X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Flift%2Fprops.ma;h=610e02018b2285469a6cbe7ab51de0f56f5fff97;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=97cd58e0ea385afc7221109be6e59778430e2899;hpb=6d1bb99e7f355d826c07285ba46b6b13a4abaefc;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma b/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma
index 97cd58e0e..610e02018 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/lift/props.ma
@@ -20,613 +20,304 @@ include "basic_1/s/props.ma".
 
 include "basic_1/T/fwd.ma".
 
-theorem lift_sort:
+lemma lift_sort:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort 
 n)) (TSort n))))
 \def
- \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(let TMP_1 \def (TSort 
-n) in (refl_equal T TMP_1)))).
+ \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort 
+n)))).
 
-theorem lift_lref_lt:
+lemma lift_lref_lt:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T 
 (lift h d (TLRef n)) (TLRef n)))))
 \def
  \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n 
-d)).(let TMP_4 \def (\lambda (b: bool).(let TMP_1 \def (match b with [true 
-\Rightarrow n | false \Rightarrow (plus n h)]) in (let TMP_2 \def (TLRef 
-TMP_1) in (let TMP_3 \def (TLRef n) in (eq T TMP_2 TMP_3))))) in (let TMP_5 
-\def (TLRef n) in (let TMP_6 \def (refl_equal T TMP_5) in (let TMP_7 \def 
-(blt n d) in (let TMP_8 \def (blt n d) in (let TMP_9 \def (lt_blt d n H) in 
-(let TMP_10 \def (sym_eq bool TMP_8 true TMP_9) in (eq_ind bool true TMP_4 
-TMP_6 TMP_7 TMP_10))))))))))).
+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_eq bool (blt n d) true (lt_blt d n H)))))).
 
-theorem lift_lref_ge:
+lemma lift_lref_ge:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T 
 (lift h d (TLRef n)) (TLRef (plus n h))))))
 \def
  \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d 
-n)).(let TMP_5 \def (\lambda (b: bool).(let TMP_1 \def (match b with [true 
-\Rightarrow n | false \Rightarrow (plus n h)]) in (let TMP_2 \def (TLRef 
-TMP_1) in (let TMP_3 \def (plus n h) in (let TMP_4 \def (TLRef TMP_3) in (eq 
-T TMP_2 TMP_4)))))) in (let TMP_6 \def (plus n h) in (let TMP_7 \def (TLRef 
-TMP_6) in (let TMP_8 \def (refl_equal T TMP_7) in (let TMP_9 \def (blt n d) 
-in (let TMP_10 \def (blt n d) in (let TMP_11 \def (le_bge d n H) in (let 
-TMP_12 \def (sym_eq bool TMP_10 false TMP_11) in (eq_ind bool false TMP_5 
-TMP_8 TMP_9 TMP_12)))))))))))).
+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_eq bool (blt n d) false 
+(le_bge d n H)))))).
 
-theorem lift_head:
+lemma lift_head:
  \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
 (d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d) 
 t)))))))
 \def
  \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
-(d: nat).(let TMP_1 \def (lift h d u) in (let TMP_2 \def (s k d) in (let 
-TMP_3 \def (lift h TMP_2 t) in (let TMP_4 \def (THead k TMP_1 TMP_3) in 
-(refl_equal T TMP_4))))))))).
+(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
 
-theorem lift_bind:
+lemma lift_bind:
  \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
 (d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u) 
 (lift h (S d) t)))))))
 \def
  \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
-(d: nat).(let TMP_1 \def (Bind b) in (let TMP_2 \def (lift h d u) in (let 
-TMP_3 \def (S d) in (let TMP_4 \def (lift h TMP_3 t) in (let TMP_5 \def 
-(THead TMP_1 TMP_2 TMP_4) in (refl_equal T TMP_5)))))))))).
+(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
 
-theorem lift_flat:
+lemma lift_flat:
  \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
 (d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u) 
 (lift h d t)))))))
 \def
  \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
-(d: nat).(let TMP_1 \def (Flat f) in (let TMP_2 \def (lift h d u) in (let 
-TMP_3 \def (lift h d t) in (let TMP_4 \def (THead TMP_1 TMP_2 TMP_3) in 
-(refl_equal T TMP_4))))))))).
+(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
 
-theorem thead_x_lift_y_y:
+lemma thead_x_lift_y_y:
  \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall 
 (d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
 \def
- \lambda (k: K).(\lambda (t: T).(let TMP_1 \def (\lambda (t0: T).(\forall (v: 
+ \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v: 
 T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0) 
-\to (\forall (P: Prop).P)))))) in (let TMP_7 \def (\lambda (n: nat).(\lambda 
-(v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v 
-(lift h d (TSort n))) (TSort n))).(\lambda (P: Prop).(let TMP_2 \def (TSort 
-n) in (let TMP_3 \def (lift h d TMP_2) in (let TMP_4 \def (THead k v TMP_3) 
-in (let TMP_5 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) in 
-(let TMP_6 \def (TSort n) in (let H0 \def (eq_ind T TMP_4 TMP_5 I TMP_6 H) in 
-(False_ind P H0))))))))))))) in (let TMP_13 \def (\lambda (n: nat).(\lambda 
-(v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v 
-(lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let TMP_8 \def (TLRef 
-n) in (let TMP_9 \def (lift h d TMP_8) in (let TMP_10 \def (THead k v TMP_9) 
-in (let TMP_11 \def (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) in 
-(let TMP_12 \def (TLRef n) in (let H0 \def (eq_ind T TMP_10 TMP_11 I TMP_12 
-H) in (False_ind P H0))))))))))))) in (let TMP_72 \def (\lambda (k0: 
-K).(\lambda (t0: T).(\lambda (_: ((\forall (v: T).(\forall (h: nat).(\forall 
-(d: nat).((eq T (THead k v (lift h d t0)) t0) \to (\forall (P: 
-Prop).P))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (v: T).(\forall (h: 
-nat).(\forall (d: nat).((eq T (THead k v (lift h d t1)) t1) \to (\forall (P: 
-Prop).P))))))).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
-(H1: (eq T (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 
-t1))).(\lambda (P: Prop).(let TMP_14 \def (\lambda (e: T).(match e with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) 
-\Rightarrow k1])) in (let TMP_15 \def (THead k0 t0 t1) in (let TMP_16 \def 
-(lift h d TMP_15) in (let TMP_17 \def (THead k v TMP_16) in (let TMP_18 \def 
-(THead k0 t0 t1) in (let H2 \def (f_equal T K TMP_14 TMP_17 TMP_18 H1) in 
-(let TMP_19 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow v | 
-(TLRef _) \Rightarrow v | (THead _ t2 _) \Rightarrow t2])) in (let TMP_20 
-\def (THead k0 t0 t1) in (let TMP_21 \def (lift h d TMP_20) in (let TMP_22 
-\def (THead k v TMP_21) in (let TMP_23 \def (THead k0 t0 t1) in (let H3 \def 
-(f_equal T T TMP_19 TMP_22 TMP_23 H1) in (let TMP_54 \def (\lambda (e: 
-T).(match e with [(TSort _) \Rightarrow (let TMP_43 \def lref_map in (let 
-TMP_44 \def (\lambda (x: nat).(plus x h)) in (let TMP_45 \def (TMP_43 TMP_44 
-d t0) in (let TMP_50 \def lref_map in (let TMP_51 \def (\lambda (x: 
-nat).(plus x h)) in (let TMP_52 \def (s k0 d) in (let TMP_53 \def (TMP_50 
-TMP_51 TMP_52 t1) in (THead k0 TMP_45 TMP_53)))))))) | (TLRef _) \Rightarrow 
-(let TMP_28 \def lref_map in (let TMP_29 \def (\lambda (x: nat).(plus x h)) 
-in (let TMP_30 \def (TMP_28 TMP_29 d t0) in (let TMP_35 \def lref_map in (let 
-TMP_36 \def (\lambda (x: nat).(plus x h)) in (let TMP_37 \def (s k0 d) in 
-(let TMP_38 \def (TMP_35 TMP_36 TMP_37 t1) in (THead k0 TMP_30 TMP_38)))))))) 
-| (THead _ _ t2) \Rightarrow t2])) in (let TMP_55 \def (THead k0 t0 t1) in 
-(let TMP_56 \def (lift h d TMP_55) in (let TMP_57 \def (THead k v TMP_56) in 
-(let TMP_58 \def (THead k0 t0 t1) in (let H4 \def (f_equal T T TMP_54 TMP_57 
-TMP_58 H1) in (let TMP_70 \def (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K 
-k k0)).(let TMP_59 \def (\lambda (k1: K).(\forall (v0: T).(\forall (h0: 
-nat).(\forall (d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall 
-(P0: Prop).P0)))))) in (let H7 \def (eq_ind K k TMP_59 H0 k0 H6) in (let 
-TMP_60 \def (THead k0 t0 t1) in (let TMP_61 \def (lift h d TMP_60) in (let 
-TMP_62 \def (\lambda (t2: T).(eq T t2 t1)) in (let TMP_63 \def (lift h d t0) 
-in (let TMP_64 \def (s k0 d) in (let TMP_65 \def (lift h TMP_64 t1) in (let 
-TMP_66 \def (THead k0 TMP_63 TMP_65) in (let TMP_67 \def (lift_head k0 t0 t1 
-h d) in (let H8 \def (eq_ind T TMP_61 TMP_62 H4 TMP_66 TMP_67) in (let TMP_68 
-\def (lift h d t0) in (let TMP_69 \def (s k0 d) in (H7 TMP_68 h TMP_69 H8 
-P)))))))))))))))) in (let TMP_71 \def (TMP_70 H3) in (TMP_71 
-H2))))))))))))))))))))))))))))))) in (T_ind TMP_1 TMP_7 TMP_13 TMP_72 t)))))).
+\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n))) 
+(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d 
+(TSort n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
+H) in (False_ind P H0)))))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TLRef n))) 
+(TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d 
+(TLRef n))) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
+H) in (False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: 
+((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift 
+h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0: 
+((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift 
+h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0 
+t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (lift h d (THead 
+k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead 
+_ t2 _) \Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 
+t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow (THead k0 (lref_map (\lambda (x: nat).(plus x h)) d 
+t0) (lref_map (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) 
+\Rightarrow (THead k0 (lref_map (\lambda (x: nat).(plus x h)) d t0) (lref_map 
+(\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2) \Rightarrow 
+t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in 
+(\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind K k 
+(\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall (d0: nat).((eq 
+T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0: Prop).P0)))))) H0 k0 
+H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t2: 
+T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) (lift_head 
+k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) H3)) H2)))))))))))) 
+t)).
 
