\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n
d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true
\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T
-(TLRef n)) (blt n d) (sym_equal bool (blt n d) true (lt_blt d n H)))))).
+(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))).
theorem lift_lref_ge:
\forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T
\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d
n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true
\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h))))
-(refl_equal T (TLRef (plus n h))) (blt n d) (sym_equal bool (blt n d) false
+(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false
(le_bge d n H)))))).
theorem lift_head:
n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H:
(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort
n)) (\lambda (H0: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0))
-(\lambda (t: T).(eq T (TSort n) t)) H (TLRef n0) (lift_lref_lt n0 h d H0)) in
-(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? ?
-t)).((eq T t (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal
-\Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef n0))).(let H2 \def (eq_ind T
-(TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow False])) I (TLRef n0) H1) in (False_ind (eq T (TLRef n0) (TSort
-n)) H2)))]) in (H2 (refl_equal T (TLRef n0)))))) (\lambda (H0: (le d
-n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T
-(TSort n) t)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d H0)) in (let H2 \def
-(match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t
-(TLRef (plus n0 h))) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal
-\Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef (plus n0 h)))).(let H2 \def
-(eq_ind T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_:
+(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d H0))
+in (let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ?
+t0)).((eq T t0 (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with
+[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (TLRef n0))).(let H3
+\def (eq_ind T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow False])) I (TLRef (plus n0 h)) H1) in (False_ind
-(eq T (TLRef n0) (TSort n)) H2)))]) in (H2 (refl_equal T (TLRef (plus n0
-h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n)
-(lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq
-T (TSort n) (lift h d t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T
-(TSort n) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d
-(THead k t0 t1)) (\lambda (t: T).(eq T (TSort n) t)) H1 (THead k (lift h d
-t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2
-in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k (lift
-h d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TSort n))))) with
-[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (THead k (lift h d t0)
-(lift h (s k d) t1)))).(let H3 \def (eq_ind T (TSort n) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k
-t0 t1) (TSort n)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h
-(s k d) t1)))))))))))) t)))).
+(THead _ _ _) \Rightarrow False])) I (TLRef n0) H2) in (False_ind (eq T
+(TLRef n0) (TSort n)) H3)))]) in (H2 (refl_equal T (TLRef n0)))))) (\lambda
+(H0: (le d n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0:
+T).(eq T (TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d H0)) in
+(let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ?
+t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TSort n))))) with
+[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (TLRef (plus n0
+h)))).(let H3 \def (eq_ind T (TSort n) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef (plus n0 h))
+H2) in (False_ind (eq T (TLRef n0) (TSort n)) H3)))]) in (H2 (refl_equal T
+(TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_:
+(((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1:
+T).(\lambda (_: (((eq T (TSort n) (lift h d t1)) \to (eq T t1 (TSort
+n))))).(\lambda (H1: (eq T (TSort n) (lift h d (THead k t0 t1)))).(let H2
+\def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TSort n)
+t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d))
+in (let H3 \def (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ?
+t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T (THead
+k t0 t1) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H3: (eq T
+(TSort n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind
+T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in
+(False_ind (eq T (THead k t0 t1) (TSort n)) H4)))]) in (H3 (refl_equal T
+(THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t)))).
theorem lift_gen_lref:
\forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T
(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i
h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda
(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def
-(eq_ind T (lift h d (TSort n)) (\lambda (t: T).(eq T (TLRef i) t)) H (TSort
+(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort
n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee:
T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or
(land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T
(TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind
-T (lift h d (TLRef n)) (\lambda (t: T).(eq T (TLRef i) t)) H (TLRef n)
+T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n)
(lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e:
T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i |
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n)
-H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n)
-(TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef
+n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef
+n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0
h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus
d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n)
(TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d
-n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t: T).(eq T (TLRef
-i) t)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (f_equal
-T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort
-_) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i]))
-(TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) (\lambda (n0:
-nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le (plus d h)
-n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n (\lambda (n0:
-nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land
-(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) (or_intror (land
-(lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land (le (plus d h)
-(plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) (plus n h)) (eq
-T (TLRef n) (TLRef n)) (plus_le_compat d n h h H0 (le_n h)) (refl_equal T
-(TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i H2)))))))))) (\lambda
-(k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: nat).(\forall (h:
-nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d)
-(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i
-h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: nat).(\forall (h:
-nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to (or (land (lt i d)
-(eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 (TLRef (minus i
-h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda
-(H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T
-(lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef i) t)) H1 (THead k
-(lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef
+i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h)
+(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le
+(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n
+(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n
+h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0)))))
+(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land
+(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h)
+(plus n h)) (eq T (TLRef n) (TLRef n)) (plus_le_compat d n h h H0 (le_n h))
+(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i
+H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d:
+nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to
+(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0
+(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d:
+nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to
+(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1
+(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i:
+nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def
+(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1
+(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let
+H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d)
t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i)))
(land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h)))))
(lift h d (TSort n0)))).(sym_eq T (TLRef n) (TSort n0) H0))) (\lambda (n0:
nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq
T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift
-h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef n0)
+h d (TLRef n0)) (\lambda (t0: T).(eq T (TLRef n) t0)) H0 (TLRef n0)
(lift_lref_lt n0 h d H1)) in (sym_eq T (TLRef n) (TLRef n0) H2))) (\lambda
-(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t:
-T).(eq T (TLRef n) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in
-(let H3 \def (match H2 in eq return (\lambda (t: T).(\lambda (_: (eq ? ?
