components: subst0

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma
index 0234ff06c57445ff15f1ee290f1ce4f7ac2d284a..a787f6872f574eb21b02206e671d08b8108c072b 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/dec.ma
@@ -14,169 +14,419 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/subst0/defs.ma".
+include "basic_1/subst0/defs.ma".
 
-include "Basic-1/lift/props.ma".
+include "basic_1/lift/props.ma".
 
 theorem dnf_dec2:
  \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: 
 T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S 
 O) d v))))))
 \def
- \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w: 
-T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda 
-(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d: 
-nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n) 
-(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d 
-v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort 
-n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T 
-(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n: 
-nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T 
-(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w: 
-T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T 
-(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: 
-T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n) 
-(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d 
-(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind 
-nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0 
-w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift 
-(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w 
-(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S 
-O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n) 
-(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w) 
-(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n 
-(lift n O w)) (lift_free w n (S O) O n (le_n (plus O n)) (le_O_n n)))))) d 
-H)) (\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T 
-(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n) 
-(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0: 
-T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred 
-n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda 
-(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w 
-t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d 
-v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w: 
-T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda 
-(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d) 
-in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 
-d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d 
-v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) 
-(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) 
-d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0 
-(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in 
-(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S 
-O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v)))) 
-(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift 
-(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d 
-v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w 
-t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq 
-T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w) 
-in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift 
-(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S 
-O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s 
-k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda 
-(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w 
-(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d 
-w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 
-t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) 
-(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2)) 
-(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6) 
-(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5)))))) 
-(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) 
-v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or 
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) 
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) 
-(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T 
-(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda 
-(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: 
-T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex 
-T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) 
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x)) 
-(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def 
-H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T 
-(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d 
-v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d 
-x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) 
-x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0) 
-(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O) 
-(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d) 
-x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1 
-H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d 
