B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1
-(Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1))
-(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (ptrans
-hds0 i) v))))))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0:
-(drop1 (PCons h d hds0) c2 c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v:
-T).(\lambda (i: nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2
-\def (match H0 in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda
-(c0: C).(\lambda (_: (drop1 p c c0)).((eq PList p (PCons h d hds0)) \to ((eq
-C c c2) \to ((eq C c0 c1) \to (ex2 C (\lambda (e2: C).(drop1 (match (blt
-(trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0
-i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) (\lambda
-(e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans
-hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b)
-(lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d
-(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)])
-v)))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil
-(PCons h d hds0))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c
-c1)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList
-return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _)
-\Rightarrow False])) I (PCons h d hds0) H2) in (False_ind ((eq C c c2) \to
-((eq C c c1) \to (ex2 C (\lambda (e2: C).(drop1 (match (blt (trans hds0 i) d)
-with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i))
-| false \Rightarrow (ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match
-(blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
-\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1 (match
-(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans
-hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))))
-H5)) H3 H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds1 H3) \Rightarrow (\lambda
-(H4: (eq PList (PCons h0 d0 hds1) (PCons h d hds0))).(\lambda (H5: (eq C c0
-c2)).(\lambda (H6: (eq C c4 c1)).((let H7 \def (f_equal PList PList (\lambda
-(e: PList).(match e in PList return (\lambda (_: PList).PList) with [PNil
-\Rightarrow hds1 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds1) (PCons h
-d hds0) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e
-in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow d0 | (PCons _
-n _) \Rightarrow n])) (PCons h0 d0 hds1) (PCons h d hds0) H4) in ((let H9
-\def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda
-(_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n]))
-(PCons h0 d0 hds1) (PCons h d hds0) H4) in (eq_ind nat h (\lambda (n:
-nat).((eq nat d0 d) \to ((eq PList hds1 hds0) \to ((eq C c0 c2) \to ((eq C c4
-c1) \to ((drop n d0 c0 c3) \to ((drop1 hds1 c3 c4) \to (ex2 C (\lambda (e2:
-C).(drop1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h
-(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
-hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with
-[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i)
-h)]) c2 (CHead e2 (Bind b) (lift1 (match (blt (trans hds0 i) d) with [true
-\Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false
-\Rightarrow (ptrans hds0 i)]) v)))))))))))) (\lambda (H10: (eq nat d0
-d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds1 hds0) \to ((eq C c0 c2)
-\to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1 hds1 c3 c4) \to (ex2 C
-(\lambda (e2: C).(drop1 (match (blt (trans hds0 i) d) with [true \Rightarrow
-(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow
-(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match (blt (trans hds0 i)
-d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans
-hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1 (match (blt (trans hds0 i) d) with
-[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) |
-false \Rightarrow (ptrans hds0 i)]) v))))))))))) (\lambda (H11: (eq PList
-hds1 hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C
-c4 c1) \to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex2 C (\lambda (e2:
-C).(drop1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h
-(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
-hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with
-[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i)
-h)]) c2 (CHead e2 (Bind b) (lift1 (match (blt (trans hds0 i) d) with [true
-\Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false
-\Rightarrow (ptrans hds0 i)]) v)))))))))) (\lambda (H12: (eq C c0
-c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d c c3) \to
-((drop1 hds0 c3 c4) \to (ex2 C (\lambda (e2: C).(drop1 (match (blt (trans
-hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i)))
-(ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) (\lambda (e2:
-C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i)
-| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1
-(match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S
-(trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)])
-v))))))))) (\lambda (H13: (eq C c4 c1)).(eq_ind C c1 (\lambda (c: C).((drop h
-d c2 c3) \to ((drop1 hds0 c3 c) \to (ex2 C (\lambda (e2: C).(drop1 (match
+i c1 (CHead e1 (Bind b) v))).(let H_y \def (drop1_gen_pnil c2 c1 H) in
+(eq_ind_r C c1 (\lambda (c: C).(ex2 C (\lambda (e2: C).(drop1 PNil e2 e1))
+(\lambda (e2: C).(getl i c (CHead e2 (Bind b) v))))) (ex_intro2 C (\lambda
+(e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b)
+v))) e1 (drop1_nil e1) H0) c2 H_y)))))))))) (\lambda (h: nat).(\lambda (d:
+nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2:
+C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v:
+T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda
+(e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i)
+c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))))))))))))).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda
+(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl
+i c1 (CHead e1 (Bind b) v))).(let H_x \def (drop1_gen_pcons c2 c1 hds0 h d
+H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h d c2 c3))
+(\lambda (c3: C).(drop1 hds0 c3 c1)) (ex2 C (\lambda (e2: C).(drop1 (match
(blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans
hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1))
(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow
(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2
(Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h
(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
-hds0 i)]) v)))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1
-hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex2
-C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons h (minus d
