X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Flift1%2Fprops.ma;h=bb6374c163e23e6d07af085837a28ee81fb4a039;hb=7b95759c5011a25f96d5171561ea79d063770db4;hp=ebda0267b48515dad8367c3aa7ee12d1145946c5;hpb=d795687ffe924872a5e36122c2bd3069d6409454;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
index ebda0267b..bb6374c16 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/lift1/props.ma
@@ -14,126 +14,231 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/lift/props.ma".
+include "basic_1/lift1/defs.ma".
 
-include "Basic-1/drop1/defs.ma".
+include "basic_1/lift/props.ma".
 
-theorem lift1_lift1:
- \forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1 
-(lift1 is2 t)) (lift1 (papp is1 is2) t))))
+theorem lift1_sort:
+ \forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
 \def
- \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: 
-PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 (papp p is2) 
-t))))) (\lambda (is2: PList).(\lambda (t: T).(refl_equal T (lift1 is2 t)))) 
-(\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: 
-((\forall (is2: PList).(\forall (t: T).(eq T (lift1 p (lift1 is2 t)) (lift1 
-(papp p is2) t)))))).(\lambda (is2: PList).(\lambda (t: T).(f_equal3 nat nat 
-T T lift n n n0 n0 (lift1 p (lift1 is2 t)) (lift1 (papp p is2) t) (refl_equal 
-nat n) (refl_equal nat n0) (H is2 t)))))))) is1).
-(* COMMENTS
-Initial nodes: 145
-END *)
+ \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: 
+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)))))).
 
-theorem lift1_xhg:
- \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) 
-(lift (S O) O (lift1 hds t))))
+theorem lift1_lref:
+ \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef 
+(trans hds i))))
 \def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T 
-(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t: 
-T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d: 
-nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p) 
-(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T 
-(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S 
-O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n: 
-nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d 
-(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda 
-(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift 
-(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1 
-p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S 
-d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds).
-(* COMMENTS
-Initial nodes: 371
-END *)
+ \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)))).
 
-theorem lifts1_xhg:
- \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts 
-(S O) O ts)) (lifts (S O) O (lifts1 hds ts))))
+theorem 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 (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq 
-TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t)))) 
-(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq 
-TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds 
-t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList 
-(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1 
-hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O 
-(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1 
-hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds 
-t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O 
-(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds) 
-(lift (S O) O t)) (lift1_xhg hds t))))) ts)).
-(* COMMENTS
-Initial nodes: 307
-END *)
+ \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))))).
 
-theorem lift1_free:
- \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds 
-(lift (S i) O t)) (lift (S (trans hds i)) O (lift1 (ptrans hds i) t)))))
+theorem 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 (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: 
-nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O 
-(lift1 (ptrans p i) t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T 
-(lift (S i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: 
-PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0 
-(lift (S i) O t)) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) 
-t))))))).(\lambda (i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 
-i)) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T (lift h d t0) (lift 
-(S (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | 
-false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match (blt (trans hds0 
-i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans 
-hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t)))) (xinduction bool (blt 
-(trans hds0 i) d) (\lambda (b: bool).(eq T (lift h d (lift (S (trans hds0 i)) 
-O (lift1 (ptrans hds0 i) t))) (lift (S (match b with [true \Rightarrow (trans 
-hds0 i) | false \Rightarrow (plus (trans hds0 i) h)])) O (lift1 (match b with 
-[true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | 
-false \Rightarrow (ptrans hds0 i)]) t)))) (\lambda (x_x: bool).(bool_ind 
-(\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b) \to (eq T (lift h d 
-(lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (match b with 
-[true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) 
-h)])) O (lift1 (match b with [true \Rightarrow (PCons h (minus d (S (trans 
-hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) t))))) 
-(\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat (plus (S 
-(trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq T (lift h 
-n (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift (S (trans hds0 
-i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) 
-(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) 
-(lift1 (ptrans hds0 i) t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 
-i)) O (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) t)))) 
-(refl_equal T (lift (S (trans hds0 i)) O (lift1 (PCons h (minus d (S (trans 
-hds0 i))) (ptrans hds0 i)) t))) (lift h (plus (S (trans hds0 i)) (minus d (S 
-(trans hds0 i)))) (lift (S (trans hds0 i)) O (lift1 (ptrans hds0 i) t))) 
-(lift_d (lift1 (ptrans hds0 i) t) h (S (trans hds0 i)) (minus d (S (trans 
-hds0 i))) O (le_O_n (minus d (S (trans hds0 i)))))) d (le_plus_minus (S 
-(trans hds0 i)) d (bge_le (S (trans hds0 i)) d (le_bge (S (trans hds0 i)) d 
-(lt_le_S (trans hds0 i) d (blt_lt d (trans hds0 i) H0))))))) (\lambda (H0: 
-(eq bool (blt (trans hds0 i) d) false)).(eq_ind_r T (lift (plus h (S (trans 
-hds0 i))) O (lift1 (ptrans hds0 i) t)) (\lambda (t0: T).(eq T t0 (lift (S 
-(plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (eq_ind nat (S (plus 
-h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n O (lift1 (ptrans hds0 i) 
-t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) 
-(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O 
-(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans 
-hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1 
-(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_sym h (trans hds0 i))) 
-(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S 
-(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0 
-i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i))) 
-(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda 
-(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d 
-(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i)) 
-(plus_sym O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans 
-hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t)))))))) 
-hds).
-(* COMMENTS
-Initial nodes: 1339
-END *)
+ \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))))).
+
+theorem 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)))))))).
+
+theorem 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)))))))).
+
+theorem 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)))).
+
+theorem 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 
+(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))))))).