-theorem lift_r:
+lemma lift_r:
  \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
 \def
- \lambda (t: T).(let TMP_2 \def (\lambda (t0: T).(\forall (d: nat).(let TMP_1 
-\def (lift O d t0) in (eq T TMP_1 t0)))) in (let TMP_4 \def (\lambda (n: 
-nat).(\lambda (_: nat).(let TMP_3 \def (TSort n) in (refl_equal T TMP_3)))) 
-in (let TMP_31 \def (\lambda (n: nat).(\lambda (d: nat).(let TMP_5 \def 
-(TLRef n) in (let TMP_6 \def (lift O d TMP_5) in (let TMP_7 \def (TLRef n) in 
-(let TMP_8 \def (eq T TMP_6 TMP_7) in (let TMP_17 \def (\lambda (H: (lt n 
-d)).(let TMP_9 \def (TLRef n) in (let TMP_11 \def (\lambda (t0: T).(let 
-TMP_10 \def (TLRef n) in (eq T t0 TMP_10))) in (let TMP_12 \def (TLRef n) in 
-(let TMP_13 \def (refl_equal T TMP_12) in (let TMP_14 \def (TLRef n) in (let 
-TMP_15 \def (lift O d TMP_14) in (let TMP_16 \def (lift_lref_lt n O d H) in 
-(eq_ind_r T TMP_9 TMP_11 TMP_13 TMP_15 TMP_16))))))))) in (let TMP_30 \def 
-(\lambda (H: (le d n)).(let TMP_18 \def (plus n O) in (let TMP_19 \def (TLRef 
-TMP_18) in (let TMP_21 \def (\lambda (t0: T).(let TMP_20 \def (TLRef n) in 
-(eq T t0 TMP_20))) in (let TMP_22 \def (plus n O) in (let TMP_23 \def (plus n 
-O) in (let TMP_24 \def (plus_n_O n) in (let TMP_25 \def (sym_eq nat n TMP_23 
-TMP_24) in (let TMP_26 \def (f_equal nat T TLRef TMP_22 n TMP_25) in (let 
-TMP_27 \def (TLRef n) in (let TMP_28 \def (lift O d TMP_27) in (let TMP_29 
-\def (lift_lref_ge n O d H) in (eq_ind_r T TMP_19 TMP_21 TMP_26 TMP_28 
-TMP_29))))))))))))) in (lt_le_e n d TMP_8 TMP_17 TMP_30))))))))) in (let 
-TMP_61 \def (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (d: 
-nat).(eq T (lift O d t0) t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: 
-nat).(eq T (lift O d t1) t1)))).(\lambda (d: nat).(let TMP_32 \def (lift O d 
-t0) in (let TMP_33 \def (s k d) in (let TMP_34 \def (lift O TMP_33 t1) in 
-(let TMP_35 \def (THead k TMP_32 TMP_34) in (let TMP_37 \def (\lambda (t2: 
-T).(let TMP_36 \def (THead k t0 t1) in (eq T t2 TMP_36))) in (let TMP_38 \def 
-(THead k t0 t1) in (let TMP_39 \def (lift O d t0) in (let TMP_40 \def (s k d) 
-in (let TMP_41 \def (lift O TMP_40 t1) in (let TMP_42 \def (THead k TMP_39 
-TMP_41) in (let TMP_43 \def (lift O d t0) in (let TMP_44 \def (s k d) in (let 
-TMP_45 \def (lift O TMP_44 t1) in (let TMP_46 \def (THead k TMP_43 TMP_45) in 
-(let TMP_47 \def (THead k t0 t1) in (let TMP_48 \def (lift O d t0) in (let 
-TMP_49 \def (s k d) in (let TMP_50 \def (lift O TMP_49 t1) in (let TMP_51 
-\def (refl_equal K k) in (let TMP_52 \def (H d) in (let TMP_53 \def (s k d) 
-in (let TMP_54 \def (H0 TMP_53) in (let TMP_55 \def (f_equal3 K T T T THead k 
-k TMP_48 t0 TMP_50 t1 TMP_51 TMP_52 TMP_54) in (let TMP_56 \def (sym_eq T 
-TMP_46 TMP_47 TMP_55) in (let TMP_57 \def (sym_eq T TMP_38 TMP_42 TMP_56) in 
-(let TMP_58 \def (THead k t0 t1) in (let TMP_59 \def (lift O d TMP_58) in 
-(let TMP_60 \def (lift_head k t0 t1 O d) in (eq_ind_r T TMP_35 TMP_37 TMP_57 
-TMP_59 TMP_60))))))))))))))))))))))))))))))))))) in (T_ind TMP_2 TMP_4 TMP_31 
-TMP_61 t))))).
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0) 
+t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda 
+(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n)) 
+(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef 
+n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H))) 
+(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T 
+t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) 
+(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k: 
+K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) 
+t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) 
+t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) 
+t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq T (THead k t0 t1) 
+(THead k (lift O d t0) (lift O (s k d) t1)) (sym_eq T (THead k (lift O d t0) 
+(lift O (s k d) t1)) (THead k t0 t1) (f_equal3 K T T T THead k k (lift O d 
+t0) t0 (lift O (s k d) t1) t1 (refl_equal K k) (H d) (H0 (s k d))))) (lift O 
+d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t).
 
