X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Flift1%2Fprops.ma;h=286dfbee36c9c19859eaa5417e785aa5705eaf8a;hp=bb6374c163e23e6d07af085837a28ee81fb4a039;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hpb=7b95759c5011a25f96d5171561ea79d063770db4

diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
index bb6374c16..286dfbee3 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
@@ -18,227 +18,123 @@ include "basic_1/lift1/defs.ma".
 
 include "basic_1/lift/props.ma".
 
-theorem lift1_sort:
+lemma lift1_sort:
  \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
 \def
- \lambda (n: nat).(\lambda (is: PList).(let TMP_4 \def (\lambda (p: 
-PList).(let TMP_1 \def (TSort n) in (let TMP_2 \def (lift1 p TMP_1) in (let 
-TMP_3 \def (TSort n) in (eq T TMP_2 TMP_3))))) in (let TMP_5 \def (TSort n) 
-in (let TMP_6 \def (refl_equal T TMP_5) in (let TMP_15 \def (\lambda (n0: 
+ \lambda (n: nat).(\lambda (is: PList).(PList_ind (\lambda (p: PList).(eq T 
+(lift1 p (TSort n)) (TSort n))) (refl_equal T (TSort n)) (\lambda (n0: 
 nat).(\lambda (n1: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 p 
-(TSort n)) (TSort n))).(let TMP_7 \def (TSort n) in (let TMP_10 \def (\lambda 
-(t: T).(let TMP_8 \def (lift n0 n1 t) in (let TMP_9 \def (TSort n) in (eq T 
-TMP_8 TMP_9)))) in (let TMP_11 \def (TSort n) in (let TMP_12 \def (refl_equal 
-T TMP_11) in (let TMP_13 \def (TSort n) in (let TMP_14 \def (lift1 p TMP_13) 
-in (eq_ind_r T TMP_7 TMP_10 TMP_12 TMP_14 H))))))))))) in (PList_ind TMP_4 
-TMP_6 TMP_15 is)))))).
+(TSort n)) (TSort n))).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (lift n0 
+n1 t) (TSort n))) (refl_equal T (TSort n)) (lift1 p (TSort n)) H))))) is)).
 
-theorem lift1_lref:
+lemma lift1_lref:
  \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef 
 (trans hds i))))
 \def
- \lambda (hds: PList).(let TMP_5 \def (\lambda (p: PList).(\forall (i: 
-nat).(let TMP_1 \def (TLRef i) in (let TMP_2 \def (lift1 p TMP_1) in (let 
-TMP_3 \def (trans p i) in (let TMP_4 \def (TLRef TMP_3) in (eq T TMP_2 
-TMP_4))))))) in (let TMP_7 \def (\lambda (i: nat).(let TMP_6 \def (TLRef i) 
-in (refl_equal T TMP_6))) in (let TMP_26 \def (\lambda (n: nat).(\lambda (n0: 
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (i: nat).(eq T (lift1 p 
-(TLRef i)) (TLRef (trans p i)))))).(\lambda (i: nat).(let TMP_8 \def (trans p 
-i) in (let TMP_9 \def (TLRef TMP_8) in (let TMP_16 \def (\lambda (t: T).(let 
-TMP_10 \def (lift n n0 t) in (let TMP_11 \def (trans p i) in (let TMP_12 \def 
-(blt TMP_11 n0) in (let TMP_14 \def (match TMP_12 with [true \Rightarrow 
-(trans p i) | false \Rightarrow (let TMP_13 \def (trans p i) in (plus TMP_13 
-n))]) in (let TMP_15 \def (TLRef TMP_14) in (eq T TMP_10 TMP_15))))))) in 
-(let TMP_17 \def (trans p i) in (let TMP_18 \def (blt TMP_17 n0) in (let 
-TMP_20 \def (match TMP_18 with [true \Rightarrow (trans p i) | false 
-\Rightarrow (let TMP_19 \def (trans p i) in (plus TMP_19 n))]) in (let TMP_21 
-\def (TLRef TMP_20) in (let TMP_22 \def (refl_equal T TMP_21) in (let TMP_23 
-\def (TLRef i) in (let TMP_24 \def (lift1 p TMP_23) in (let TMP_25 \def (H i) 
-in (eq_ind_r T TMP_9 TMP_16 TMP_22 TMP_24 TMP_25))))))))))))))))) in 
-(PList_ind TMP_5 TMP_7 TMP_26 hds)))).