-t)).((eq T t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with
-[refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef n) (TLRef (plus n0
-h)))).(let H3 \def (f_equal T nat (\lambda (e: T).(match e in T return
-(\lambda (_: T).nat) with [(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n
-| (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H2) in (eq_ind
-nat (plus n0 h) (\lambda (n: nat).(eq T (TLRef n0) (TLRef n))) (let H0 \def
-(eq_ind nat n (\lambda (n: nat).(lt n d)) H (plus n0 h) H3) in (le_false d n0
-(eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d (le_lt_trans n0 (plus
-n0 h) d (le_plus_l n0 h) H0)))) n (sym_eq nat n (plus n0 h) H3))))]) in (H3
-(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0:
-T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef
-n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (eq
-T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d (THead k t0
-t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq
-T (TLRef n) t)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k
-t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t: T).(\lambda
-(_: (eq ? ? t)).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq
-T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H3:
-(eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def
-(eq_ind T (TLRef n) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d)
-t1)) H3) in (False_ind (eq T (THead k t0 t1) (TLRef n)) H4)))]) in (H4
-(refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))).
+(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0:
+T).(eq T (TLRef n) t0)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in
+(let H3 \def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ?
+t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with
+[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef n) (TLRef (plus n0
+h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e in T return
+(\lambda (_: T).nat) with [(TSort _) \Rightarrow n | (TLRef n1) \Rightarrow
+n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H3) in
+(eq_ind nat (plus n0 h) (\lambda (n1: nat).(eq T (TLRef n0) (TLRef n1))) (let
+H5 \def (eq_ind nat n (\lambda (n1: nat).(lt n1 d)) H (plus n0 h) H4) in
+(le_false d n0 (eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d
+(le_lt_trans n0 (plus n0 h) d (le_plus_l n0 h) H5)))) n (sym_eq nat n (plus
+n0 h) H4))))]) in (H3 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k:
+K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T
+t0 (TLRef n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d
+t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d
+(THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda
+(t2: T).(eq T (TLRef n) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1))
+(lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2:
+T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d)
+t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow
+(\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d)
+t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d
+t0) (lift h (s k d) t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n))
+H5)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d)
+t1)))))))))))) t))))).
theorem lift_gen_lref_false:
\forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n
n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(T_ind (\lambda (t0:
T).((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P))) (\lambda (n0:
nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TSort n0)))).(\lambda (P:
-Prop).(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ?
-? t)).((eq T t (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow
+Prop).(let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ?
+? t0)).((eq T t0 (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow
(\lambda (H2: (eq T (TLRef n) (lift h d (TSort n0)))).(let H3 \def (eq_ind T
(TLRef n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
(H2 (refl_equal T (lift h d (TSort n0)))))))) (\lambda (n0: nat).(\lambda
(H1: (eq T (TLRef n) (lift h d (TLRef n0)))).(\lambda (P: Prop).(lt_le_e n0 d
P (\lambda (H2: (lt n0 d)).(let H3 \def (eq_ind T (lift h d (TLRef n0))
-(\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef n0) (lift_lref_lt n0 h d H2))
-in (let H4 \def (match H3 in eq return (\lambda (t: T).(\lambda (_: (eq ? ?
-t)).((eq T t (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda (H3:
-(eq T (TLRef n) (TLRef n0))).(let H4 \def (f_equal T nat (\lambda (e:
+(\lambda (t0: T).(eq T (TLRef n) t0)) H1 (TLRef n0) (lift_lref_lt n0 h d H2))
+in (let H4 \def (match H3 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ?