-theorem lift_lref_gt:
+lemma lift_lref_gt:
  \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef 
 (pred n))) (TLRef n))))
 \def
- \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(let TMP_1 \def 
-(pred n) in (let TMP_2 \def (S O) in (let TMP_3 \def (plus TMP_1 TMP_2) in 
-(let TMP_4 \def (TLRef TMP_3) in (let TMP_6 \def (\lambda (t: T).(let TMP_5 
-\def (TLRef n) in (eq T t TMP_5))) in (let TMP_7 \def (S O) in (let TMP_8 
-\def (pred n) in (let TMP_9 \def (plus TMP_7 TMP_8) in (let TMP_12 \def 
-(\lambda (n0: nat).(let TMP_10 \def (TLRef n0) in (let TMP_11 \def (TLRef n) 
-in (eq T TMP_10 TMP_11)))) in (let TMP_15 \def (\lambda (n0: nat).(let TMP_13 
-\def (TLRef n0) in (let TMP_14 \def (TLRef n) in (eq T TMP_13 TMP_14)))) in 
-(let TMP_16 \def (TLRef n) in (let TMP_17 \def (refl_equal T TMP_16) in (let 
-TMP_18 \def (pred n) in (let TMP_19 \def (S TMP_18) in (let TMP_20 \def 
-(S_pred n d H) in (let TMP_21 \def (eq_ind nat n TMP_15 TMP_17 TMP_19 TMP_20) 
-in (let TMP_22 \def (pred n) in (let TMP_23 \def (S O) in (let TMP_24 \def 
-(plus TMP_22 TMP_23) in (let TMP_25 \def (S O) in (let TMP_26 \def (pred n) 
-in (let TMP_27 \def (plus_sym TMP_25 TMP_26) in (let TMP_28 \def (eq_ind nat 
-TMP_9 TMP_12 TMP_21 TMP_24 TMP_27) in (let TMP_29 \def (S O) in (let TMP_30 
-\def (pred n) in (let TMP_31 \def (TLRef TMP_30) in (let TMP_32 \def (lift 
-TMP_29 d TMP_31) in (let TMP_33 \def (pred n) in (let TMP_34 \def (S O) in 
-(let TMP_35 \def (pred n) in (let TMP_37 \def (\lambda (n0: nat).(let TMP_36 
-\def (S d) in (le TMP_36 n0))) in (let TMP_38 \def (pred n) in (let TMP_39 
-\def (S TMP_38) in (let TMP_40 \def (S_pred n d H) in (let TMP_41 \def 
-(eq_ind nat n TMP_37 H TMP_39 TMP_40) in (let TMP_42 \def (le_S_n d TMP_35 
-TMP_41) in (let TMP_43 \def (lift_lref_ge TMP_33 TMP_34 d TMP_42) in 
-(eq_ind_r T TMP_4 TMP_6 TMP_28 TMP_32 
-TMP_43)))))))))))))))))))))))))))))))))))))))).
+ \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef 
+(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus 
+(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n 
+(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S 
+(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n))) 
+(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d 
+(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n)) 
+(S_pred n d H))))))).
 
-theorem lift_tle:
+lemma lift_tle:
  \forall (t: T).(\forall (h: nat).(\forall (d: nat).(tle t (lift h d t))))
 \def
- \lambda (t: T).(let TMP_4 \def (\lambda (t0: T).(\forall (h: nat).(\forall 
-(d: nat).(let TMP_1 \def (tweight t0) in (let TMP_2 \def (lift h d t0) in 
-(let TMP_3 \def (tweight TMP_2) in (le TMP_1 TMP_3))))))) in (let TMP_6 \def 
-(\lambda (_: nat).(\lambda (_: nat).(\lambda (_: nat).(let TMP_5 \def (S O) 
-in (le_n TMP_5))))) in (let TMP_8 \def (\lambda (_: nat).(\lambda (_: 
-nat).(\lambda (_: nat).(let TMP_7 \def (S O) in (le_n TMP_7))))) in (let 
-TMP_31 \def (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: 
-nat).(\forall (d: nat).(le (tweight t0) (tweight (lift h d t0))))))).(\lambda 
-(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).(le (tweight t1) 
-(tweight (lift h d t1))))))).(\lambda (h: nat).(\lambda (d: nat).(let H_y 
-\def (H h d) in (let TMP_9 \def (s k d) in (let H_y0 \def (H0 h TMP_9) in 
-(let TMP_10 \def (tweight t0) in (let TMP_11 \def (tweight t1) in (let TMP_12 
-\def (plus TMP_10 TMP_11) in (let TMP_13 \def (\lambda (x: nat).(plus x h)) 
-in (let TMP_14 \def (lref_map TMP_13 d t0) in (let TMP_15 \def (tweight 
-TMP_14) in (let TMP_16 \def (\lambda (x: nat).(plus x h)) in (let TMP_17 \def 
-(s k d) in (let TMP_18 \def (lref_map TMP_16 TMP_17 t1) in (let TMP_19 \def 
-(tweight TMP_18) in (let TMP_20 \def (plus TMP_15 TMP_19) in (let TMP_21 \def 
-(tweight t0) in (let TMP_22 \def (\lambda (x: nat).(plus x h)) in (let TMP_23 
-\def (lref_map TMP_22 d t0) in (let TMP_24 \def (tweight TMP_23) in (let 
-TMP_25 \def (tweight t1) in (let TMP_26 \def (\lambda (x: nat).(plus x h)) in 
-(let TMP_27 \def (s k d) in (let TMP_28 \def (lref_map TMP_26 TMP_27 t1) in 
-(let TMP_29 \def (tweight TMP_28) in (let TMP_30 \def (le_plus_plus TMP_21 
-TMP_24 TMP_25 TMP_29 H_y H_y0) in (le_n_S TMP_12 TMP_20 
-TMP_30)))))))))))))))))))))))))))))))) in (T_ind TMP_4 TMP_6 TMP_8 TMP_31 
-t))))).
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: 
+nat).(le (tweight t0) (tweight (lift h d t0)))))) (\lambda (_: nat).(\lambda 
+(_: nat).(\lambda (_: nat).(le_n (S O))))) (\lambda (_: nat).(\lambda (_: 
+nat).(\lambda (_: nat).(le_n (S O))))) (\lambda (k: K).(\lambda (t0: 
+T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(le (tweight t0) 
+(tweight (lift h d t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: 
+nat).(\forall (d: nat).(le (tweight t1) (tweight (lift h d t1))))))).(\lambda 
+(h: nat).(\lambda (d: nat).(let H_y \def (H h d) in (let H_y0 \def (H0 h (s k 
+d)) in (le_n_S (plus (tweight t0) (tweight t1)) (plus (tweight (lref_map 
+(\lambda (x: nat).(plus x h)) d t0)) (tweight (lref_map (\lambda (x: 
+nat).(plus x h)) (s k d) t1))) (le_plus_plus (tweight t0) (tweight (lref_map 
+(\lambda (x: nat).(plus x h)) d t0)) (tweight t1) (tweight (lref_map (\lambda 
+(x: nat).(plus x h)) (s k d) t1)) H_y H_y0))))))))))) t).
 
-theorem lifts_tapp:
+lemma lifts_tapp:
  \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq 
 TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
 \def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: TList).(let 
-TMP_6 \def (\lambda (t: TList).(let TMP_1 \def (TApp t v) in (let TMP_2 \def 
-(lifts h d TMP_1) in (let TMP_3 \def (lifts h d t) in (let TMP_4 \def (lift h 
-d v) in (let TMP_5 \def (TApp TMP_3 TMP_4) in (eq TList TMP_2 TMP_5))))))) in 
-(let TMP_7 \def (lift h d v) in (let TMP_8 \def (TCons TMP_7 TNil) in (let 
-TMP_9 \def (refl_equal TList TMP_8) in (let TMP_29 \def (\lambda (t: 
-T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp t0 v)) (TApp 
-(lifts h d t0) (lift h d v)))).(let TMP_10 \def (lifts h d t0) in (let TMP_11 
-\def (lift h d v) in (let TMP_12 \def (TApp TMP_10 TMP_11) in (let TMP_20 
-\def (\lambda (t1: TList).(let TMP_13 \def (lift h d t) in (let TMP_14 \def 
-(TCons TMP_13 t1) in (let TMP_15 \def (lift h d t) in (let TMP_16 \def (lifts 
-h d t0) in (let TMP_17 \def (lift h d v) in (let TMP_18 \def (TApp TMP_16 
-TMP_17) in (let TMP_19 \def (TCons TMP_15 TMP_18) in (eq TList TMP_14 
-TMP_19))))))))) in (let TMP_21 \def (lift h d t) in (let TMP_22 \def (lifts h 
-d t0) in (let TMP_23 \def (lift h d v) in (let TMP_24 \def (TApp TMP_22 
-TMP_23) in (let TMP_25 \def (TCons TMP_21 TMP_24) in (let TMP_26 \def 
-(refl_equal TList TMP_25) in (let TMP_27 \def (TApp t0 v) in (let TMP_28 \def 
-(lifts h d TMP_27) in (eq_ind_r TList TMP_12 TMP_20 TMP_26 TMP_28 
-H)))))))))))))))) in (TList_ind TMP_6 TMP_9 TMP_29 vs))))))))).
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs: 
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp 
+(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil)) 
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp 
+t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d 
+t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1) 
+(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList 
+(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0 
+v)) H)))) vs)))).
 