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T 
+(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T 
+(TLRef i))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
+(H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef (trans p 
+i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda (t: T).(eq 
+T (lift n n0 t) (TLRef (match (blt (trans p i) n0) with [true \Rightarrow 
+(trans p i) | false \Rightarrow (plus (trans p i) n)])))) (refl_equal T 
+(TLRef (match (blt (trans p i) n0) with [true \Rightarrow (trans p i) | false 
+\Rightarrow (plus (trans p i) n)]))) (lift1 p (TLRef i)) (H i))))))) hds).
 
-theorem lift1_bind:
+lemma lift1_bind:
  \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T 
 (lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss 
 hds) t))))))
 \def
- \lambda (b: B).(\lambda (hds: PList).(let TMP_9 \def (\lambda (p: 
-PList).(\forall (u: T).(\forall (t: T).(let TMP_1 \def (Bind b) in (let TMP_2 
-\def (THead TMP_1 u t) in (let TMP_3 \def (lift1 p TMP_2) in (let TMP_4 \def 
-(Bind b) in (let TMP_5 \def (lift1 p u) in (let TMP_6 \def (Ss p) in (let 
-TMP_7 \def (lift1 TMP_6 t) in (let TMP_8 \def (THead TMP_4 TMP_5 TMP_7) in 
-(eq T TMP_3 TMP_8)))))))))))) in (let TMP_12 \def (\lambda (u: T).(\lambda 
-(t: T).(let TMP_10 \def (Bind b) in (let TMP_11 \def (THead TMP_10 u t) in 
-(refl_equal T TMP_11))))) in (let TMP_69 \def (\lambda (n: nat).(\lambda (n0: 
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T 
-(lift1 p (THead (Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) 
-t))))))).(\lambda (u: T).(\lambda (t: T).(let TMP_13 \def (Bind b) in (let 
-TMP_14 \def (lift1 p u) in (let TMP_15 \def (Ss p) in (let TMP_16 \def (lift1 
-TMP_15 t) in (let TMP_17 \def (THead TMP_13 TMP_14 TMP_16) in (let TMP_27 
-\def (\lambda (t0: T).(let TMP_18 \def (lift n n0 t0) in (let TMP_19 \def 
-(Bind b) in (let TMP_20 \def (lift1 p u) in (let TMP_21 \def (lift n n0 
-TMP_20) in (let TMP_22 \def (S n0) in (let TMP_23 \def (Ss p) in (let TMP_24 
-\def (lift1 TMP_23 t) in (let TMP_25 \def (lift n TMP_22 TMP_24) in (let 
-TMP_26 \def (THead TMP_19 TMP_21 TMP_25) in (eq T TMP_18 TMP_26))))))))))) in 
-(let TMP_28 \def (Bind b) in (let TMP_29 \def (lift1 p u) in (let TMP_30 \def 
-(lift n n0 TMP_29) in (let TMP_31 \def (S n0) in (let TMP_32 \def (Ss p) in 
-(let TMP_33 \def (lift1 TMP_32 t) in (let TMP_34 \def (lift n TMP_31 TMP_33) 
-in (let TMP_35 \def (THead TMP_28 TMP_30 TMP_34) in (let TMP_44 \def (\lambda 
-(t0: T).(let TMP_36 \def (Bind b) in (let TMP_37 \def (lift1 p u) in (let 
-TMP_38 \def (lift n n0 TMP_37) in (let TMP_39 \def (S n0) in (let TMP_40 \def 
-(Ss p) in (let TMP_41 \def (lift1 TMP_40 t) in (let TMP_42 \def (lift n 
-TMP_39 TMP_41) in (let TMP_43 \def (THead TMP_36 TMP_38 TMP_42) in (eq T t0 
-TMP_43)))))))))) in (let TMP_45 \def (Bind b) in (let TMP_46 \def (lift1 p u) 
-in (let TMP_47 \def (lift n n0 TMP_46) in (let TMP_48 \def (S n0) in (let 
-TMP_49 \def (Ss p) in (let TMP_50 \def (lift1 TMP_49 t) in (let TMP_51 \def 
-(lift n TMP_48 TMP_50) in (let TMP_52 \def (THead TMP_45 TMP_47 TMP_51) in 
-(let TMP_53 \def (refl_equal T TMP_52) in (let TMP_54 \def (Bind b) in (let 
-TMP_55 \def (lift1 p u) in (let TMP_56 \def (Ss p) in (let TMP_57 \def (lift1 
-TMP_56 t) in (let TMP_58 \def (THead TMP_54 TMP_55 TMP_57) in (let TMP_59 
-\def (lift n n0 TMP_58) in (let TMP_60 \def (lift1 p u) in (let TMP_61 \def 
-(Ss p) in (let TMP_62 \def (lift1 TMP_61 t) in (let TMP_63 \def (lift_bind b 
-TMP_60 TMP_62 n n0) in (let TMP_64 \def (eq_ind_r T TMP_35 TMP_44 TMP_53 
-TMP_59 TMP_63) in (let TMP_65 \def (Bind b) in (let TMP_66 \def (THead TMP_65 
-u t) in (let TMP_67 \def (lift1 p TMP_66) in (let TMP_68 \def (H u t) in 
-(eq_ind_r T TMP_17 TMP_27 TMP_64 TMP_67 
-TMP_68)))))))))))))))))))))))))))))))))))))))))))))) in (PList_ind TMP_9 
-TMP_12 TMP_69 hds))))).