+t0)).((eq T t0 (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda
+(H4: (eq T (TLRef n) (TLRef n0))).(let H5 \def (f_equal T nat (\lambda (e:
T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0)
-H3) in (eq_ind nat n0 (\lambda (_: nat).P) (let H1 \def (eq_ind_r nat n0
-(\lambda (n: nat).(lt n d)) H2 n H4) in (le_false d n P H H1)) n (sym_eq nat
-n n0 H4))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d
-n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T
-(TLRef n) t)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4
-\def (match H3 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t
-(TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H3: (eq
-T (TLRef n) (TLRef (plus n0 h)))).(let H4 \def (f_equal T nat (\lambda (e:
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
-(plus n0 h)) H3) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let H1 \def
-(eq_ind nat n (\lambda (n: nat).(lt n (plus d h))) H0 (plus n0 h) H4) in
-(le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H1)))) n (sym_eq
-nat n (plus n0 h) H4))))]) in (H4 (refl_equal T (TLRef (plus n0 h)))))))))))
-(\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d
-t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: (((eq T
-(TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: (eq T
-(TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def
-(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef n) t)) H3
+(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
+n0) H4) in (eq_ind nat n0 (\lambda (_: nat).P) (let H6 \def (eq_ind_r nat n0
+(\lambda (n1: nat).(lt n1 d)) H2 n H5) in (le_false d n P H H6)) n (sym_eq
+nat n n0 H5))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d
+n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T
+(TLRef n) t0)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4
+\def (match H3 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T
+t0 (TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H4:
+(eq T (TLRef n) (TLRef (plus n0 h)))).(let H5 \def (f_equal T nat (\lambda
+(e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow
+n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n)
+(TLRef (plus n0 h)) H4) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let
+H6 \def (eq_ind nat n (\lambda (n1: nat).(lt n1 (plus d h))) H0 (plus n0 h)
+H5) in (le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H6)))) n
+(sym_eq nat n (plus n0 h) H5))))]) in (H4 (refl_equal T (TLRef (plus n0
+h))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n)
+(lift h d t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_:
+(((eq T (TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3:
+(eq T (TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def
+(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef n) t2)) H3
(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let
-H5 \def (match H4 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq
-T t (THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with [refl_equal
-\Rightarrow (\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k
-d) t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d
-t0) (lift h (s k d) t1)) H4) in (False_ind P H5)))]) in (H5 (refl_equal T
-(THead k (lift h d t0) (lift h (s k d) t1))))))))))))) t)))))).
+H5 \def (match H4 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ?
+t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with
+[refl_equal \Rightarrow (\lambda (H5: (eq T (TLRef n) (THead k (lift h d t0)
+(lift h (s k d) t1)))).(let H6 \def (eq_ind T (TLRef n) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead k (lift h d t0) (lift h (s k d) t1)) H5) in (False_ind P H6)))]) in
+(H5 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))))
+t)))))).
theorem lift_gen_lref_ge:
\forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall
n)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef (plus n h)) (lift h
d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T
(TLRef (plus n h)) (lift h d (TSort n0)))).(let H1 \def (match H0 in eq
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (lift h d (TSort
+return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort
n0))) \to (eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow
(\lambda (H1: (eq T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H2 \def
(eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e in T return (\lambda
(eq T (TSort n0) (TLRef n)) H2)))]) in (H1 (refl_equal T (lift h d (TSort
n0))))))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d
(TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0
-d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef
-(plus n h)) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 \def
-(match H2 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t
-(TLRef n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow
-(\lambda (H2: (eq T (TLRef (plus n h)) (TLRef n0))).(let H3 \def (f_equal T
-nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
-\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m:
-nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in
-plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec
-plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O
-\Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef
-(plus n h)) (TLRef n0) H2) in (eq_ind nat (plus n h) (\lambda (n0: nat).(eq T
-(TLRef n0) (TLRef n))) (let H0 \def (eq_ind_r nat n0 (\lambda (n: nat).(lt n
-d)) H1 (plus n h) H3) in (le_false d n (eq T (TLRef (plus n h)) (TLRef n)) H
-(lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l n h) H0)))) n0 H3)))]) in
-(H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le d n0)).(let H2 \def
-(eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef (plus n h)) t))
-H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2 in
-eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (TLRef (plus n0
-h))) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda
-(H2: (eq T (TLRef (plus n h)) (TLRef (plus n0 h)))).(let H3 \def (f_equal T
-nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
-\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m:
-nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in
-plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec
-plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O
-\Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef
-(plus n h)) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n h) (\lambda (_:
-nat).(eq T (TLRef n0) (TLRef n))) (f_equal nat T TLRef n0 n (simpl_plus_r h
-n0 n (sym_eq nat (plus n h) (plus n0 h) H3))) (plus n0 h) H3)))]) in (H3
-(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0:
-T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef
-n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d
-t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift
-h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1))
-(\lambda (t: T).(eq T (TLRef (plus n h)) t)) H2 (THead k (lift h d t0) (lift
-h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k (lift h d
-t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with
-[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (THead k (lift
-h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind T (TLRef (plus n h))
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in (False_ind (eq
-T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T (THead k (lift h d
-t0) (lift h (s k d) t1)))))))))))) t))))).