-theorem lift_free:
+lemma lift_free:
  \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: 
 nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e 
 (lift h d t)) (lift (plus k h) d t))))))))
 \def
- \lambda (t: T).(let TMP_5 \def (\lambda (t0: T).(\forall (h: nat).(\forall 
-(k: nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) 
-\to (let TMP_1 \def (lift h d t0) in (let TMP_2 \def (lift k e TMP_1) in (let 
-TMP_3 \def (plus k h) in (let TMP_4 \def (lift TMP_3 d t0) in (eq T TMP_2 
-TMP_4)))))))))))) in (let TMP_35 \def (\lambda (n: nat).(\lambda (h: 
-nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_: (le e 
-(plus d h))).(\lambda (_: (le d e)).(let TMP_6 \def (TSort n) in (let TMP_11 
-\def (\lambda (t0: T).(let TMP_7 \def (lift k e t0) in (let TMP_8 \def (plus 
-k h) in (let TMP_9 \def (TSort n) in (let TMP_10 \def (lift TMP_8 d TMP_9) in 
-(eq T TMP_7 TMP_10)))))) in (let TMP_12 \def (TSort n) in (let TMP_16 \def 
-(\lambda (t0: T).(let TMP_13 \def (plus k h) in (let TMP_14 \def (TSort n) in 
-(let TMP_15 \def (lift TMP_13 d TMP_14) in (eq T t0 TMP_15))))) in (let 
-TMP_17 \def (TSort n) in (let TMP_19 \def (\lambda (t0: T).(let TMP_18 \def 
-(TSort n) in (eq T TMP_18 t0))) in (let TMP_20 \def (TSort n) in (let TMP_21 
-\def (refl_equal T TMP_20) in (let TMP_22 \def (plus k h) in (let TMP_23 \def 
-(TSort n) in (let TMP_24 \def (lift TMP_22 d TMP_23) in (let TMP_25 \def 
-(plus k h) in (let TMP_26 \def (lift_sort n TMP_25 d) in (let TMP_27 \def 
-(eq_ind_r T TMP_17 TMP_19 TMP_21 TMP_24 TMP_26) in (let TMP_28 \def (TSort n) 
-in (let TMP_29 \def (lift k e TMP_28) in (let TMP_30 \def (lift_sort n k e) 
-in (let TMP_31 \def (eq_ind_r T TMP_12 TMP_16 TMP_27 TMP_29 TMP_30) in (let 
-TMP_32 \def (TSort n) in (let TMP_33 \def (lift h d TMP_32) in (let TMP_34 
-\def (lift_sort n h d) in (eq_ind_r T TMP_6 TMP_11 TMP_31 TMP_33 
-TMP_34))))))))))))))))))))))))))))) in (let TMP_122 \def (\lambda (n: 
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: 
+nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to 
+(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n: 
 nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: 
-nat).(\lambda (H: (le e (plus d h))).(\lambda (H0: (le d e)).(let TMP_36 \def 
-(TLRef n) in (let TMP_37 \def (lift h d TMP_36) in (let TMP_38 \def (lift k e 
-TMP_37) in (let TMP_39 \def (plus k h) in (let TMP_40 \def (TLRef n) in (let 
-TMP_41 \def (lift TMP_39 d TMP_40) in (let TMP_42 \def (eq T TMP_38 TMP_41) 
-in (let TMP_73 \def (\lambda (H1: (lt n d)).(let TMP_43 \def (TLRef n) in 
-(let TMP_48 \def (\lambda (t0: T).(let TMP_44 \def (lift k e t0) in (let 
-TMP_45 \def (plus k h) in (let TMP_46 \def (TLRef n) in (let TMP_47 \def 
-(lift TMP_45 d TMP_46) in (eq T TMP_44 TMP_47)))))) in (let TMP_49 \def 
-(TLRef n) in (let TMP_53 \def (\lambda (t0: T).(let TMP_50 \def (plus k h) in 
-(let TMP_51 \def (TLRef n) in (let TMP_52 \def (lift TMP_50 d TMP_51) in (eq 
-T t0 TMP_52))))) in (let TMP_54 \def (TLRef n) in (let TMP_56 \def (\lambda 
-(t0: T).(let TMP_55 \def (TLRef n) in (eq T TMP_55 t0))) in (let TMP_57 \def 
-(TLRef n) in (let TMP_58 \def (refl_equal T TMP_57) in (let TMP_59 \def (plus 
-k h) in (let TMP_60 \def (TLRef n) in (let TMP_61 \def (lift TMP_59 d TMP_60) 
-in (let TMP_62 \def (plus k h) in (let TMP_63 \def (lift_lref_lt n TMP_62 d 
-H1) in (let TMP_64 \def (eq_ind_r T TMP_54 TMP_56 TMP_58 TMP_61 TMP_63) in 
-(let TMP_65 \def (TLRef n) in (let TMP_66 \def (lift k e TMP_65) in (let 
-TMP_67 \def (lt_le_trans n d e H1 H0) in (let TMP_68 \def (lift_lref_lt n k e 
-TMP_67) in (let TMP_69 \def (eq_ind_r T TMP_49 TMP_53 TMP_64 TMP_66 TMP_68) 
-in (let TMP_70 \def (TLRef n) in (let TMP_71 \def (lift h d TMP_70) in (let 
-TMP_72 \def (lift_lref_lt n h d H1) in (eq_ind_r T TMP_43 TMP_48 TMP_69 
-TMP_71 TMP_72)))))))))))))))))))))))) in (let TMP_121 \def (\lambda (H1: (le 
-d n)).(let TMP_74 \def (plus n h) in (let TMP_75 \def (TLRef TMP_74) in (let 
-TMP_80 \def (\lambda (t0: T).(let TMP_76 \def (lift k e t0) in (let TMP_77 
-\def (plus k h) in (let TMP_78 \def (TLRef n) in (let TMP_79 \def (lift 
-TMP_77 d TMP_78) in (eq T TMP_76 TMP_79)))))) in (let TMP_81 \def (plus n h) 
-in (let TMP_82 \def (plus TMP_81 k) in (let TMP_83 \def (TLRef TMP_82) in 
-(let TMP_87 \def (\lambda (t0: T).(let TMP_84 \def (plus k h) in (let TMP_85 
-\def (TLRef n) in (let TMP_86 \def (lift TMP_84 d TMP_85) in (eq T t0 
-TMP_86))))) in (let TMP_88 \def (plus k h) in (let TMP_89 \def (plus n 
-TMP_88) in (let TMP_90 \def (TLRef TMP_89) in (let TMP_94 \def (\lambda (t0: 
-T).(let TMP_91 \def (plus n h) in (let TMP_92 \def (plus TMP_91 k) in (let 
-TMP_93 \def (TLRef TMP_92) in (eq T TMP_93 t0))))) in (let TMP_95 \def (plus 
-n h) in (let TMP_96 \def (plus TMP_95 k) in (let TMP_97 \def (plus k h) in 
-(let TMP_98 \def (plus n TMP_97) in (let TMP_99 \def 
-(plus_permute_2_in_3_assoc n h k) in (let TMP_100 \def (f_equal nat T TLRef 
-TMP_96 TMP_98 TMP_99) in (let TMP_101 \def (plus k h) in (let TMP_102 \def 
-(TLRef n) in (let TMP_103 \def (lift TMP_101 d TMP_102) in (let TMP_104 \def 
-(plus k h) in (let TMP_105 \def (lift_lref_ge n TMP_104 d H1) in (let TMP_106 
-\def (eq_ind_r T TMP_90 TMP_94 TMP_100 TMP_103 TMP_105) in (let TMP_107 \def 
-(plus n h) in (let TMP_108 \def (TLRef TMP_107) in (let TMP_109 \def (lift k 
-e TMP_108) in (let TMP_110 \def (plus n h) in (let TMP_111 \def (plus d h) in 
-(let TMP_112 \def (plus n h) in (let TMP_113 \def (le_n h) in (let TMP_114 
-\def (le_plus_plus d n h h H1 TMP_113) in (let TMP_115 \def (le_trans e 
-TMP_111 TMP_112 H TMP_114) in (let TMP_116 \def (lift_lref_ge TMP_110 k e 
-TMP_115) in (let TMP_117 \def (eq_ind_r T TMP_83 TMP_87 TMP_106 TMP_109 
-TMP_116) in (let TMP_118 \def (TLRef n) in (let TMP_119 \def (lift h d 
-TMP_118) in (let TMP_120 \def (lift_lref_ge n h d H1) in (eq_ind_r T TMP_75 
-TMP_80 TMP_117 TMP_119 TMP_120))))))))))))))))))))))))))))))))))))))) in 
-(lt_le_e n d TMP_42 TMP_73 TMP_121))))))))))))))))) in (let TMP_204 \def 
-(\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall 
-(k0: nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d 
-e) \to (eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d 
-t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: 
+nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T 
+(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort 