-v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w: 
-T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex 
-T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x: 
-T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in 
-(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s 
-k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S 
-O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead 
-k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) 
-(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O) 
-d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w 
-(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2 
-t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: 
-T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T 
-(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v)))) 
-(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T 
-(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda 
-(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda 
-(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d) 
-x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1) 
-(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift 
-(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1) 
-t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift 
-(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3)) 
-(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) 
-v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or 
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) 
-d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) 
-(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T 
-(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda 
-(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v: 
-T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x) 
-(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead 
-k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq 
-T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror 
-(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) 
-(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T 
-(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v)))) 
-(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k 
-d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d 
-x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x) 
-(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift 
-(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) 
-d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t).
-(* COMMENTS
-Initial nodes: 3549
-END *)
+ \lambda (t: T).(let TMP_9 \def (\lambda (t0: T).(\forall (d: nat).(let TMP_4 
+\def (\forall (w: T).(let TMP_3 \def (\lambda (v: T).(let TMP_1 \def (S O) in 
+(let TMP_2 \def (lift TMP_1 d v) in (subst0 d w t0 TMP_2)))) in (ex T 
+TMP_3))) in (let TMP_7 \def (\lambda (v: T).(let TMP_5 \def (S O) in (let 
+TMP_6 \def (lift TMP_5 d v) in (eq T t0 TMP_6)))) in (let TMP_8 \def (ex T 
+TMP_7) in (or TMP_4 TMP_8)))))) in (let TMP_37 \def (\lambda (n: 
+nat).(\lambda (d: nat).(let TMP_14 \def (\forall (w: T).(let TMP_13 \def 
+(\lambda (v: T).(let TMP_10 \def (TSort n) in (let TMP_11 \def (S O) in (let 
+TMP_12 \def (lift TMP_11 d v) in (subst0 d w TMP_10 TMP_12))))) in (ex T 
+TMP_13))) in (let TMP_18 \def (\lambda (v: T).(let TMP_15 \def (TSort n) in 
+(let TMP_16 \def (S O) in (let TMP_17 \def (lift TMP_16 d v) in (eq T TMP_15 
+TMP_17))))) in (let TMP_19 \def (ex T TMP_18) in (let TMP_23 \def (\lambda 
+(v: T).(let TMP_20 \def (TSort n) in (let TMP_21 \def (S O) in (let TMP_22 
+\def (lift TMP_21 d v) in (eq T TMP_20 TMP_22))))) in (let TMP_24 \def (TSort 
+n) in (let TMP_25 \def (TSort n) in (let TMP_27 \def (\lambda (t0: T).(let 
+TMP_26 \def (TSort n) in (eq T TMP_26 t0))) in (let TMP_28 \def (TSort n) in 
+(let TMP_29 \def (refl_equal T TMP_28) in (let TMP_30 \def (S O) in (let 
+TMP_31 \def (TSort n) in (let TMP_32 \def (lift TMP_30 d TMP_31) in (let 
+TMP_33 \def (S O) in (let TMP_34 \def (lift_sort n TMP_33 d) in (let TMP_35 
+\def (eq_ind_r T TMP_25 TMP_27 TMP_29 TMP_32 TMP_34) in (let TMP_36 \def 
+(ex_intro T TMP_23 TMP_24 TMP_35) in (or_intror TMP_14 TMP_19 
+TMP_36))))))))))))))))))) in (let TMP_149 \def (\lambda (n: nat).(\lambda (d: 
+nat).(let TMP_42 \def (\forall (w: T).(let TMP_41 \def (\lambda (v: T).(let 
+TMP_38 \def (TLRef n) in (let TMP_39 \def (S O) in (let TMP_40 \def (lift 
+TMP_39 d v) in (subst0 d w TMP_38 TMP_40))))) in (ex T TMP_41))) in (let 
+TMP_46 \def (\lambda (v: T).(let TMP_43 \def (TLRef n) in (let TMP_44 \def (S 
+O) in (let TMP_45 \def (lift TMP_44 d v) in (eq T TMP_43 TMP_45))))) in (let 
+TMP_47 \def (ex T TMP_46) in (let TMP_48 \def (or TMP_42 TMP_47) in (let 
+TMP_76 \def (\lambda (H: (lt n d)).(let TMP_53 \def (\forall (w: T).(let 
+TMP_52 \def (\lambda (v: T).(let TMP_49 \def (TLRef n) in (let TMP_50 \def (S 
+O) in (let TMP_51 \def (lift TMP_50 d v) in (subst0 d w TMP_49 TMP_51))))) in 
+(ex T TMP_52))) in (let TMP_57 \def (\lambda (v: T).(let TMP_54 \def (TLRef 
+n) in (let TMP_55 \def (S O) in (let TMP_56 \def (lift TMP_55 d v) in (eq T 