+d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T
+(TLRef (plus n h)) t0)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3
+\def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T
+t0 (TLRef n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal
+\Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (TLRef n0))).(let H4 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort _) \Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def
+(\lambda (m: nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S
+(plus p m))])) in plus) n h) | (TLRef n1) \Rightarrow n1 | (THead _ _ _)
+\Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def (\lambda (m:
+nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in
+plus) n h)])) (TLRef (plus n h)) (TLRef n0) H3) in (eq_ind nat (plus n h)
+(\lambda (n1: nat).(eq T (TLRef n1) (TLRef n))) (let H5 \def (eq_ind_r nat n0
+(\lambda (n1: nat).(lt n1 d)) H1 (plus n h) H4) in (le_false d n (eq T (TLRef
+(plus n h)) (TLRef n)) H (lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l
+n h) H5)))) n0 H4)))]) in (H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le
+d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T
+(TLRef (plus n h)) t0)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in
+(let H3 \def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ?
+t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with
+[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (TLRef (plus
+n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e in T return
+(\lambda (_: T).nat) with [(TSort _) \Rightarrow ((let rec plus (n1: nat) on
+n1: (nat \to nat) \def (\lambda (m: nat).(match n1 with [O \Rightarrow m | (S
+p) \Rightarrow (S (plus p m))])) in plus) n h) | (TLRef n1) \Rightarrow n1 |
+(THead _ _ _) \Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def
+(\lambda (m: nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S
+(plus p m))])) in plus) n h)])) (TLRef (plus n h)) (TLRef (plus n0 h)) H3) in
+(eq_ind nat (plus n h) (\lambda (_: nat).(eq T (TLRef n0) (TLRef n)))
+(f_equal nat T TLRef n0 n (simpl_plus_r h n0 n (sym_eq nat (plus n h) (plus
+n0 h) H4))) (plus n0 h) H4)))]) in (H3 (refl_equal T (TLRef (plus n0
+h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef
+(plus n h)) (lift h d t0)) \to (eq T t0 (TLRef n))))).(\lambda (t1:
+T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t1)) \to (eq T t1 (TLRef
+n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift h d (THead k t0
+t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq
+T (TLRef (plus n h)) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1))
+(lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2:
+T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d)
+t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow
+(\lambda (H4: (eq T (TLRef (plus n h)) (THead k (lift h d t0) (lift h (s k d)
+t1)))).(let H5 \def (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e in
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d
+t0) (lift h (s k d) t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n))
+H5)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d)
+t1)))))))))))) t))))).