+n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d 
+(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) 
+(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k 
+h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n)) 
+(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k: 
+nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d 
+h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef 
+n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T 
+(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef 
+n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d 
+(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) 
+(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus 
+k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1 
+H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d 
+n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift 
+(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda 
+(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n 
+(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal 
+nat T TLRef (plus (plus n h) k) (plus n (plus k h)) 
+(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n)) 
+(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge 
+(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1 
+(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda 
+(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: 
 nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to 
-(eq T (lift k0 e (lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: 
-nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le 
-e (plus d h))).(\lambda (H2: (le d e)).(let TMP_123 \def (lift h d t0) in 
-(let TMP_124 \def (s k d) in (let TMP_125 \def (lift h TMP_124 t1) in (let 
-TMP_126 \def (THead k TMP_123 TMP_125) in (let TMP_131 \def (\lambda (t2: 
-T).(let TMP_127 \def (lift k0 e t2) in (let TMP_128 \def (plus k0 h) in (let 
-TMP_129 \def (THead k t0 t1) in (let TMP_130 \def (lift TMP_128 d TMP_129) in 
-(eq T TMP_127 TMP_130)))))) in (let TMP_132 \def (lift h d t0) in (let 
-TMP_133 \def (lift k0 e TMP_132) in (let TMP_134 \def (s k e) in (let TMP_135 
-\def (s k d) in (let TMP_136 \def (lift h TMP_135 t1) in (let TMP_137 \def 
-(lift k0 TMP_134 TMP_136) in (let TMP_138 \def (THead k TMP_133 TMP_137) in 
-(let TMP_142 \def (\lambda (t2: T).(let TMP_139 \def (plus k0 h) in (let 
-TMP_140 \def (THead k t0 t1) in (let TMP_141 \def (lift TMP_139 d TMP_140) in 
-(eq T t2 TMP_141))))) in (let TMP_143 \def (plus k0 h) in (let TMP_144 \def 
-(lift TMP_143 d t0) in (let TMP_145 \def (plus k0 h) in (let TMP_146 \def (s 
-k d) in (let TMP_147 \def (lift TMP_145 TMP_146 t1) in (let TMP_148 \def 
-(THead k TMP_144 TMP_147) in (let TMP_156 \def (\lambda (t2: T).(let TMP_149 
-\def (lift h d t0) in (let TMP_150 \def (lift k0 e TMP_149) in (let TMP_151 
-\def (s k e) in (let TMP_152 \def (s k d) in (let TMP_153 \def (lift h 
-TMP_152 t1) in (let TMP_154 \def (lift k0 TMP_151 TMP_153) in (let TMP_155 
-\def (THead k TMP_150 TMP_154) in (eq T TMP_155 t2))))))))) in (let TMP_157 
-\def (lift h d t0) in (let TMP_158 \def (lift k0 e TMP_157) in (let TMP_159 
-\def (plus k0 h) in (let TMP_160 \def (lift TMP_159 d t0) in (let TMP_161 
-\def (s k e) in (let TMP_162 \def (s k d) in (let TMP_163 \def (lift h 
-TMP_162 t1) in (let TMP_164 \def (lift k0 TMP_161 TMP_163) in (let TMP_165 
-\def (plus k0 h) in (let TMP_166 \def (s k d) in (let TMP_167 \def (lift 
-TMP_165 TMP_166 t1) in (let TMP_168 \def (refl_equal K k) in (let TMP_169 
-\def (H h k0 d e H1 H2) in (let TMP_170 \def (s k d) in (let TMP_171 \def (s 
-k e) in (let TMP_172 \def (plus d h) in (let TMP_173 \def (s k TMP_172) in 
-(let TMP_175 \def (\lambda (n: nat).(let TMP_174 \def (s k e) in (le TMP_174 
-n))) in (let TMP_176 \def (plus d h) in (let TMP_177 \def (s_le k e TMP_176 
-H1) in (let TMP_178 \def (s k d) in (let TMP_179 \def (plus TMP_178 h) in 
-(let TMP_180 \def (s_plus k d h) in (let TMP_181 \def (eq_ind nat TMP_173 
-TMP_175 TMP_177 TMP_179 TMP_180) in (let TMP_182 \def (s_le k d e H2) in (let 
-TMP_183 \def (H0 h k0 TMP_170 TMP_171 TMP_181 TMP_182) in (let TMP_184 \def 
-(f_equal3 K T T T THead k k TMP_158 TMP_160 TMP_164 TMP_167 TMP_168 TMP_169 
-TMP_183) in (let TMP_185 \def (plus k0 h) in (let TMP_186 \def (THead k t0 
-t1) in (let TMP_187 \def (lift TMP_185 d TMP_186) in (let TMP_188 \def (plus 
-k0 h) in (let TMP_189 \def (lift_head k t0 t1 TMP_188 d) in (let TMP_190 \def 
-(eq_ind_r T TMP_148 TMP_156 TMP_184 TMP_187 TMP_189) in (let TMP_191 \def 
-(lift h d t0) in (let TMP_192 \def (s k d) in (let TMP_193 \def (lift h 
-TMP_192 t1) in (let TMP_194 \def (THead k TMP_191 TMP_193) in (let TMP_195 
-\def (lift k0 e TMP_194) in (let TMP_196 \def (lift h d t0) in (let TMP_197 
-\def (s k d) in (let TMP_198 \def (lift h TMP_197 t1) in (let TMP_199 \def 
-(lift_head k TMP_196 TMP_198 k0 e) in (let TMP_200 \def (eq_ind_r T TMP_138 
-TMP_142 TMP_190 TMP_195 TMP_199) in (let TMP_201 \def (THead k t0 t1) in (let 
-TMP_202 \def (lift h d TMP_201) in (let TMP_203 \def (lift_head k t0 t1 h d) 
-in (eq_ind_r T TMP_126 TMP_131 TMP_200 TMP_202 
-TMP_203)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
-))))))) in (T_ind TMP_5 TMP_35 TMP_122 TMP_204 t))))).
+(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda 
+(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: 
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e 
+(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda 
+(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d 
+h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k 
+d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0 
+t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift 
+h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0 
+t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d) 
+t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k 
+e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h 
+d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift 
+(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d) 
+(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le 
+k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift 
+(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e 
+(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift 
+h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h 
+d))))))))))))) t).
 