+TMP_54 TMP_56))))) in (let TMP_58 \def (ex T TMP_57) in (let TMP_62 \def 
+(\lambda (v: T).(let TMP_59 \def (TLRef n) in (let TMP_60 \def (S O) in (let 
+TMP_61 \def (lift TMP_60 d v) in (eq T TMP_59 TMP_61))))) in (let TMP_63 \def 
+(TLRef n) in (let TMP_64 \def (TLRef n) in (let TMP_66 \def (\lambda (t0: 
+T).(let TMP_65 \def (TLRef n) in (eq T TMP_65 t0))) in (let TMP_67 \def 
+(TLRef n) in (let TMP_68 \def (refl_equal T TMP_67) in (let TMP_69 \def (S O) 
+in (let TMP_70 \def (TLRef n) in (let TMP_71 \def (lift TMP_69 d TMP_70) in 
+(let TMP_72 \def (S O) in (let TMP_73 \def (lift_lref_lt n TMP_72 d H) in 
+(let TMP_74 \def (eq_ind_r T TMP_64 TMP_66 TMP_68 TMP_71 TMP_73) in (let 
+TMP_75 \def (ex_intro T TMP_62 TMP_63 TMP_74) in (or_intror TMP_53 TMP_58 
+TMP_75)))))))))))))))))) in (let TMP_119 \def (\lambda (H: (eq nat n d)).(let 
+TMP_87 \def (\lambda (n0: nat).(let TMP_81 \def (\forall (w: T).(let TMP_80 
+\def (\lambda (v: T).(let TMP_77 \def (TLRef n) in (let TMP_78 \def (S O) in 
+(let TMP_79 \def (lift TMP_78 n0 v) in (subst0 n0 w TMP_77 TMP_79))))) in (ex 
+T TMP_80))) in (let TMP_85 \def (\lambda (v: T).(let TMP_82 \def (TLRef n) in 
+(let TMP_83 \def (S O) in (let TMP_84 \def (lift TMP_83 n0 v) in (eq T TMP_82 
+TMP_84))))) in (let TMP_86 \def (ex T TMP_85) in (or TMP_81 TMP_86))))) in 
+(let TMP_92 \def (\forall (w: T).(let TMP_91 \def (\lambda (v: T).(let TMP_88 
+\def (TLRef n) in (let TMP_89 \def (S O) in (let TMP_90 \def (lift TMP_89 n 
+v) in (subst0 n w TMP_88 TMP_90))))) in (ex T TMP_91))) in (let TMP_96 \def 
+(\lambda (v: T).(let TMP_93 \def (TLRef n) in (let TMP_94 \def (S O) in (let 
+TMP_95 \def (lift TMP_94 n v) in (eq T TMP_93 TMP_95))))) in (let TMP_97 \def 
+(ex T TMP_96) in (let TMP_117 \def (\lambda (w: T).(let TMP_101 \def (\lambda 
+(v: T).(let TMP_98 \def (TLRef n) in (let TMP_99 \def (S O) in (let TMP_100 
+\def (lift TMP_99 n v) in (subst0 n w TMP_98 TMP_100))))) in (let TMP_102 
+\def (lift n O w) in (let TMP_103 \def (S O) in (let TMP_104 \def (plus 
+TMP_103 n) in (let TMP_105 \def (lift TMP_104 O w) in (let TMP_107 \def 
+(\lambda (t0: T).(let TMP_106 \def (TLRef n) in (subst0 n w TMP_106 t0))) in 
+(let TMP_108 \def (subst0_lref w n) in (let TMP_109 \def (S O) in (let 
+TMP_110 \def (lift n O w) in (let TMP_111 \def (lift TMP_109 n TMP_110) in 
+(let TMP_112 \def (S O) in (let TMP_113 \def (le_plus_r O n) in (let TMP_114 
+\def (le_O_n n) in (let TMP_115 \def (lift_free w n TMP_112 O n TMP_113 
+TMP_114) in (let TMP_116 \def (eq_ind_r T TMP_105 TMP_107 TMP_108 TMP_111 
+TMP_115) in (ex_intro T TMP_101 TMP_102 TMP_116))))))))))))))))) in (let 
+TMP_118 \def (or_introl TMP_92 TMP_97 TMP_117) in (eq_ind nat n TMP_87 
+TMP_118 d H)))))))) in (let TMP_148 \def (\lambda (H: (lt d n)).(let TMP_124 
+\def (\forall (w: T).(let TMP_123 \def (\lambda (v: T).(let TMP_120 \def 
+(TLRef n) in (let TMP_121 \def (S O) in (let TMP_122 \def (lift TMP_121 d v) 
+in (subst0 d w TMP_120 TMP_122))))) in (ex T TMP_123))) in (let TMP_128 \def 
+(\lambda (v: T).(let TMP_125 \def (TLRef n) in (let TMP_126 \def (S O) in 
+(let TMP_127 \def (lift TMP_126 d v) in (eq T TMP_125 TMP_127))))) in (let 
+TMP_129 \def (ex T TMP_128) in (let TMP_133 \def (\lambda (v: T).(let TMP_130 
+\def (TLRef n) in (let TMP_131 \def (S O) in (let TMP_132 \def (lift TMP_131 
+d v) in (eq T TMP_130 TMP_132))))) in (let TMP_134 \def (pred n) in (let 
+TMP_135 \def (TLRef TMP_134) in (let TMP_136 \def (TLRef n) in (let TMP_138 
+\def (\lambda (t0: T).(let TMP_137 \def (TLRef n) in (eq T TMP_137 t0))) in 
+(let TMP_139 \def (TLRef n) in (let TMP_140 \def (refl_equal T TMP_139) in 
+(let TMP_141 \def (S O) in (let TMP_142 \def (pred n) in (let TMP_143 \def 
+(TLRef TMP_142) in (let TMP_144 \def (lift TMP_141 d TMP_143) in (let TMP_145 
+\def (lift_lref_gt d n H) in (let TMP_146 \def (eq_ind_r T TMP_136 TMP_138 
+TMP_140 TMP_144 TMP_145) in (let TMP_147 \def (ex_intro T TMP_133 TMP_135 
+TMP_146) in (or_intror TMP_124 TMP_129 TMP_147))))))))))))))))))) in 
+(lt_eq_gt_e n d TMP_48 TMP_76 TMP_119 TMP_148)))))))))) in (let TMP_562 \def 
+(\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(or (\forall 
+(w: T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T 
+(\lambda (v: T).(eq T t0 (lift (S O) d v)))))))).(\lambda (t1: T).(\lambda 
+(H0: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w 
+t1 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) d 
+v)))))))).(\lambda (d: nat).(let H_x \def (H d) in (let H1 \def H_x in (let 
+TMP_153 \def (\forall (w: T).(let TMP_152 \def (\lambda (v: T).(let TMP_150 
+\def (S O) in (let TMP_151 \def (lift TMP_150 d v) in (subst0 d w t0 
+TMP_151)))) in (ex T TMP_152))) in (let TMP_156 \def (\lambda (v: T).(let 
+TMP_154 \def (S O) in (let TMP_155 \def (lift TMP_154 d v) in (eq T t0 
+TMP_155)))) in (let TMP_157 \def (ex T TMP_156) in (let TMP_162 \def (\forall 
+(w: T).(let TMP_161 \def (\lambda (v: T).(let TMP_158 \def (THead k t0 t1) in 
+(let TMP_159 \def (S O) in (let TMP_160 \def (lift TMP_159 d v) in (subst0 d 
+w TMP_158 TMP_160))))) in (ex T TMP_161))) in (let TMP_166 \def (\lambda (v: 
+T).(let TMP_163 \def (THead k t0 t1) in (let TMP_164 \def (S O) in (let 