theorem lift_gen_head:
\forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z))))
(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with [refl_equal
-\Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef n))).(let H2 \def
+\Rightarrow (\lambda (H2: (eq T (THead k u t) (TLRef n))).(let H3 \def
(eq_ind T (THead k u t) (\lambda (e: T).(match e in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T
(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z))))) H2)))]) in (H2 (refl_equal T (TLRef n))))))
+T).(eq T t (lift h (s k d) z))))) H3)))]) in (H2 (refl_equal T (TLRef n))))))
(\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda
(t0: T).(eq T (THead k u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d
H0)) in (let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq
? ? t0)).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y:
T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda
(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
-h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead k
-u t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead k u t) (\lambda (e:
+h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead k
+u t) (TLRef (plus n h)))).(let H3 \def (eq_ind T (THead k u t) (\lambda (e:
T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
-(TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z:
+(TLRef (plus n h)) H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z:
T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d)
-z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda
+z))))) H3)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda
(k0: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d:
nat).((eq T (THead k u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y:
T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_:
T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s
k d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T
(THead k u t) (lift h d (THead k0 t0 t1)))).(let H2 \def (eq_ind T (lift h d
-(THead k0 t0 t1)) (\lambda (t0: T).(eq T (THead k u t) t0)) H1 (THead k0
+(THead k0 t0 t1)) (\lambda (t2: T).(eq T (THead k u t) t2)) H1 (THead k0
(lift h d t0) (lift h (s k0 d) t1)) (lift_head k0 t0 t1 h d)) in (let H3 \def
(match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq T t2
(THead k0 (lift h d t0) (lift h (s k0 d) t1))) \to (ex3_2 T T (\lambda (y:
T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y:
T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
T).(eq T t (lift h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda
-(H2: (eq T (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)))).(let
-H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t)
-\Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1))
-H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return
+(H3: (eq T (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)))).(let
+H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2)
+\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1))
+H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return
(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
-(THead _ t _) \Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h
-(s k0 d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e in T
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u t) (THead k0 (lift
-h d t0) (lift h (s k0 d) t1)) H2) in (eq_ind K k0 (\lambda (k: K).((eq T u
-(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y:
+(THead _ t2 _) \Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift
+h (s k0 d) t1)) H3) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e in
+T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0
+(lift h d t0) (lift h (s k0 d) t1)) H3) in (eq_ind K k0 (\lambda (k1: K).((eq
+T u (lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda
+(y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k1 y z)))) (\lambda (y:
T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h (s k d) z)))))))) (\lambda (H6: (eq T u (lift h d
+T).(eq T t (lift h (s k1 d) z)))))))) (\lambda (H7: (eq T u (lift h d
t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s k0 d) t1))
\to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead
k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda
-(_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H7: (eq T
-t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t:
+(_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H8: (eq T
+t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t2:
T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead
k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k0 d) z))))))
(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1)
(THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h
d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k0 d) t1) (lift h (s
k0 d) z)))) t0 t1 (refl_equal T (THead k0 t0 t1)) (refl_equal T (lift h d
t0)) (refl_equal T (lift h (s k0 d) t1))) t (sym_eq T t (lift h (s k0 d) t1)
-H7))) u (sym_eq T u (lift h d t0) H6))) k (sym_eq K k k0 H5))) H4)) H3)))])
+H8))) u (sym_eq T u (lift h d t0) H7))) k (sym_eq K k k0 H6))) H5)) H4)))])
in (H3 (refl_equal T (THead k0 (lift h d t0) (lift h (s k0 d)
t1)))))))))))))) x)))).
T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z))))
(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal
-\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef n))).(let H2 \def
+\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (TLRef n))).(let H3 \def
(eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in T return (\lambda
(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T
+| (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T
(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z))))
(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in (H2 (refl_equal T
+T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H3)))]) in (H2 (refl_equal T
(TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d
(TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef (plus n
h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda
T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z))))
(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal
-\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef (plus n
-h)))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in
+\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (TLRef (plus n
+h)))).(let H3 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in
T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h))
-H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
+H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H3)))]) in
(H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0:
T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u
t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0
(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
h (S d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T
(THead (Bind b) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T
-(lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0))
+(lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (THead (Bind b) u t) t2))
H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in
(let H3 \def (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ?
t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T
(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Bind b) y z))))
(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal
-\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (THead k (lift h d t0)
-(lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in
+\Rightarrow (\lambda (H3: (eq T (THead (Bind b) u t) (THead k (lift h d t0)
+(lift h (s k d) t1)))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in
T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
-\Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u t) (THead k
-(lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u
-| (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u
-t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H5 \def (f_equal
-T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | (THead k _ _)
-\Rightarrow k])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d)
-t1)) H2) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u (lift h d t0)) \to
-((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z:
-T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_:
-T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S
-d) z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind T (lift h d t0)
-(\lambda (t2: T).((eq T t (lift h (s (Bind b) d) t1)) \to (ex3_2 T T (\lambda
-(y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) (\lambda (H7: (eq T t (lift
-h (s (Bind b) d) t1))).(eq_ind T (lift h (s (Bind b) d) t1) (\lambda (t:
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1)
-(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0)
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))
-(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1)
-(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0)
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s (Bind b) d)
-t1) (lift h (S d) z)))) t0 t1 (refl_equal T (THead (Bind b) t0 t1))
-(refl_equal T (lift h d t0)) (refl_equal T (lift h (S d) t1))) t (sym_eq T t
-(lift h (s (Bind b) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5))
-H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d)
-t1)))))))))))))) x)))).