+ \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b) 
+(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal 
+T (THead (Bind b) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: 
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead 
+(Bind b) u t)) (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))))))).(\lambda 
+(u: T).(\lambda (t: T).(eq_ind_r T (THead (Bind b) (lift1 p u) (lift1 (Ss p) 
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (THead (Bind b) (lift n n0 (lift1 p 
+u)) (lift n (S n0) (lift1 (Ss p) t))))) (eq_ind_r T (THead (Bind b) (lift n 
+n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 
+(THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 (Ss p) t))))) 
+(refl_equal T (THead (Bind b) (lift n n0 (lift1 p u)) (lift n (S n0) (lift1 
+(Ss p) t)))) (lift n n0 (THead (Bind b) (lift1 p u) (lift1 (Ss p) t))) 
+(lift_bind b (lift1 p u) (lift1 (Ss p) t) n n0)) (lift1 p (THead (Bind b) u 
+t)) (H u t)))))))) hds)).
 
-theorem lift1_flat:
+lemma lift1_flat:
  \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T 
 (lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds 
 t))))))
 \def
- \lambda (f: F).(\lambda (hds: PList).(let TMP_8 \def (\lambda (p: 
-PList).(\forall (u: T).(\forall (t: T).(let TMP_1 \def (Flat f) in (let TMP_2 
-\def (THead TMP_1 u t) in (let TMP_3 \def (lift1 p TMP_2) in (let TMP_4 \def 
-(Flat f) in (let TMP_5 \def (lift1 p u) in (let TMP_6 \def (lift1 p t) in 
-(let TMP_7 \def (THead TMP_4 TMP_5 TMP_6) in (eq T TMP_3 TMP_7))))))))))) in 
-(let TMP_11 \def (\lambda (u: T).(\lambda (t: T).(let TMP_9 \def (Flat f) in 
-(let TMP_10 \def (THead TMP_9 u t) in (refl_equal T TMP_10))))) in (let 
-TMP_57 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda 
-(H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) 
-(THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u: T).(\lambda (t: 
-T).(let TMP_12 \def (Flat f) in (let TMP_13 \def (lift1 p u) in (let TMP_14 
-\def (lift1 p t) in (let TMP_15 \def (THead TMP_12 TMP_13 TMP_14) in (let 
-TMP_23 \def (\lambda (t0: T).(let TMP_16 \def (lift n n0 t0) in (let TMP_17 
-\def (Flat f) in (let TMP_18 \def (lift1 p u) in (let TMP_19 \def (lift n n0 
-TMP_18) in (let TMP_20 \def (lift1 p t) in (let TMP_21 \def (lift n n0 
-TMP_20) in (let TMP_22 \def (THead TMP_17 TMP_19 TMP_21) in (eq T TMP_16 
-TMP_22))))))))) in (let TMP_24 \def (Flat f) in (let TMP_25 \def (lift1 p u) 
-in (let TMP_26 \def (lift n n0 TMP_25) in (let TMP_27 \def (lift1 p t) in 
-(let TMP_28 \def (lift n n0 TMP_27) in (let TMP_29 \def (THead TMP_24 TMP_26 
-TMP_28) in (let TMP_36 \def (\lambda (t0: T).(let TMP_30 \def (Flat f) in 
-(let TMP_31 \def (lift1 p u) in (let TMP_32 \def (lift n n0 TMP_31) in (let 
-TMP_33 \def (lift1 p t) in (let TMP_34 \def (lift n n0 TMP_33) in (let TMP_35 
-\def (THead TMP_30 TMP_32 TMP_34) in (eq T t0 TMP_35)))))))) in (let TMP_37 
-\def (Flat f) in (let TMP_38 \def (lift1 p u) in (let TMP_39 \def (lift n n0 
-TMP_38) in (let TMP_40 \def (lift1 p t) in (let TMP_41 \def (lift n n0 