+TMP_165 \def (lift TMP_164 d v) in (eq T TMP_163 TMP_165))))) in (let TMP_167 
+\def (ex T TMP_166) in (let TMP_168 \def (or TMP_162 TMP_167) in (let TMP_341 
+\def (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0 
+(lift (S O) d v))))))).(let TMP_169 \def (s k d) in (let H_x0 \def (H0 
+TMP_169) in (let H3 \def H_x0 in (let TMP_175 \def (\forall (w: T).(let 
+TMP_174 \def (\lambda (v: T).(let TMP_170 \def (s k d) in (let TMP_171 \def 
+(S O) in (let TMP_172 \def (s k d) in (let TMP_173 \def (lift TMP_171 TMP_172 
+v) in (subst0 TMP_170 w t1 TMP_173)))))) in (ex T TMP_174))) in (let TMP_179 
+\def (\lambda (v: T).(let TMP_176 \def (S O) in (let TMP_177 \def (s k d) in 
+(let TMP_178 \def (lift TMP_176 TMP_177 v) in (eq T t1 TMP_178))))) in (let 
+TMP_180 \def (ex T TMP_179) in (let TMP_185 \def (\forall (w: T).(let TMP_184 
+\def (\lambda (v: T).(let TMP_181 \def (THead k t0 t1) in (let TMP_182 \def 
+(S O) in (let TMP_183 \def (lift TMP_182 d v) in (subst0 d w TMP_181 
+TMP_183))))) in (ex T TMP_184))) in (let TMP_189 \def (\lambda (v: T).(let 
+TMP_186 \def (THead k t0 t1) in (let TMP_187 \def (S O) in (let TMP_188 \def 
+(lift TMP_187 d v) in (eq T TMP_186 TMP_188))))) in (let TMP_190 \def (ex T 
+TMP_189) in (let TMP_191 \def (or TMP_185 TMP_190) in (let TMP_248 \def 
+(\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 
+(lift (S O) (s k d) v))))))).(let TMP_196 \def (\forall (w: T).(let TMP_195 
+\def (\lambda (v: T).(let TMP_192 \def (THead k t0 t1) in (let TMP_193 \def 
+(S O) in (let TMP_194 \def (lift TMP_193 d v) in (subst0 d w TMP_192 
+TMP_194))))) in (ex T TMP_195))) in (let TMP_200 \def (\lambda (v: T).(let 
+TMP_197 \def (THead k t0 t1) in (let TMP_198 \def (S O) in (let TMP_199 \def 
+(lift TMP_198 d v) in (eq T TMP_197 TMP_199))))) in (let TMP_201 \def (ex T 
+TMP_200) in (let TMP_247 \def (\lambda (w: T).(let H_x1 \def (H4 w) in (let 
+H5 \def H_x1 in (let TMP_206 \def (\lambda (v: T).(let TMP_202 \def (s k d) 
+in (let TMP_203 \def (S O) in (let TMP_204 \def (s k d) in (let TMP_205 \def 
+(lift TMP_203 TMP_204 v) in (subst0 TMP_202 w t1 TMP_205)))))) in (let 
+TMP_210 \def (\lambda (v: T).(let TMP_207 \def (THead k t0 t1) in (let 
+TMP_208 \def (S O) in (let TMP_209 \def (lift TMP_208 d v) in (subst0 d w 
+TMP_207 TMP_209))))) in (let TMP_211 \def (ex T TMP_210) in (let TMP_246 \def 
+(\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s k d) 
+x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (let TMP_214 \def (\lambda 
+(v: T).(let TMP_212 \def (S O) in (let TMP_213 \def (lift TMP_212 d v) in 
+(subst0 d w t0 TMP_213)))) in (let TMP_218 \def (\lambda (v: T).(let TMP_215 
+\def (THead k t0 t1) in (let TMP_216 \def (S O) in (let TMP_217 \def (lift 
+TMP_216 d v) in (subst0 d w TMP_215 TMP_217))))) in (let TMP_219 \def (ex T 
+TMP_218) in (let TMP_245 \def (\lambda (x0: T).(\lambda (H8: (subst0 d w t0 
+(lift (S O) d x0))).(let TMP_223 \def (\lambda (v: T).(let TMP_220 \def 
+(THead k t0 t1) in (let TMP_221 \def (S O) in (let TMP_222 \def (lift TMP_221 
+d v) in (subst0 d w TMP_220 TMP_222))))) in (let TMP_224 \def (THead k x0 x) 
+in (let TMP_225 \def (S O) in (let TMP_226 \def (lift TMP_225 d x0) in (let 
+TMP_227 \def (S O) in (let TMP_228 \def (s k d) in (let TMP_229 \def (lift 
+TMP_227 TMP_228 x) in (let TMP_230 \def (THead k TMP_226 TMP_229) in (let 
+TMP_232 \def (\lambda (t2: T).(let TMP_231 \def (THead k t0 t1) in (subst0 d 
+w TMP_231 t2))) in (let TMP_233 \def (S O) in (let TMP_234 \def (lift TMP_233 
+d x0) in (let TMP_235 \def (S O) in (let TMP_236 \def (s k d) in (let TMP_237 
+\def (lift TMP_235 TMP_236 x) in (let TMP_238 \def (subst0_both w t0 TMP_234 
+d H8 k t1 TMP_237 H6) in (let TMP_239 \def (S O) in (let TMP_240 \def (THead 
+k x0 x) in (let TMP_241 \def (lift TMP_239 d TMP_240) in (let TMP_242 \def (S 
+O) in (let TMP_243 \def (lift_head k x0 x TMP_242 d) in (let TMP_244 \def 
+(eq_ind_r T TMP_230 TMP_232 TMP_238 TMP_241 TMP_243) in (ex_intro T TMP_223 
+TMP_224 TMP_244)))))))))))))))))))))))) in (ex_ind T TMP_214 TMP_219 TMP_245 
+H7))))))))) in (ex_ind T TMP_206 TMP_211 TMP_246 H5)))))))) in (or_introl 
+TMP_196 TMP_201 TMP_247)))))) in (let TMP_340 \def (\lambda (H4: (ex T 
+(\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))))).(let TMP_252 \def 
+(\lambda (v: T).(let TMP_249 \def (S O) in (let TMP_250 \def (s k d) in (let 
+TMP_251 \def (lift TMP_249 TMP_250 v) in (eq T t1 TMP_251))))) in (let 
+TMP_257 \def (\forall (w: T).(let TMP_256 \def (\lambda (v: T).(let TMP_253 
+\def (THead k t0 t1) in (let TMP_254 \def (S O) in (let TMP_255 \def (lift 
+TMP_254 d v) in (subst0 d w TMP_253 TMP_255))))) in (ex T TMP_256))) in (let 
+TMP_261 \def (\lambda (v: T).(let TMP_258 \def (THead k t0 t1) in (let 
+TMP_259 \def (S O) in (let TMP_260 \def (lift TMP_259 d v) in (eq T TMP_258 
+TMP_260))))) in (let TMP_262 \def (ex T TMP_261) in (let TMP_263 \def (or 
+TMP_257 TMP_262) in (let TMP_339 \def (\lambda (x: T).(\lambda (H5: (eq T t1 
+(lift (S O) (s k d) x))).(let TMP_264 \def (S O) in (let TMP_265 \def (s k d) 