+\Rightarrow t | (THead _ _ t2) \Rightarrow t2])) (THead (Bind b) u t) (THead
+k (lift h d t0) (lift h (s k d) t1)) H3) in ((let H5 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2]))
+(THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let
+H6 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) |
+(THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u t) (THead k (lift h d t0)
+(lift h (s k d) t1)) H3) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u
+(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead (Bind b) y z)))) (\lambda
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (S d) z)))))))) (\lambda (H7: (eq T u (lift h d
+t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Bind b)
+d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b)
+t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift
+h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))
+(\lambda (H8: (eq T t (lift h (s (Bind b) d) t1))).(eq_ind T (lift h (s (Bind
+b) d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
+(THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
+t2 (lift h (S d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq
+T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
+(lift h (s (Bind b) d) t1) (lift h (S d) z)))) t0 t1 (refl_equal T (THead
+(Bind b) t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h (S d)
+t1))) t (sym_eq T t (lift h (s (Bind b) d) t1) H8))) u (sym_eq T u (lift h d
+t0) H7))) k H6)) H5)) H4)))]) in (H3 (refl_equal T (THead k (lift h d t0)
+(lift h (s k d) t1)))))))))))))) x)))).
theorem lift_gen_flat:
\forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y
z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow
-(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef n))).(let H2 \def (eq_ind T
+(\lambda (H2: (eq T (THead (Flat f) u t) (TLRef n))).(let H3 \def (eq_ind T
(THead (Flat f) u t) (\lambda (e: T).(match e in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T
(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z))))
(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T
+T).(\lambda (z: T).(eq T t (lift h d z))))) H3)))]) in (H2 (refl_equal T
(TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d
(TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef (plus n
h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda
T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z))))
(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow
-(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H2 \def
+(\lambda (H2: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H3 \def
(eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e in T return (\lambda
(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind
+| (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H2) in (False_ind
(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y
z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T
+T).(\lambda (z: T).(eq T t (lift h d z))))) H3)))]) in (H2 (refl_equal T
(TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_:
((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d
t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f)
(_: T).(\lambda (z: T).(eq T t (lift h d z)))))))))).(\lambda (h:
nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead (Flat f) u t) (lift h d
(THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda
-(t0: T).(eq T (THead (Flat f) u t) t0)) H1 (THead k (lift h d t0) (lift h (s
+(t2: T).(eq T (THead (Flat f) u t) t2)) H1 (THead k (lift h d t0) (lift h (s
k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 in eq return
(\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0)
(lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
(THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))))
-with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead (Flat f) u t) (THead
-k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e:
+with [refl_equal \Rightarrow (\lambda (H3: (eq T (THead (Flat f) u t) (THead
+k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (f_equal T T (\lambda (e:
T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t |
-(TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Flat f) u t)
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t]))
-(THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let
-H5 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+(TLRef _) \Rightarrow t | (THead _ _ t2) \Rightarrow t2])) (THead (Flat f) u
+t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let H5 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2]))
+(THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let
+H6 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
with [(TSort _) \Rightarrow (Flat f) | (TLRef _) \Rightarrow (Flat f) |
-(THead k _ _) \Rightarrow k])) (THead (Flat f) u t) (THead k (lift h d t0)
-(lift h (s k d) t1)) H2) in (eq_ind K (Flat f) (\lambda (k: K).((eq T u (lift
-h d t0)) \to ((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T t (lift h d z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind
+(THead k0 _ _) \Rightarrow k0])) (THead (Flat f) u t) (THead k (lift h d t0)
+(lift h (s k d) t1)) H3) in (eq_ind K (Flat f) (\lambda (k0: K).((eq T u
+(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead (Flat f) y z)))) (\lambda
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h d z)))))))) (\lambda (H7: (eq T u (lift h d t0))).(eq_ind
T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Flat f) d) t1)) \to
(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1)
(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d
y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) (\lambda
-(H7: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d)
-t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
+(H8: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d)
+t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
+(Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
+(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift
+h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
(Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
-(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h
-d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat
-f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift
-h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s
-(Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0 t1))
-(refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t
-(lift h (s (Flat f) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5))
-H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d)
+(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h
+(s (Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0
+t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t
+(lift h (s (Flat f) d) t1) H8))) u (sym_eq T u (lift h d t0) H7))) k H6))
+H5)) H4)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d)
t1)))))))))))))) x)))).