-theorem lift_free_sym:
+lemma lift_free_sym:
  \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: 
 nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e 
 (lift h d t)) (lift (plus h k) d t))))))))
 \def
  \lambda (t: T).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: 
 nat).(\lambda (e: nat).(\lambda (H: (le e (plus d h))).(\lambda (H0: (le d 
-e)).(let TMP_1 \def (plus k h) in (let TMP_5 \def (\lambda (n: nat).(let 
-TMP_2 \def (lift h d t) in (let TMP_3 \def (lift k e TMP_2) in (let TMP_4 
-\def (lift n d t) in (eq T TMP_3 TMP_4))))) in (let TMP_6 \def (lift_free t h 
-k d e H H0) in (let TMP_7 \def (plus h k) in (let TMP_8 \def (plus_sym h k) 
-in (eq_ind_r nat TMP_1 TMP_5 TMP_6 TMP_7 TMP_8)))))))))))).
+e)).(eq_ind_r nat (plus k h) (\lambda (n: nat).(eq T (lift k e (lift h d t)) 
+(lift n d t))) (lift_free t h k d e H H0) (plus h k) (plus_sym h k)))))))).
 
-theorem lift_d:
+lemma lift_d:
  \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: 
 nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t)) 
 (lift k e (lift h d t))))))))
 \def
- \lambda (t: T).(let TMP_6 \def (\lambda (t0: T).(\forall (h: nat).(\forall 
-(k: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (let TMP_1 \def 
-(plus k d) in (let TMP_2 \def (lift k e t0) in (let TMP_3 \def (lift h TMP_1 
-TMP_2) in (let TMP_4 \def (lift h d t0) in (let TMP_5 \def (lift k e TMP_4) 
-in (eq T TMP_3 TMP_5)))))))))))) in (let TMP_45 \def (\lambda (n: 
-nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: 
-nat).(\lambda (_: (le e d)).(let TMP_7 \def (TSort n) in (let TMP_13 \def 
-(\lambda (t0: T).(let TMP_8 \def (plus k d) in (let TMP_9 \def (lift h TMP_8 
-t0) in (let TMP_10 \def (TSort n) in (let TMP_11 \def (lift h d TMP_10) in 
-(let TMP_12 \def (lift k e TMP_11) in (eq T TMP_9 TMP_12))))))) in (let 
-TMP_14 \def (TSort n) in (let TMP_18 \def (\lambda (t0: T).(let TMP_15 \def 
-(TSort n) in (let TMP_16 \def (lift h d TMP_15) in (let TMP_17 \def (lift k e 
-TMP_16) in (eq T t0 TMP_17))))) in (let TMP_19 \def (TSort n) in (let TMP_22 
-\def (\lambda (t0: T).(let TMP_20 \def (TSort n) in (let TMP_21 \def (lift k 
-e t0) in (eq T TMP_20 TMP_21)))) in (let TMP_23 \def (TSort n) in (let TMP_25 
-\def (\lambda (t0: T).(let TMP_24 \def (TSort n) in (eq T TMP_24 t0))) in 
-(let TMP_26 \def (TSort n) in (let TMP_27 \def (refl_equal T TMP_26) in (let 
-TMP_28 \def (TSort n) in (let TMP_29 \def (lift k e TMP_28) in (let TMP_30 
-\def (lift_sort n k e) in (let TMP_31 \def (eq_ind_r T TMP_23 TMP_25 TMP_27 
-TMP_29 TMP_30) in (let TMP_32 \def (TSort n) in (let TMP_33 \def (lift h d 
-TMP_32) in (let TMP_34 \def (lift_sort n h d) in (let TMP_35 \def (eq_ind_r T 
-TMP_19 TMP_22 TMP_31 TMP_33 TMP_34) in (let TMP_36 \def (plus k d) in (let 
-TMP_37 \def (TSort n) in (let TMP_38 \def (lift h TMP_36 TMP_37) in (let 
-TMP_39 \def (plus k d) in (let TMP_40 \def (lift_sort n h TMP_39) in (let 
-TMP_41 \def (eq_ind_r T TMP_14 TMP_18 TMP_35 TMP_38 TMP_40) in (let TMP_42 
-\def (TSort n) in (let TMP_43 \def (lift k e TMP_42) in (let TMP_44 \def 
-(lift_sort n k e) in (eq_ind_r T TMP_7 TMP_13 TMP_41 TMP_43 
-TMP_44)))))))))))))))))))))))))))))))))) in (let TMP_212 \def (\lambda (n: 
-nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: 
-nat).(\lambda (H: (le e d)).(let TMP_46 \def (plus k d) in (let TMP_47 \def 
-(TLRef n) in (let TMP_48 \def (lift k e TMP_47) in (let TMP_49 \def (lift h 
-TMP_46 TMP_48) in (let TMP_50 \def (TLRef n) in (let TMP_51 \def (lift h d 
-TMP_50) in (let TMP_52 \def (lift k e TMP_51) in (let TMP_53 \def (eq T 
-TMP_49 TMP_52) in (let TMP_95 \def (\lambda (H0: (lt n e)).(let H1 \def 
-(lt_le_trans n e d H0 H) in (let TMP_54 \def (TLRef n) in (let TMP_60 \def 
-(\lambda (t0: T).(let TMP_55 \def (plus k d) in (let TMP_56 \def (lift h 
-TMP_55 t0) in (let TMP_57 \def (TLRef n) in (let TMP_58 \def (lift h d 
-TMP_57) in (let TMP_59 \def (lift k e TMP_58) in (eq T TMP_56 TMP_59))))))) 
-in (let TMP_61 \def (TLRef n) in (let TMP_65 \def (\lambda (t0: T).(let 
-TMP_62 \def (TLRef n) in (let TMP_63 \def (lift h d TMP_62) in (let TMP_64 
-\def (lift k e TMP_63) in (eq T t0 TMP_64))))) in (let TMP_66 \def (TLRef n) 
-in (let TMP_69 \def (\lambda (t0: T).(let TMP_67 \def (TLRef n) in (let 
-TMP_68 \def (lift k e t0) in (eq T TMP_67 TMP_68)))) in (let TMP_70 \def 
-(TLRef n) in (let TMP_72 \def (\lambda (t0: T).(let TMP_71 \def (TLRef n) in 
-(eq T TMP_71 t0))) in (let TMP_73 \def (TLRef n) in (let TMP_74 \def 
-(refl_equal T TMP_73) in (let TMP_75 \def (TLRef n) in (let TMP_76 \def (lift 
-k e TMP_75) in (let TMP_77 \def (lift_lref_lt n k e H0) in (let TMP_78 \def 
-(eq_ind_r T TMP_70 TMP_72 TMP_74 TMP_76 TMP_77) in (let TMP_79 \def (TLRef n) 
-in (let TMP_80 \def (lift h d TMP_79) in (let TMP_81 \def (lift_lref_lt n h d 
-H1) in (let TMP_82 \def (eq_ind_r T TMP_66 TMP_69 TMP_78 TMP_80 TMP_81) in 
-(let TMP_83 \def (plus k d) in (let TMP_84 \def (TLRef n) in (let TMP_85 \def 
-(lift h TMP_83 TMP_84) in (let TMP_86 \def (plus k d) in (let TMP_87 \def 
-(plus k d) in (let TMP_88 \def (le_plus_r k d) in (let TMP_89 \def 
-(lt_le_trans n d TMP_87 H1 TMP_88) in (let TMP_90 \def (lift_lref_lt n h 
-TMP_86 TMP_89) in (let TMP_91 \def (eq_ind_r T TMP_61 TMP_65 TMP_82 TMP_85 
-TMP_90) in (let TMP_92 \def (TLRef n) in (let TMP_93 \def (lift k e TMP_92) 
-in (let TMP_94 \def (lift_lref_lt n k e H0) in (eq_ind_r T TMP_54 TMP_60 
-TMP_91 TMP_93 TMP_94))))))))))))))))))))))))))))))))) in (let TMP_211 \def 
-(\lambda (H0: (le e n)).(let TMP_96 \def (plus n k) in (let TMP_97 \def 
-(TLRef TMP_96) in (let TMP_103 \def (\lambda (t0: T).(let TMP_98 \def (plus k 
-d) in (let TMP_99 \def (lift h TMP_98 t0) in (let TMP_100 \def (TLRef n) in 
-(let TMP_101 \def (lift h d TMP_100) in (let TMP_102 \def (lift k e TMP_101) 
-in (eq T TMP_99 TMP_102))))))) in (let TMP_104 \def (plus d k) in (let 
-TMP_111 \def (\lambda (n0: nat).(let TMP_105 \def (plus n k) in (let TMP_106 
-\def (TLRef TMP_105) in (let TMP_107 \def (lift h n0 TMP_106) in (let TMP_108 
-\def (TLRef n) in (let TMP_109 \def (lift h d TMP_108) in (let TMP_110 \def 
-(lift k e TMP_109) in (eq T TMP_107 TMP_110)))))))) in (let TMP_112 \def 
-(plus d k) in (let TMP_113 \def (plus n k) in (let TMP_114 \def (TLRef 
-TMP_113) in (let TMP_115 \def (lift h TMP_112 TMP_114) in (let TMP_116 \def 
-(TLRef n) in (let TMP_117 \def (lift h d TMP_116) in (let TMP_118 \def (lift 
-k e TMP_117) in (let TMP_119 \def (eq T TMP_115 TMP_118) in (let TMP_155 \def 
-(\lambda (H1: (lt n d)).(let TMP_120 \def (plus n k) in (let TMP_121 \def 
-(TLRef TMP_120) in (let TMP_125 \def (\lambda (t0: T).(let TMP_122 \def 
-(TLRef n) in (let TMP_123 \def (lift h d TMP_122) in (let TMP_124 \def (lift 
-k e TMP_123) in (eq T t0 TMP_124))))) in (let TMP_126 \def (TLRef n) in (let 
-TMP_130 \def (\lambda (t0: T).(let TMP_127 \def (plus n k) in (let TMP_128 
-\def (TLRef TMP_127) in (let TMP_129 \def (lift k e t0) in (eq T TMP_128 
-TMP_129))))) in (let TMP_131 \def (plus n k) in (let TMP_132 \def (TLRef 