+in (let TMP_266 \def (lift TMP_264 TMP_265 x) in (let TMP_277 \def (\lambda 
+(t2: T).(let TMP_271 \def (\forall (w: T).(let TMP_270 \def (\lambda (v: 
+T).(let TMP_267 \def (THead k t0 t2) in (let TMP_268 \def (S O) in (let 
+TMP_269 \def (lift TMP_268 d v) in (subst0 d w TMP_267 TMP_269))))) in (ex T 
+TMP_270))) in (let TMP_275 \def (\lambda (v: T).(let TMP_272 \def (THead k t0 
+t2) in (let TMP_273 \def (S O) in (let TMP_274 \def (lift TMP_273 d v) in (eq 
+T TMP_272 TMP_274))))) in (let TMP_276 \def (ex T TMP_275) in (or TMP_271 
+TMP_276))))) in (let TMP_285 \def (\forall (w: T).(let TMP_284 \def (\lambda 
+(v: T).(let TMP_278 \def (S O) in (let TMP_279 \def (s k d) in (let TMP_280 
+\def (lift TMP_278 TMP_279 x) in (let TMP_281 \def (THead k t0 TMP_280) in 
+(let TMP_282 \def (S O) in (let TMP_283 \def (lift TMP_282 d v) in (subst0 d 
+w TMP_281 TMP_283)))))))) in (ex T TMP_284))) in (let TMP_292 \def (\lambda 
+(v: T).(let TMP_286 \def (S O) in (let TMP_287 \def (s k d) in (let TMP_288 
+\def (lift TMP_286 TMP_287 x) in (let TMP_289 \def (THead k t0 TMP_288) in 
+(let TMP_290 \def (S O) in (let TMP_291 \def (lift TMP_290 d v) in (eq T 
+TMP_289 TMP_291)))))))) in (let TMP_293 \def (ex T TMP_292) in (let TMP_337 
+\def (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def H_x1 in (let 
+TMP_296 \def (\lambda (v: T).(let TMP_294 \def (S O) in (let TMP_295 \def 
+(lift TMP_294 d v) in (subst0 d w t0 TMP_295)))) in (let TMP_303 \def 
+(\lambda (v: T).(let TMP_297 \def (S O) in (let TMP_298 \def (s k d) in (let 
+TMP_299 \def (lift TMP_297 TMP_298 x) in (let TMP_300 \def (THead k t0 
+TMP_299) in (let TMP_301 \def (S O) in (let TMP_302 \def (lift TMP_301 d v) 
+in (subst0 d w TMP_300 TMP_302)))))))) in (let TMP_304 \def (ex T TMP_303) in 
+(let TMP_336 \def (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d 
+x0))).(let TMP_311 \def (\lambda (v: T).(let TMP_305 \def (S O) in (let 
+TMP_306 \def (s k d) in (let TMP_307 \def (lift TMP_305 TMP_306 x) in (let 
+TMP_308 \def (THead k t0 TMP_307) in (let TMP_309 \def (S O) in (let TMP_310 
+\def (lift TMP_309 d v) in (subst0 d w TMP_308 TMP_310)))))))) in (let 
+TMP_312 \def (THead k x0 x) in (let TMP_313 \def (S O) in (let TMP_314 \def 
+(lift TMP_313 d x0) in (let TMP_315 \def (S O) in (let TMP_316 \def (s k d) 
+in (let TMP_317 \def (lift TMP_315 TMP_316 x) in (let TMP_318 \def (THead k 
+TMP_314 TMP_317) in (let TMP_323 \def (\lambda (t2: T).(let TMP_319 \def (S 
+O) in (let TMP_320 \def (s k d) in (let TMP_321 \def (lift TMP_319 TMP_320 x) 
+in (let TMP_322 \def (THead k t0 TMP_321) in (subst0 d w TMP_322 t2)))))) in 
+(let TMP_324 \def (S O) in (let TMP_325 \def (lift TMP_324 d x0) in (let 
+TMP_326 \def (S O) in (let TMP_327 \def (s k d) in (let TMP_328 \def (lift 
+TMP_326 TMP_327 x) in (let TMP_329 \def (subst0_fst w TMP_325 t0 d H7 TMP_328 
+k) in (let TMP_330 \def (S O) in (let TMP_331 \def (THead k x0 x) in (let 
+TMP_332 \def (lift TMP_330 d TMP_331) in (let TMP_333 \def (S O) in (let 
+TMP_334 \def (lift_head k x0 x TMP_333 d) in (let TMP_335 \def (eq_ind_r T 
+TMP_318 TMP_323 TMP_329 TMP_332 TMP_334) in (ex_intro T TMP_311 TMP_312 
+TMP_335)))))))))))))))))))))))) in (ex_ind T TMP_296 TMP_304 TMP_336 
+H6)))))))) in (let TMP_338 \def (or_introl TMP_285 TMP_293 TMP_337) in 
+(eq_ind_r T TMP_266 TMP_277 TMP_338 t1 H5)))))))))))) in (ex_ind T TMP_252 
+TMP_263 TMP_339 H4)))))))) in (or_ind TMP_175 TMP_180 TMP_191 TMP_248 TMP_340 
+H3)))))))))))))) in (let TMP_561 \def (\lambda (H2: (ex T (\lambda (v: T).(eq 
+T t0 (lift (S O) d v))))).(let TMP_344 \def (\lambda (v: T).(let TMP_342 \def 
+(S O) in (let TMP_343 \def (lift TMP_342 d v) in (eq T t0 TMP_343)))) in (let 
+TMP_349 \def (\forall (w: T).(let TMP_348 \def (\lambda (v: T).(let TMP_345 
+\def (THead k t0 t1) in (let TMP_346 \def (S O) in (let TMP_347 \def (lift 
+TMP_346 d v) in (subst0 d w TMP_345 TMP_347))))) in (ex T TMP_348))) in (let 
+TMP_353 \def (\lambda (v: T).(let TMP_350 \def (THead k t0 t1) in (let 
+TMP_351 \def (S O) in (let TMP_352 \def (lift TMP_351 d v) in (eq T TMP_350 
+TMP_352))))) in (let TMP_354 \def (ex T TMP_353) in (let TMP_355 \def (or 
+TMP_349 TMP_354) in (let TMP_560 \def (\lambda (x: T).(\lambda (H3: (eq T t0 
+(lift (S O) d x))).(let TMP_356 \def (s k d) in (let H_x0 \def (H0 TMP_356) 
+in (let H4 \def H_x0 in (let TMP_362 \def (\forall (w: T).(let TMP_361 \def 
+(\lambda (v: T).(let TMP_357 \def (s k d) in (let TMP_358 \def (S O) in (let 
+TMP_359 \def (s k d) in (let TMP_360 \def (lift TMP_358 TMP_359 v) in (subst0 
+TMP_357 w t1 TMP_360)))))) in (ex T TMP_361))) in (let TMP_366 \def (\lambda 
+(v: T).(let TMP_363 \def (S O) in (let TMP_364 \def (s k d) in (let TMP_365 
+\def (lift TMP_363 TMP_364 v) in (eq T t1 TMP_365))))) in (let TMP_367 \def 
+(ex T TMP_366) in (let TMP_372 \def (\forall (w: T).(let TMP_371 \def 
+(\lambda (v: T).(let TMP_368 \def (THead k t0 t1) in (let TMP_369 \def (S O) 
+in (let TMP_370 \def (lift TMP_369 d v) in (subst0 d w TMP_368 TMP_370))))) 
+in (ex T TMP_371))) in (let TMP_376 \def (\lambda (v: T).(let TMP_373 \def 
+(THead k t0 t1) in (let TMP_374 \def (S O) in (let TMP_375 \def (lift TMP_374 