-TMP_40) in (let TMP_42 \def (THead TMP_37 TMP_39 TMP_41) in (let TMP_43 \def 
-(refl_equal T TMP_42) in (let TMP_44 \def (Flat f) in (let TMP_45 \def (lift1 
-p u) in (let TMP_46 \def (lift1 p t) in (let TMP_47 \def (THead TMP_44 TMP_45 
-TMP_46) in (let TMP_48 \def (lift n n0 TMP_47) in (let TMP_49 \def (lift1 p 
-u) in (let TMP_50 \def (lift1 p t) in (let TMP_51 \def (lift_flat f TMP_49 
-TMP_50 n n0) in (let TMP_52 \def (eq_ind_r T TMP_29 TMP_36 TMP_43 TMP_48 
-TMP_51) in (let TMP_53 \def (Flat f) in (let TMP_54 \def (THead TMP_53 u t) 
-in (let TMP_55 \def (lift1 p TMP_54) in (let TMP_56 \def (H u t) in (eq_ind_r 
-T TMP_15 TMP_23 TMP_52 TMP_55 TMP_56))))))))))))))))))))))))))))))))))))))) 
-in (PList_ind TMP_8 TMP_11 TMP_57 hds))))).
+ \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall 
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) 
+(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T 
+(THead (Flat f) u t)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: 
+PList).(\lambda (H: ((\forall (u: T).(\forall (t: T).(eq T (lift1 p (THead 
+(Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p t))))))).(\lambda (u: 
+T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p u) (lift1 p t)) 
+(\lambda (t0: T).(eq T (lift n n0 t0) (THead (Flat f) (lift n n0 (lift1 p u)) 
+(lift n n0 (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift n n0 (lift1 p 
+u)) (lift n n0 (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat f) (lift 
+n n0 (lift1 p u)) (lift n n0 (lift1 p t))))) (refl_equal T (THead (Flat f) 
+(lift n n0 (lift1 p u)) (lift n n0 (lift1 p t)))) (lift n n0 (THead (Flat f) 
+(lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) n n0)) (lift1 
+p (THead (Flat f) u t)) (H u t)))))))) hds)).
 
-theorem lift1_cons_tail:
+lemma lift1_cons_tail:
  \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq 
 T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t))))))
 \def
  \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds: 
-PList).(let TMP_5 \def (\lambda (p: PList).(let TMP_1 \def (PConsTail p h d) 
-in (let TMP_2 \def (lift1 TMP_1 t) in (let TMP_3 \def (lift h d t) in (let 
-TMP_4 \def (lift1 p TMP_3) in (eq T TMP_2 TMP_4)))))) in (let TMP_6 \def 
-(lift h d t) in (let TMP_7 \def (refl_equal T TMP_6) in (let TMP_21 \def 
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T 
-(lift1 (PConsTail p h d) t) (lift1 p (lift h d t)))).(let TMP_8 \def (lift h 
-d t) in (let TMP_9 \def (lift1 p TMP_8) in (let TMP_14 \def (\lambda (t0: 
-T).(let TMP_10 \def (lift n n0 t0) in (let TMP_11 \def (lift h d t) in (let 
-TMP_12 \def (lift1 p TMP_11) in (let TMP_13 \def (lift n n0 TMP_12) in (eq T 
-TMP_10 TMP_13)))))) in (let TMP_15 \def (lift h d t) in (let TMP_16 \def 
-(lift1 p TMP_15) in (let TMP_17 \def (lift n n0 TMP_16) in (let TMP_18 \def 
-(refl_equal T TMP_17) in (let TMP_19 \def (PConsTail p h d) in (let TMP_20 
-\def (lift1 TMP_19 t) in (eq_ind_r T TMP_9 TMP_14 TMP_18 TMP_20 
-H)))))))))))))) in (PList_ind TMP_5 TMP_7 TMP_21 hds)))))))).
+PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t) 
+(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1 
+(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d 
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d 
+t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p 
+h d) t) H))))) hds)))).
 
-theorem lifts1_flat:
+lemma lifts1_flat:
  \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts: 
 TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds 
 ts) (lift1 hds t))))))
 \def
  \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts: 
-TList).(let TMP_8 \def (\lambda (t0: TList).(let TMP_1 \def (Flat f) in (let 
-TMP_2 \def (THeads TMP_1 t0 t) in (let TMP_3 \def (lift1 hds TMP_2) in (let 
-TMP_4 \def (Flat f) in (let TMP_5 \def (lifts1 hds t0) in (let TMP_6 \def 
-(lift1 hds t) in (let TMP_7 \def (THeads TMP_4 TMP_5 TMP_6) in (eq T TMP_3 
-TMP_7))))))))) in (let TMP_9 \def (lift1 hds t) in (let TMP_10 \def 
-(refl_equal T TMP_9) in (let TMP_60 \def (\lambda (t0: T).(\lambda (t1: 
-TList).(\lambda (H: (eq T (lift1 hds (THeads (Flat f) t1 t)) (THeads (Flat f) 
-(lifts1 hds t1) (lift1 hds t)))).(let TMP_11 \def (Flat f) in (let TMP_12 
-\def (lift1 hds t0) in (let TMP_13 \def (Flat f) in (let TMP_14 \def (THeads 
-TMP_13 t1 t) in (let TMP_15 \def (lift1 hds TMP_14) in (let TMP_16 \def 
-(THead TMP_11 TMP_12 TMP_15) in (let TMP_24 \def (\lambda (t2: T).(let TMP_17 
-\def (Flat f) in (let TMP_18 \def (lift1 hds t0) in (let TMP_19 \def (Flat f) 
-in (let TMP_20 \def (lifts1 hds t1) in (let TMP_21 \def (lift1 hds t) in (let 
-TMP_22 \def (THeads TMP_19 TMP_20 TMP_21) in (let TMP_23 \def (THead TMP_17 
-TMP_18 TMP_22) in (eq T t2 TMP_23))))))))) in (let TMP_25 \def (Flat f) in 
-(let TMP_26 \def (lifts1 hds t1) in (let TMP_27 \def (lift1 hds t) in (let 
-TMP_28 \def (THeads TMP_25 TMP_26 TMP_27) in (let TMP_39 \def (\lambda (t2: 
-T).(let TMP_29 \def (Flat f) in (let TMP_30 \def (lift1 hds t0) in (let 
-TMP_31 \def (THead TMP_29 TMP_30 t2) in (let TMP_32 \def (Flat f) in (let 
-TMP_33 \def (lift1 hds t0) in (let TMP_34 \def (Flat f) in (let TMP_35 \def 
-(lifts1 hds t1) in (let TMP_36 \def (lift1 hds t) in (let TMP_37 \def (THeads 
-TMP_34 TMP_35 TMP_36) in (let TMP_38 \def (THead TMP_32 TMP_33 TMP_37) in (eq 
-T TMP_31 TMP_38)))))))))))) in (let TMP_40 \def (Flat f) in (let TMP_41 \def 
-(lift1 hds t0) in (let TMP_42 \def (Flat f) in (let TMP_43 \def (lifts1 hds 
-t1) in (let TMP_44 \def (lift1 hds t) in (let TMP_45 \def (THeads TMP_42 
-TMP_43 TMP_44) in (let TMP_46 \def (THead TMP_40 TMP_41 TMP_45) in (let 
-TMP_47 \def (refl_equal T TMP_46) in (let TMP_48 \def (Flat f) in (let TMP_49 
-\def (THeads TMP_48 t1 t) in (let TMP_50 \def (lift1 hds TMP_49) in (let 
-TMP_51 \def (eq_ind_r T TMP_28 TMP_39 TMP_47 TMP_50 H) in (let TMP_52 \def 
-(Flat f) in (let TMP_53 \def (Flat f) in (let TMP_54 \def (THeads TMP_53 t1 
-t) in (let TMP_55 \def (THead TMP_52 t0 TMP_54) in (let TMP_56 \def (lift1 
-hds TMP_55) in (let TMP_57 \def (Flat f) in (let TMP_58 \def (THeads TMP_57 
-t1 t) in (let TMP_59 \def (lift1_flat f hds t0 TMP_58) in (eq_ind_r T TMP_16 
-TMP_24 TMP_51 TMP_56 TMP_59)))))))))))))))))))))))))))))))))))) in (TList_ind 
-TMP_8 TMP_10 TMP_60 ts)))))))).
+TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0 
+t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1 
+hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds 
+(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds 
+t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f) 
+t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads 
+(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f) 
+(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1 
+hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1) 
+(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat 
+f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H) 
+(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0 
+(THeads (Flat f) t1 t)))))) ts)))).
 
-theorem lifts1_nil:
+lemma lifts1_nil:
  \forall (ts: TList).(eq TList (lifts1 PNil ts) ts)
 \def
- \lambda (ts: TList).(let TMP_2 \def (\lambda (t: TList).(let TMP_1 \def 
-(lifts1 PNil t) in (eq TList TMP_1 t))) in (let TMP_3 \def (refl_equal TList 
-TNil) in (let TMP_10 \def (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: 
-(eq TList (lifts1 PNil t0) t0)).(let TMP_6 \def (\lambda (t1: TList).(let 
-TMP_4 \def (TCons t t1) in (let TMP_5 \def (TCons t t0) in (eq TList TMP_4 
-TMP_5)))) in (let TMP_7 \def (TCons t t0) in (let TMP_8 \def (refl_equal 
-TList TMP_7) in (let TMP_9 \def (lifts1 PNil t0) in (eq_ind_r TList t0 TMP_6 
-TMP_8 TMP_9 H)))))))) in (TList_ind TMP_2 TMP_3 TMP_10 ts)))).
+ \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t) 
+t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: 
+(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq 
+TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1 
+PNil t0) H)))) ts).
 
-theorem lifts1_cons:
+lemma lifts1_cons:
  \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts: 
 TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts))))))
 \def
  \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts: 
-TList).(let TMP_5 \def (\lambda (t: TList).(let TMP_1 \def (PCons h d hds) in 
-(let TMP_2 \def (lifts1 TMP_1 t) in (let TMP_3 \def (lifts1 hds t) in (let 
-TMP_4 \def (lifts h d TMP_3) in (eq TList TMP_2 TMP_4)))))) in (let TMP_6 
-\def (refl_equal TList TNil) in (let TMP_26 \def (\lambda (t: T).(\lambda 
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t) 
+(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda 
 (t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d 
-(lifts1 hds t0)))).(let TMP_7 \def (lifts1 hds t0) in (let TMP_8 \def (lifts 
-h d TMP_7) in (let TMP_17 \def (\lambda (t1: TList).(let TMP_9 \def (lift1 
-hds t) in (let TMP_10 \def (lift h d TMP_9) in (let TMP_11 \def (TCons TMP_10 
-t1) in (let TMP_12 \def (lift1 hds t) in (let TMP_13 \def (lift h d TMP_12) 
-in (let TMP_14 \def (lifts1 hds t0) in (let TMP_15 \def (lifts h d TMP_14) in 
-(let TMP_16 \def (TCons TMP_13 TMP_15) in (eq TList TMP_11 TMP_16)))))))))) 
-in (let TMP_18 \def (lift1 hds t) in (let TMP_19 \def (lift h d TMP_18) in 
-(let TMP_20 \def (lifts1 hds t0) in (let TMP_21 \def (lifts h d TMP_20) in 
-(let TMP_22 \def (TCons TMP_19 TMP_21) in (let TMP_23 \def (refl_equal TList 
-TMP_22) in (let TMP_24 \def (PCons h d hds) in (let TMP_25 \def (lifts1 
-TMP_24 t0) in (eq_ind_r TList TMP_8 TMP_17 TMP_23 TMP_25 H))))))))))))))) in 
-(TList_ind TMP_5 TMP_6 TMP_26 ts))))))).
+(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1: 
+TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1 
+hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d 
+(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0) 
+H)))) ts)))).