(* This file was automatically generated: do not edit *********************)
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/lift/fwd".
-
-include "lift/defs.ma".
+include "LambdaDelta-1/lift/defs.ma".
theorem lift_sort:
\forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort
(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort
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 (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 (_:
+n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0))
+(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let
+H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
+(lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
+ee 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)))) (\lambda (_: (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 (let H1 \def
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
-(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)))).
+(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind
+(le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match
+ee 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))))))) (\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 (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee 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))))))))) t)))).
theorem lift_gen_lref:
\forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T
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))
+(plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus 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
(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
\def
\lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n
-d)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0))
-\to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef n)
-(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 (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 (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
-(lt_le_trans n0 (S (plus n0 h)) d (le_lt_n_Sm n0 (plus n0 h) (le_plus_l n0
-h)) (lt_le_S (plus n0 h) d 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))))).
+d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x
+\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d)
+(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq
+T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind
+(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n
+d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2:
+(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d
+h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus
+d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef
+(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq
+T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S
+n) d h H))) t H4))) H2)) H1)))))))).
theorem lift_gen_lref_false:
\forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n
(P: Prop).P)))))))
\def
\lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
-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 (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 _ _ _)
-\Rightarrow False])) I (lift h d (TSort n0)) H2) in (False_ind P H3)))]) in
-(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 (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 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 (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)))))).
+n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T
+(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d
+h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n)))
+(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land
+(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P
+(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H
+H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n
+h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda
+(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false
+(plus d h) n P H4 H0))) H3)) H2)))))))))).
theorem lift_gen_lref_ge:
\forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall
(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
\def
\lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
-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 (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
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (lift h d (TSort n0)) H1) in (False_ind
-(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 (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 (simpl_lt_plus_r h n d (lt_le_trans
-(plus n h) d (plus d h) H5 (le_plus_l d h)))))) 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))))).
+n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d
+t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in
+(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus
+d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n))
+(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind
+(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda
+(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r
+T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq
+T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d
+(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2))
+(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
+h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
+h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda
+(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n
+h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus
+(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))).
theorem lift_gen_head:
\forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
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 (n:
nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t)
-(lift h d (TSort n)))).(let H0 \def (match H in eq return (\lambda (t0:
-T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T
-(\lambda (y: T).(\lambda (z: T).(eq T (TSort 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
-(H0: (eq T (THead k u t) (lift h d (TSort n)))).(let H1 \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 (lift h d (TSort n)) H0) in (False_ind (ex3_2 T T
+(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda
+(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1
+\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T
(\lambda (y: T).(\lambda (z: T).(eq T (TSort 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))))) H1)))]) in (H0 (refl_equal T (lift h d
-(TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef n)))).(lt_le_e n d
-(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))))) (\lambda (H0: (lt n
-d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead
-k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 in
-eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \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 (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).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef
+n)))).(lt_le_e n d (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))))) (\lambda (H0:
+(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T
+(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def
+(eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) 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))))) 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 (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)) 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))))) 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 (t1: T).(\lambda (_: ((\forall (h: nat).(\forall
-(d: nat).((eq T (THead k u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T 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)))))))))).(\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 (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).(eq T t (lift h (s k d) z))))) H2)))) (\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 (eq_ind T (THead
+k u t) (\lambda (ee: T).(match ee 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) (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
-(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)
+T).(eq T t (lift h (s k d) z))))) H2))))))))) (\lambda (k0: K).(\lambda (t0:
+T).(\lambda (H: ((\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 (t1:
+T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
+(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T 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)))))))))).(\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 (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 (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)) 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 _ t2 _) \Rightarrow t2]))
+(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5
+\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 _ 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 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 (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 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)
-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)))).
+H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let
+H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k
+H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z:
+T).(eq T (THead k1 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)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0:
+nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let
+H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0:
+nat).((eq T (THead k u t2) (lift h0 d0 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 h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift
+h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k
+d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
+(THead k 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 t2 (lift h (s k d)
+z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
+nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
+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 t2 (lift h0 d0 y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10
+(lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0:
+nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0
+t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9
+(lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z))))))
+(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead
+k 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 k d) t1) (lift h (s k d)
+z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0))
+(refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4))
+H3))))))))))) x)))).
theorem lift_gen_bind:
\forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
T).(eq T t (lift h (S d) z)))))))))))
\def
- \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
-(\lambda (t0: T).(\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
-(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 (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T
-(THead (Bind b) u t) (lift h d (TSort n)))).(let H0 \def (match H in eq
-return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort
-n))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort 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 (H0: (eq T (THead (Bind b) u t) (lift h d
-(TSort n)))).(let H1 \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
-(lift h d (TSort n)) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda
-(z: T).(eq T (TSort 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))))) H1)))]) in (H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda
-(n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind
-b) u t) (lift h d (TLRef n)))).(lt_le_e n d (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))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind
-T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H
-(TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 in eq return
-(\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \to (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)))))))) with [refl_equal
-\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) 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))))) 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
-(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 (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) (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))
-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))))) 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
-(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 (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d:
-nat).((eq T (THead (Bind b) u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y:
-T).(\lambda (z: T).(eq T t1 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d
+x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (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)))) (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T x (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 (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 (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 (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 _ _ 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)))).
+h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
+(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
+(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (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)))))) (eq_ind_r T (lift h (S d)
+x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
+(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
+u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d)
+z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro
+T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind
+b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d)
+z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d
+x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
theorem lift_gen_flat:
\forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
T).(eq T t (lift h d z)))))))))))
\def
- \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
-(\lambda (t0: T).(\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) 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
-(n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat
-f) u t) (lift h d (TSort n)))).(let H0 \def (match H in eq return (\lambda
-(t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n))) \to (ex3_2
-T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort 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 (H0: (eq T (THead (Flat f) u t) (lift h d (TSort n)))).(let H1 \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 (lift h d (TSort n)) H0) in (False_ind
-(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Flat f) y
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d
+x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (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))))) H1)))]) in (H0 (refl_equal T
-(lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d (TLRef n)))).(lt_le_e
-n d (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))))) (\lambda (H0: (lt n d)).(let
-H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f)
-u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 in
-eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \to
-(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 (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) 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))))) 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
-(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 (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) (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)) 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))))) 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)
-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 (t1: T).(\lambda
-(_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h
-d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T 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 (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
-(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 (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 _ _ 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 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
+T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T x (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 (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead
+(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t
+(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T
+T (\lambda (y: T).(\lambda (z: T).(eq T t0 (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
-(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 (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)))).
+T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0:
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1)
+(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T
+(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq
+T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d
+x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T
+(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
+(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1))
+(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x
+H1)))))) H0))))))))).
projects / helm.git / blobdiff