-(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2
-e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) |
-false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1
-(match b0 with [true \Rightarrow (PCons h (minus d (S (trans hds0 i)))
-(ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))) (\lambda (x_x:
-bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 i) d) b0) \to
-(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons h
-(minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
+hds0 i)]) v))))) (\lambda (x: C).(\lambda (H3: (drop h d c2 x)).(\lambda (H4:
+(drop1 hds0 x c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0:
+bool).(ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons
+h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow
(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2
(Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans
-hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))))
-(\lambda (H16: (eq bool (blt (trans hds0 i) d) true)).(let H_x \def (H c1 c3
-H15 b e1 v i H1) in (let H17 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1
-(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2
-(Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2: C).(drop1 (PCons h
-(minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl
-(trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0
-i))) (ptrans hds0 i)) v))))) (\lambda (x: C).(\lambda (H18: (drop1 (ptrans
-hds0 i) x e1)).(\lambda (H19: (getl (trans hds0 i) c3 (CHead x (Bind b)
-(lift1 (ptrans hds0 i) v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0
-i) d (blt_lt d (trans hds0 i) H16) c2 c3 h H14 b x (lift1 (ptrans hds0 i) v)
-H19) in (let H20 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans hds0
-i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans
-hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S (trans hds0 i))) e2 x))
-(ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans
-hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b)
-(lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda
-(x0: C).(\lambda (H21: (getl (trans hds0 i) c2 (CHead x0 (Bind b) (lift h
-(minus d (S (trans hds0 i))) (lift1 (ptrans hds0 i) v))))).(\lambda (H22:
-(drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro2 C (\lambda (e2:
+hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v))))))
+(\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0
+i) d) b0) \to (ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow
+(PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow
+(ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true
+\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2
+(CHead e2 (Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d
+(S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)])
+v))))))) (\lambda (H5: (eq bool (blt (trans hds0 i) d) true)).(let H_x0 \def
+(H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in (ex2_ind C (\lambda (e2:
+C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) x
+(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2:
C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1))
(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h
-(minus d (S (trans hds0 i))) (ptrans hds0 i)) v)))) x0 (drop1_cons x0 x h
-(minus d (S (trans hds0 i))) H22 e1 (ptrans hds0 i) H18) H21)))) H20))))))
-H17)))) (\lambda (H16: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def
-(H c1 c3 H15 b e1 v i H1) in (let H17 \def H_x in (ex2_ind C (\lambda (e2:
-C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c3
-(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2:
-C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i)
-h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))) (\lambda (x:
-C).(\lambda (H18: (drop1 (ptrans hds0 i) x e1)).(\lambda (H19: (getl (trans
-hds0 i) c3 (CHead x (Bind b) (lift1 (ptrans hds0 i) v)))).(let H20 \def
-(drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x (Bind b) (lift1
-(ptrans hds0 i) v)) H19) in (ex_intro2 C (\lambda (e2: C).(drop1 (ptrans hds0
-i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind
-b) (lift1 (ptrans hds0 i) v)))) x H18 (H20 (bge_le d (trans hds0 i)
-H16))))))) H17)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq C c0 c2
-H12))) hds1 (sym_eq PList hds1 hds0 H11))) d0 (sym_eq nat d0 d H10))) h0
-(sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList
-(PCons h d hds0)) (refl_equal C c2) (refl_equal C c1))))))))))))))) hds).
+(minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda (x0:
+C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8: (getl (trans
+hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let H_x1 \def
+(drop_getl_trans_lt (trans hds0 i) d (blt_lt d (trans hds0 i) H5) c2 x h H3 b
+x0 (lift1 (ptrans hds0 i) v) H8) in (let H9 \def H_x1 in (ex2_ind C (\lambda
+(e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans
+hds0 i))) (lift1 (ptrans hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S
+(trans hds0 i))) e2 x0)) (ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S
+(trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0
+i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans
+hds0 i)) v))))) (\lambda (x1: C).(\lambda (H10: (getl (trans hds0 i) c2
+(CHead x1 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans hds0
+i) v))))).(\lambda (H11: (drop h (minus d (S (trans hds0 i))) x1
+x0)).(ex_intro2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0
+i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead
+e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i))
+v)))) x1 (drop1_cons x1 x0 h (minus d (S (trans hds0 i))) H11 e1 (ptrans hds0
+i) H7) H10)))) H9)))))) H6)))) (\lambda (H5: (eq bool (blt (trans hds0 i) d)
+false)).(let H_x0 \def (H c1 x H4 b e1 v i H1) in (let H6 \def H_x0 in
+(ex2_ind C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2:
+C).(getl (trans hds0 i) x (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))
+(ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl
+(plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))))
+(\lambda (x0: C).(\lambda (H7: (drop1 (ptrans hds0 i) x0 e1)).(\lambda (H8:
+(getl (trans hds0 i) x (CHead x0 (Bind b) (lift1 (ptrans hds0 i) v)))).(let
+H9 \def (drop_getl_trans_ge (trans hds0 i) c2 x d h H3 (CHead x0 (Bind b)
+(lift1 (ptrans hds0 i) v)) H8) in (ex_intro2 C (\lambda (e2: C).(drop1
+(ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2
+(CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) x0 H7 (H9 (bge_le d (trans
+hds0 i) H5))))))) H6)))) x_x)))))) H2))))))))))))))) hds).
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