-TMP_131) in (let TMP_135 \def (\lambda (t0: T).(let TMP_133 \def (plus n k) 
-in (let TMP_134 \def (TLRef TMP_133) in (eq T TMP_134 t0)))) in (let TMP_136 
-\def (plus n k) in (let TMP_137 \def (TLRef TMP_136) in (let TMP_138 \def 
-(refl_equal T TMP_137) in (let TMP_139 \def (TLRef n) in (let TMP_140 \def 
-(lift k e TMP_139) in (let TMP_141 \def (lift_lref_ge n k e H0) in (let 
-TMP_142 \def (eq_ind_r T TMP_132 TMP_135 TMP_138 TMP_140 TMP_141) in (let 
-TMP_143 \def (TLRef n) in (let TMP_144 \def (lift h d TMP_143) in (let 
-TMP_145 \def (lift_lref_lt n h d H1) in (let TMP_146 \def (eq_ind_r T TMP_126 
-TMP_130 TMP_142 TMP_144 TMP_145) in (let TMP_147 \def (plus d k) in (let 
-TMP_148 \def (plus n k) in (let TMP_149 \def (TLRef TMP_148) in (let TMP_150 
-\def (lift h TMP_147 TMP_149) in (let TMP_151 \def (plus n k) in (let TMP_152 
-\def (plus d k) in (let TMP_153 \def (lt_reg_r n d k H1) in (let TMP_154 \def 
-(lift_lref_lt TMP_151 h TMP_152 TMP_153) in (eq_ind_r T TMP_121 TMP_125 
-TMP_146 TMP_150 TMP_154))))))))))))))))))))))))))))) in (let TMP_203 \def 
-(\lambda (H1: (le d n)).(let TMP_156 \def (plus n k) in (let TMP_157 \def 
-(plus TMP_156 h) in (let TMP_158 \def (TLRef TMP_157) in (let TMP_162 \def 
-(\lambda (t0: T).(let TMP_159 \def (TLRef n) in (let TMP_160 \def (lift h d 
-TMP_159) in (let TMP_161 \def (lift k e TMP_160) in (eq T t0 TMP_161))))) in 
-(let TMP_163 \def (plus n h) in (let TMP_164 \def (TLRef TMP_163) in (let 
-TMP_169 \def (\lambda (t0: T).(let TMP_165 \def (plus n k) in (let TMP_166 
-\def (plus TMP_165 h) in (let TMP_167 \def (TLRef TMP_166) in (let TMP_168 
-\def (lift k e t0) in (eq T TMP_167 TMP_168)))))) in (let TMP_170 \def (plus 
-n h) in (let TMP_171 \def (plus TMP_170 k) in (let TMP_172 \def (TLRef 
-TMP_171) in (let TMP_176 \def (\lambda (t0: T).(let TMP_173 \def (plus n k) 
-in (let TMP_174 \def (plus TMP_173 h) in (let TMP_175 \def (TLRef TMP_174) in 
-(eq T TMP_175 t0))))) in (let TMP_177 \def (plus n k) in (let TMP_178 \def 
-(plus TMP_177 h) in (let TMP_179 \def (plus n h) in (let TMP_180 \def (plus 
-TMP_179 k) in (let TMP_181 \def (plus_permute_2_in_3 n k h) in (let TMP_182 
-\def (f_equal nat T TLRef TMP_178 TMP_180 TMP_181) in (let TMP_183 \def (plus 
-n h) in (let TMP_184 \def (TLRef TMP_183) in (let TMP_185 \def (lift k e 
-TMP_184) in (let TMP_186 \def (plus n h) in (let TMP_187 \def (le_plus_trans 
-e n h H0) in (let TMP_188 \def (lift_lref_ge TMP_186 k e TMP_187) in (let 
-TMP_189 \def (eq_ind_r T TMP_172 TMP_176 TMP_182 TMP_185 TMP_188) in (let 
-TMP_190 \def (TLRef n) in (let TMP_191 \def (lift h d TMP_190) in (let 
-TMP_192 \def (lift_lref_ge n h d H1) in (let TMP_193 \def (eq_ind_r T TMP_164 
-TMP_169 TMP_189 TMP_191 TMP_192) in (let TMP_194 \def (plus d k) in (let 
-TMP_195 \def (plus n k) in (let TMP_196 \def (TLRef TMP_195) in (let TMP_197 
-\def (lift h TMP_194 TMP_196) in (let TMP_198 \def (plus n k) in (let TMP_199 
-\def (plus d k) in (let TMP_200 \def (le_n k) in (let TMP_201 \def 
-(le_plus_plus d n k k H1 TMP_200) in (let TMP_202 \def (lift_lref_ge TMP_198 
-h TMP_199 TMP_201) in (eq_ind_r T TMP_158 TMP_162 TMP_193 TMP_197 
-TMP_202))))))))))))))))))))))))))))))))))))))) in (let TMP_204 \def (lt_le_e 
-n d TMP_119 TMP_155 TMP_203) in (let TMP_205 \def (plus k d) in (let TMP_206 
-\def (plus_sym k d) in (let TMP_207 \def (eq_ind_r nat TMP_104 TMP_111 
-TMP_204 TMP_205 TMP_206) in (let TMP_208 \def (TLRef n) in (let TMP_209 \def 
-(lift k e TMP_208) in (let TMP_210 \def (lift_lref_ge n k e H0) in (eq_ind_r 
-T TMP_97 TMP_103 TMP_207 TMP_209 TMP_210)))))))))))))))))))))))) in (lt_le_e 
-n e TMP_53 TMP_95 TMP_211))))))))))))))))) in (let TMP_339 \def (\lambda (k: 
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k: 
+nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k 
+d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda 
+(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_: 
+(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0) 
+(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq 
+T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: 
+T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq 
+T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k 
+e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n)) 
+(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e)))))))) 
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: 
+nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h 
+(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda 
+(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n) 
+(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef 
+n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d 
+(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k 
+e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0)) 
+(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift 
+h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n)) 
+(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k 
+d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e 
+n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d) 
+t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0: 
+nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n))))) 
+(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d 
+(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda 
+(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n) 
+(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef 
+(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T 
+(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d 
+(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k))) 
+(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1: 
+(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0 
+(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda 
+(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef 
+(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) 
+(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) 
+(plus_permute_2_in_3 n k h)) (lift k e (TLRef (plus n h))) (lift_lref_ge 
+(plus n h) k e (le_plus_trans e n h H0))) (lift h d (TLRef n)) (lift_lref_ge 
+n h d H1)) (lift h (plus d k) (TLRef (plus n k))) (lift_lref_ge (plus n k) h 
+(plus d k) (le_plus_plus d n k k H1 (le_n k)))))) (plus k d) (plus_sym k d)) 
+(lift k e (TLRef n)) (lift_lref_ge n k e H0)))))))))) (\lambda (k: 
 K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: 
 nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 
 d) (lift k0 e t0)) (lift k0 e (lift h d t0)))))))))).(\lambda (t1: 
 T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: 
 nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k0 d) (lift k0 e 
 t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda (k0: 
-nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e d)).(let TMP_213 
-\def (lift k0 e t0) in (let TMP_214 \def (s k e) in (let TMP_215 \def (lift 
-k0 TMP_214 t1) in (let TMP_216 \def (THead k TMP_213 TMP_215) in (let TMP_222 