+d v) in (eq T TMP_373 TMP_375))))) in (let TMP_377 \def (ex T TMP_376) in 
+(let TMP_378 \def (or TMP_372 TMP_377) in (let TMP_450 \def (\lambda (H5: 
+((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k 
+d) v))))))).(let TMP_379 \def (S O) in (let TMP_380 \def (lift TMP_379 d x) 
+in (let TMP_391 \def (\lambda (t2: T).(let TMP_385 \def (\forall (w: T).(let 
+TMP_384 \def (\lambda (v: T).(let TMP_381 \def (THead k t2 t1) in (let 
+TMP_382 \def (S O) in (let TMP_383 \def (lift TMP_382 d v) in (subst0 d w 
+TMP_381 TMP_383))))) in (ex T TMP_384))) in (let TMP_389 \def (\lambda (v: 
+T).(let TMP_386 \def (THead k t2 t1) in (let TMP_387 \def (S O) in (let 
+TMP_388 \def (lift TMP_387 d v) in (eq T TMP_386 TMP_388))))) in (let TMP_390 
+\def (ex T TMP_389) in (or TMP_385 TMP_390))))) in (let TMP_398 \def (\forall 
+(w: T).(let TMP_397 \def (\lambda (v: T).(let TMP_392 \def (S O) in (let 
+TMP_393 \def (lift TMP_392 d x) in (let TMP_394 \def (THead k TMP_393 t1) in 
+(let TMP_395 \def (S O) in (let TMP_396 \def (lift TMP_395 d v) in (subst0 d 
+w TMP_394 TMP_396))))))) in (ex T TMP_397))) in (let TMP_404 \def (\lambda 
+(v: T).(let TMP_399 \def (S O) in (let TMP_400 \def (lift TMP_399 d x) in 
+(let TMP_401 \def (THead k TMP_400 t1) in (let TMP_402 \def (S O) in (let 
+TMP_403 \def (lift TMP_402 d v) in (eq T TMP_401 TMP_403))))))) in (let 
+TMP_405 \def (ex T TMP_404) in (let TMP_448 \def (\lambda (w: T).(let H_x1 
+\def (H5 w) in (let H6 \def H_x1 in (let TMP_410 \def (\lambda (v: T).(let 
+TMP_406 \def (s k d) in (let TMP_407 \def (S O) in (let TMP_408 \def (s k d) 
+in (let TMP_409 \def (lift TMP_407 TMP_408 v) in (subst0 TMP_406 w t1 
+TMP_409)))))) in (let TMP_416 \def (\lambda (v: T).(let TMP_411 \def (S O) in 
+(let TMP_412 \def (lift TMP_411 d x) in (let TMP_413 \def (THead k TMP_412 
+t1) in (let TMP_414 \def (S O) in (let TMP_415 \def (lift TMP_414 d v) in 
+(subst0 d w TMP_413 TMP_415))))))) in (let TMP_417 \def (ex T TMP_416) in 
+(let TMP_447 \def (\lambda (x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift 
+(S O) (s k d) x0))).(let TMP_423 \def (\lambda (v: T).(let TMP_418 \def (S O) 
+in (let TMP_419 \def (lift TMP_418 d x) in (let TMP_420 \def (THead k TMP_419 
+t1) in (let TMP_421 \def (S O) in (let TMP_422 \def (lift TMP_421 d v) in 
+(subst0 d w TMP_420 TMP_422))))))) in (let TMP_424 \def (THead k x x0) in 
+(let TMP_425 \def (S O) in (let TMP_426 \def (lift TMP_425 d x) in (let 
+TMP_427 \def (S O) in (let TMP_428 \def (s k d) in (let TMP_429 \def (lift 
+TMP_427 TMP_428 x0) in (let TMP_430 \def (THead k TMP_426 TMP_429) in (let 
+TMP_434 \def (\lambda (t2: T).(let TMP_431 \def (S O) in (let TMP_432 \def 
+(lift TMP_431 d x) in (let TMP_433 \def (THead k TMP_432 t1) in (subst0 d w 
+TMP_433 t2))))) in (let TMP_435 \def (S O) in (let TMP_436 \def (s k d) in 
+(let TMP_437 \def (lift TMP_435 TMP_436 x0) in (let TMP_438 \def (S O) in 
+(let TMP_439 \def (lift TMP_438 d x) in (let TMP_440 \def (subst0_snd k w 
+TMP_437 t1 d H7 TMP_439) in (let TMP_441 \def (S O) in (let TMP_442 \def 
+(THead k x x0) in (let TMP_443 \def (lift TMP_441 d TMP_442) in (let TMP_444 
+\def (S O) in (let TMP_445 \def (lift_head k x x0 TMP_444 d) in (let TMP_446 
+\def (eq_ind_r T TMP_430 TMP_434 TMP_440 TMP_443 TMP_445) in (ex_intro T 
+TMP_423 TMP_424 TMP_446)))))))))))))))))))))))) in (ex_ind T TMP_410 TMP_417 
+TMP_447 H6)))))))) in (let TMP_449 \def (or_introl TMP_398 TMP_405 TMP_448) 
+in (eq_ind_r T TMP_380 TMP_391 TMP_449 t0 H3)))))))))) in (let TMP_559 \def 
+(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))))).(let 
+TMP_454 \def (\lambda (v: T).(let TMP_451 \def (S O) in (let TMP_452 \def (s 
+k d) in (let TMP_453 \def (lift TMP_451 TMP_452 v) in (eq T t1 TMP_453))))) 
+in (let TMP_459 \def (\forall (w: T).(let TMP_458 \def (\lambda (v: T).(let 
+TMP_455 \def (THead k t0 t1) in (let TMP_456 \def (S O) in (let TMP_457 \def 
+(lift TMP_456 d v) in (subst0 d w TMP_455 TMP_457))))) in (ex T TMP_458))) in 
+(let TMP_463 \def (\lambda (v: T).(let TMP_460 \def (THead k t0 t1) in (let 
+TMP_461 \def (S O) in (let TMP_462 \def (lift TMP_461 d v) in (eq T TMP_460 
+TMP_462))))) in (let TMP_464 \def (ex T TMP_463) in (let TMP_465 \def (or 
+TMP_459 TMP_464) in (let TMP_558 \def (\lambda (x0: T).(\lambda (H6: (eq T t1 
+(lift (S O) (s k d) x0))).(let TMP_466 \def (S O) in (let TMP_467 \def (s k 
+d) in (let TMP_468 \def (lift TMP_466 TMP_467 x0) in (let TMP_479 \def 
+(\lambda (t2: T).(let TMP_473 \def (\forall (w: T).(let TMP_472 \def (\lambda 
+(v: T).(let TMP_469 \def (THead k t0 t2) in (let TMP_470 \def (S O) in (let 
+TMP_471 \def (lift TMP_470 d v) in (subst0 d w TMP_469 TMP_471))))) in (ex T 
+TMP_472))) in (let TMP_477 \def (\lambda (v: T).(let TMP_474 \def (THead k t0 
+t2) in (let TMP_475 \def (S O) in (let TMP_476 \def (lift TMP_475 d v) in (eq 
+T TMP_474 TMP_476))))) in (let TMP_478 \def (ex T TMP_477) in (or TMP_473 
+TMP_478))))) in (let TMP_480 \def (S O) in (let TMP_481 \def (lift TMP_480 d 
+x) in (let TMP_498 \def (\lambda (t2: T).(let TMP_489 \def (\forall (w: 