-\def (\lambda (t2: T).(let TMP_217 \def (plus k0 d) in (let TMP_218 \def 
-(lift h TMP_217 t2) in (let TMP_219 \def (THead k t0 t1) in (let TMP_220 \def 
-(lift h d TMP_219) in (let TMP_221 \def (lift k0 e TMP_220) in (eq T TMP_218 
-TMP_221))))))) in (let TMP_223 \def (plus k0 d) in (let TMP_224 \def (lift k0 
-e t0) in (let TMP_225 \def (lift h TMP_223 TMP_224) in (let TMP_226 \def 
-(plus k0 d) in (let TMP_227 \def (s k TMP_226) in (let TMP_228 \def (s k e) 
-in (let TMP_229 \def (lift k0 TMP_228 t1) in (let TMP_230 \def (lift h 
-TMP_227 TMP_229) in (let TMP_231 \def (THead k TMP_225 TMP_230) in (let 
-TMP_235 \def (\lambda (t2: T).(let TMP_232 \def (THead k t0 t1) in (let 
-TMP_233 \def (lift h d TMP_232) in (let TMP_234 \def (lift k0 e TMP_233) in 
-(eq T t2 TMP_234))))) in (let TMP_236 \def (lift h d t0) in (let TMP_237 \def 
-(s k d) in (let TMP_238 \def (lift h TMP_237 t1) in (let TMP_239 \def (THead 
-k TMP_236 TMP_238) in (let TMP_250 \def (\lambda (t2: T).(let TMP_240 \def 
-(plus k0 d) in (let TMP_241 \def (lift k0 e t0) in (let TMP_242 \def (lift h 
-TMP_240 TMP_241) in (let TMP_243 \def (plus k0 d) in (let TMP_244 \def (s k 
-TMP_243) in (let TMP_245 \def (s k e) in (let TMP_246 \def (lift k0 TMP_245 
-t1) in (let TMP_247 \def (lift h TMP_244 TMP_246) in (let TMP_248 \def (THead 
-k TMP_242 TMP_247) in (let TMP_249 \def (lift k0 e t2) in (eq T TMP_248 
-TMP_249)))))))))))) in (let TMP_251 \def (lift h d t0) in (let TMP_252 \def 
-(lift k0 e TMP_251) in (let TMP_253 \def (s k e) in (let TMP_254 \def (s k d) 
-in (let TMP_255 \def (lift h TMP_254 t1) in (let TMP_256 \def (lift k0 
-TMP_253 TMP_255) in (let TMP_257 \def (THead k TMP_252 TMP_256) in (let 
-TMP_267 \def (\lambda (t2: T).(let TMP_258 \def (plus k0 d) in (let TMP_259 
-\def (lift k0 e t0) in (let TMP_260 \def (lift h TMP_258 TMP_259) in (let 
-TMP_261 \def (plus k0 d) in (let TMP_262 \def (s k TMP_261) in (let TMP_263 
-\def (s k e) in (let TMP_264 \def (lift k0 TMP_263 t1) in (let TMP_265 \def 
-(lift h TMP_262 TMP_264) in (let TMP_266 \def (THead k TMP_260 TMP_265) in 
-(eq T TMP_266 t2))))))))))) in (let TMP_268 \def (s k d) in (let TMP_269 \def 
-(plus k0 TMP_268) in (let TMP_284 \def (\lambda (n: nat).(let TMP_270 \def 
-(plus k0 d) in (let TMP_271 \def (lift k0 e t0) in (let TMP_272 \def (lift h 
-TMP_270 TMP_271) in (let TMP_273 \def (s k e) in (let TMP_274 \def (lift k0 
-TMP_273 t1) in (let TMP_275 \def (lift h n TMP_274) in (let TMP_276 \def 
-(THead k TMP_272 TMP_275) in (let TMP_277 \def (lift h d t0) in (let TMP_278 
-\def (lift k0 e TMP_277) in (let TMP_279 \def (s k e) in (let TMP_280 \def (s 
-k d) in (let TMP_281 \def (lift h TMP_280 t1) in (let TMP_282 \def (lift k0 
-TMP_279 TMP_281) in (let TMP_283 \def (THead k TMP_278 TMP_282) in (eq T 
-TMP_276 TMP_283)))))))))))))))) in (let TMP_285 \def (plus k0 d) in (let 
-TMP_286 \def (lift k0 e t0) in (let TMP_287 \def (lift h TMP_285 TMP_286) in 
-(let TMP_288 \def (lift h d t0) in (let TMP_289 \def (lift k0 e TMP_288) in 
-(let TMP_290 \def (s k d) in (let TMP_291 \def (plus k0 TMP_290) in (let 
-TMP_292 \def (s k e) in (let TMP_293 \def (lift k0 TMP_292 t1) in (let 
-TMP_294 \def (lift h TMP_291 TMP_293) in (let TMP_295 \def (s k e) in (let 
-TMP_296 \def (s k d) in (let TMP_297 \def (lift h TMP_296 t1) in (let TMP_298 
-\def (lift k0 TMP_295 TMP_297) in (let TMP_299 \def (refl_equal K k) in (let 
-TMP_300 \def (H h k0 d e H1) in (let TMP_301 \def (s k d) in (let TMP_302 
-\def (s k e) in (let TMP_303 \def (s_le k e d H1) in (let TMP_304 \def (H0 h 
-k0 TMP_301 TMP_302 TMP_303) in (let TMP_305 \def (f_equal3 K T T T THead k k 
-TMP_287 TMP_289 TMP_294 TMP_298 TMP_299 TMP_300 TMP_304) in (let TMP_306 \def 
-(plus k0 d) in (let TMP_307 \def (s k TMP_306) in (let TMP_308 \def 
-(s_plus_sym k k0 d) in (let TMP_309 \def (eq_ind_r nat TMP_269 TMP_284 
-TMP_305 TMP_307 TMP_308) in (let TMP_310 \def (lift h d t0) in (let TMP_311 
-\def (s k d) in (let TMP_312 \def (lift h TMP_311 t1) in (let TMP_313 \def 
-(THead k TMP_310 TMP_312) in (let TMP_314 \def (lift k0 e TMP_313) in (let 
-TMP_315 \def (lift h d t0) in (let TMP_316 \def (s k d) in (let TMP_317 \def 
-(lift h TMP_316 t1) in (let TMP_318 \def (lift_head k TMP_315 TMP_317 k0 e) 
-in (let TMP_319 \def (eq_ind_r T TMP_257 TMP_267 TMP_309 TMP_314 TMP_318) in 
-(let TMP_320 \def (THead k t0 t1) in (let TMP_321 \def (lift h d TMP_320) in 
-(let TMP_322 \def (lift_head k t0 t1 h d) in (let TMP_323 \def (eq_ind_r T 
-TMP_239 TMP_250 TMP_319 TMP_321 TMP_322) in (let TMP_324 \def (plus k0 d) in 
-(let TMP_325 \def (lift k0 e t0) in (let TMP_326 \def (s k e) in (let TMP_327 
-\def (lift k0 TMP_326 t1) in (let TMP_328 \def (THead k TMP_325 TMP_327) in 
-(let TMP_329 \def (lift h TMP_324 TMP_328) in (let TMP_330 \def (lift k0 e 
-t0) in (let TMP_331 \def (s k e) in (let TMP_332 \def (lift k0 TMP_331 t1) in 
-(let TMP_333 \def (plus k0 d) in (let TMP_334 \def (lift_head k TMP_330 
-TMP_332 h TMP_333) in (let TMP_335 \def (eq_ind_r T TMP_231 TMP_235 TMP_323 
-TMP_329 TMP_334) in (let TMP_336 \def (THead k t0 t1) in (let TMP_337 \def 
-(lift k0 e TMP_336) in (let TMP_338 \def (lift_head k t0 t1 k0 e) in 
-(eq_ind_r T TMP_216 TMP_222 TMP_335 TMP_337 
-TMP_338)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
-))))))))))))))))))))))))) in (T_ind TMP_6 TMP_45 TMP_212 TMP_339 t))))).
+nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e d)).(eq_ind_r T 
+(THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: T).(eq T (lift h 
+(plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) (eq_ind_r T (THead k 
+(lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) (lift k0 (s k 
+e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d (THead k t0 t1))))) 
+(eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1)) (\lambda (t2: T).(eq 
+T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) 
+(lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead k (lift k0 e (lift 
+h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda (t2: T).(eq T (THead 
+k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) (lift k0 (s k 
+e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda (n: nat).(eq T (THead 
+k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift k0 (s k e) t1))) (THead 
+k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))))) 
+(f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e 
+(lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e) t1)) (lift k0 (s k 
+e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1) (H0 h k0 (s k d) (s 
+k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0 d)) (lift k0 e 
+(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift 
+h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h d)) 
+(lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0 (s k e) t1))) (lift_head 
+k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0 d))) (lift k0 e (THead k t0 
+t1)) (lift_head k t0 t1 k0 e)))))))))))) t).