+T).(let TMP_488 \def (\lambda (v: T).(let TMP_482 \def (S O) in (let TMP_483 
+\def (s k d) in (let TMP_484 \def (lift TMP_482 TMP_483 x0) in (let TMP_485 
+\def (THead k t2 TMP_484) in (let TMP_486 \def (S O) in (let TMP_487 \def 
+(lift TMP_486 d v) in (subst0 d w TMP_485 TMP_487)))))))) in (ex T TMP_488))) 
+in (let TMP_496 \def (\lambda (v: T).(let TMP_490 \def (S O) in (let TMP_491 
+\def (s k d) in (let TMP_492 \def (lift TMP_490 TMP_491 x0) in (let TMP_493 
+\def (THead k t2 TMP_492) in (let TMP_494 \def (S O) in (let TMP_495 \def 
+(lift TMP_494 d v) in (eq T TMP_493 TMP_495)))))))) in (let TMP_497 \def (ex 
+T TMP_496) in (or TMP_489 TMP_497))))) in (let TMP_508 \def (\forall (w: 
+T).(let TMP_507 \def (\lambda (v: T).(let TMP_499 \def (S O) in (let TMP_500 
+\def (lift TMP_499 d x) in (let TMP_501 \def (S O) in (let TMP_502 \def (s k 
+d) in (let TMP_503 \def (lift TMP_501 TMP_502 x0) in (let TMP_504 \def (THead 
+k TMP_500 TMP_503) in (let TMP_505 \def (S O) in (let TMP_506 \def (lift 
+TMP_505 d v) in (subst0 d w TMP_504 TMP_506)))))))))) in (ex T TMP_507))) in 
+(let TMP_517 \def (\lambda (v: T).(let TMP_509 \def (S O) in (let TMP_510 
+\def (lift TMP_509 d x) in (let TMP_511 \def (S O) in (let TMP_512 \def (s k 
+d) in (let TMP_513 \def (lift TMP_511 TMP_512 x0) in (let TMP_514 \def (THead 
+k TMP_510 TMP_513) in (let TMP_515 \def (S O) in (let TMP_516 \def (lift 
+TMP_515 d v) in (eq T TMP_514 TMP_516)))))))))) in (let TMP_518 \def (ex T 
+TMP_517) in (let TMP_527 \def (\lambda (v: T).(let TMP_519 \def (S O) in (let 
+TMP_520 \def (lift TMP_519 d x) in (let TMP_521 \def (S O) in (let TMP_522 
+\def (s k d) in (let TMP_523 \def (lift TMP_521 TMP_522 x0) in (let TMP_524 
+\def (THead k TMP_520 TMP_523) in (let TMP_525 \def (S O) in (let TMP_526 
+\def (lift TMP_525 d v) in (eq T TMP_524 TMP_526)))))))))) in (let TMP_528 
+\def (THead k x x0) in (let TMP_529 \def (S O) in (let TMP_530 \def (lift 
+TMP_529 d x) in (let TMP_531 \def (S O) in (let TMP_532 \def (s k d) in (let 
+TMP_533 \def (lift TMP_531 TMP_532 x0) in (let TMP_534 \def (THead k TMP_530 
+TMP_533) in (let TMP_541 \def (\lambda (t2: T).(let TMP_535 \def (S O) in 
+(let TMP_536 \def (lift TMP_535 d x) in (let TMP_537 \def (S O) in (let 
+TMP_538 \def (s k d) in (let TMP_539 \def (lift TMP_537 TMP_538 x0) in (let 
+TMP_540 \def (THead k TMP_536 TMP_539) in (eq T TMP_540 t2)))))))) in (let 
+TMP_542 \def (S O) in (let TMP_543 \def (lift TMP_542 d x) in (let TMP_544 
+\def (S O) in (let TMP_545 \def (s k d) in (let TMP_546 \def (lift TMP_544 
+TMP_545 x0) in (let TMP_547 \def (THead k TMP_543 TMP_546) in (let TMP_548 
+\def (refl_equal T TMP_547) in (let TMP_549 \def (S O) in (let TMP_550 \def 
+(THead k x x0) in (let TMP_551 \def (lift TMP_549 d TMP_550) in (let TMP_552 
+\def (S O) in (let TMP_553 \def (lift_head k x x0 TMP_552 d) in (let TMP_554 
+\def (eq_ind_r T TMP_534 TMP_541 TMP_548 TMP_551 TMP_553) in (let TMP_555 
+\def (ex_intro T TMP_527 TMP_528 TMP_554) in (let TMP_556 \def (or_intror 
+TMP_508 TMP_518 TMP_555) in (let TMP_557 \def (eq_ind_r T TMP_481 TMP_498 
+TMP_556 t0 H3) in (eq_ind_r T TMP_468 TMP_479 TMP_557 t1 
+H6)))))))))))))))))))))))))))))))))))))) in (ex_ind T TMP_454 TMP_465 TMP_558 
+H5)))))))) in (or_ind TMP_362 TMP_367 TMP_378 TMP_450 TMP_559 
+H4))))))))))))))) in (ex_ind T TMP_344 TMP_355 TMP_560 H2)))))))) in (or_ind 
+TMP_153 TMP_157 TMP_168 TMP_341 TMP_561 H1)))))))))))))))))) in (T_ind TMP_9 
+TMP_37 TMP_149 TMP_562 t))))).
 
 theorem dnf_dec:
  \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or 
 (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v)))))))
 \def
  \lambda (w: T).(\lambda (t: T).(\lambda (d: nat).(let H_x \def (dnf_dec2 t 
-d) in (let H \def H_x in (or_ind (\forall (w0: T).(ex T (\lambda (v: 
-T).(subst0 d w0 t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S 
-O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t 
-(lift (S O) d v))))) (\lambda (H0: ((\forall (w0: T).(ex T (\lambda (v: 
-T).(subst0 d w0 t (lift (S O) d v))))))).(let H_x0 \def (H0 w) in (let H1 
-\def H_x0 in (ex_ind T (\lambda (v: T).(subst0 d w t (lift (S O) d v))) (ex T 
-(\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d 
-v))))) (\lambda (x: T).(\lambda (H2: (subst0 d w t (lift (S O) d 
-x))).(ex_intro T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t 
-(lift (S O) d v)))) x (or_introl (subst0 d w t (lift (S O) d x)) (eq T t 
-(lift (S O) d x)) H2)))) H1)))) (\lambda (H0: (ex T (\lambda (v: T).(eq T t 
-(lift (S O) d v))))).(ex_ind T (\lambda (v: T).(eq T t (lift (S O) d v))) (ex 
-T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d 
-v))))) (\lambda (x: T).(\lambda (H1: (eq T t (lift (S O) d x))).(eq_ind_r T 
-(lift (S O) d x) (\lambda (t0: T).(ex T (\lambda (v: T).(or (subst0 d w t0 
-(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))) (ex_intro T (\lambda (v: 
-T).(or (subst0 d w (lift (S O) d x) (lift (S O) d v)) (eq T (lift (S O) d x) 
-(lift (S O) d v)))) x (or_intror (subst0 d w (lift (S O) d x) (lift (S O) d 
-x)) (eq T (lift (S O) d x) (lift (S O) d x)) (refl_equal T (lift (S O) d 
-x)))) t H1))) H0)) H))))).
-(* COMMENTS
-Initial nodes: 603
-END *)
+d) in (let H \def H_x in (let TMP_4 \def (\forall (w0: T).(let TMP_3 \def 
+(\lambda (v: T).(let TMP_1 \def (S O) in (let TMP_2 \def (lift TMP_1 d v) in 
+(subst0 d w0 t TMP_2)))) in (ex T TMP_3))) in (let TMP_7 \def (\lambda (v: 
+T).(let TMP_5 \def (S O) in (let TMP_6 \def (lift TMP_5 d v) in (eq T t 
+TMP_6)))) in (let TMP_8 \def (ex T TMP_7) in (let TMP_15 \def (\lambda (v: 
+T).(let TMP_9 \def (S O) in (let TMP_10 \def (lift TMP_9 d v) in (let TMP_11 
+\def (subst0 d w t TMP_10) in (let TMP_12 \def (S O) in (let TMP_13 \def 
+(lift TMP_12 d v) in (let TMP_14 \def (eq T t TMP_13) in (or TMP_11 
+TMP_14)))))))) in (let TMP_16 \def (ex T TMP_15) in (let TMP_43 \def (\lambda 
+(H0: ((\forall (w0: T).(ex T (\lambda (v: T).(subst0 d w0 t (lift (S O) d 
+v))))))).(let H_x0 \def (H0 w) in (let H1 \def H_x0 in (let TMP_19 \def 
+(\lambda (v: T).(let TMP_17 \def (S O) in (let TMP_18 \def (lift TMP_17 d v) 
+in (subst0 d w t TMP_18)))) in (let TMP_26 \def (\lambda (v: T).(let TMP_20 
+\def (S O) in (let TMP_21 \def (lift TMP_20 d v) in (let TMP_22 \def (subst0 
+d w t TMP_21) in (let TMP_23 \def (S O) in (let TMP_24 \def (lift TMP_23 d v) 
+in (let TMP_25 \def (eq T t TMP_24) in (or TMP_22 TMP_25)))))))) in (let 
+TMP_27 \def (ex T TMP_26) in (let TMP_42 \def (\lambda (x: T).(\lambda (H2: 
+(subst0 d w t (lift (S O) d x))).(let TMP_34 \def (\lambda (v: T).(let TMP_28 
+\def (S O) in (let TMP_29 \def (lift TMP_28 d v) in (let TMP_30 \def (subst0 
+d w t TMP_29) in (let TMP_31 \def (S O) in (let TMP_32 \def (lift TMP_31 d v) 
+in (let TMP_33 \def (eq T t TMP_32) in (or TMP_30 TMP_33)))))))) in (let 
+TMP_35 \def (S O) in (let TMP_36 \def (lift TMP_35 d x) in (let TMP_37 \def 
+(subst0 d w t TMP_36) in (let TMP_38 \def (S O) in (let TMP_39 \def (lift 
+TMP_38 d x) in (let TMP_40 \def (eq T t TMP_39) in (let TMP_41 \def 
+(or_introl TMP_37 TMP_40 H2) in (ex_intro T TMP_34 x TMP_41))))))))))) in 
+(ex_ind T TMP_19 TMP_27 TMP_42 H1)))))))) in (let TMP_92 \def (\lambda (H0: 
+(ex T (\lambda (v: T).(eq T t (lift (S O) d v))))).(let TMP_46 \def (\lambda 
+(v: T).(let TMP_44 \def (S O) in (let TMP_45 \def (lift TMP_44 d v) in (eq T 
+t TMP_45)))) in (let TMP_53 \def (\lambda (v: T).(let TMP_47 \def (S O) in 
+(let TMP_48 \def (lift TMP_47 d v) in (let TMP_49 \def (subst0 d w t TMP_48) 
+in (let TMP_50 \def (S O) in (let TMP_51 \def (lift TMP_50 d v) in (let 
+TMP_52 \def (eq T t TMP_51) in (or TMP_49 TMP_52)))))))) in (let TMP_54 \def 
+(ex T TMP_53) in (let TMP_91 \def (\lambda (x: T).(\lambda (H1: (eq T t (lift 
+(S O) d x))).(let TMP_55 \def (S O) in (let TMP_56 \def (lift TMP_55 d x) in 
+(let TMP_64 \def (\lambda (t0: T).(let TMP_63 \def (\lambda (v: T).(let 
+TMP_57 \def (S O) in (let TMP_58 \def (lift TMP_57 d v) in (let TMP_59 \def 
+(subst0 d w t0 TMP_58) in (let TMP_60 \def (S O) in (let TMP_61 \def (lift 
+TMP_60 d v) in (let TMP_62 \def (eq T t0 TMP_61) in (or TMP_59 TMP_62)))))))) 
+in (ex T TMP_63))) in (let TMP_75 \def (\lambda (v: T).(let TMP_65 \def (S O) 
+in (let TMP_66 \def (lift TMP_65 d x) in (let TMP_67 \def (S O) in (let 
+TMP_68 \def (lift TMP_67 d v) in (let TMP_69 \def (subst0 d w TMP_66 TMP_68) 
+in (let TMP_70 \def (S O) in (let TMP_71 \def (lift TMP_70 d x) in (let 
+TMP_72 \def (S O) in (let TMP_73 \def (lift TMP_72 d v) in (let TMP_74 \def 
+(eq T TMP_71 TMP_73) in (or TMP_69 TMP_74)))))))))))) in (let TMP_76 \def (S 
+O) in (let TMP_77 \def (lift TMP_76 d x) in (let TMP_78 \def (S O) in (let 
+TMP_79 \def (lift TMP_78 d x) in (let TMP_80 \def (subst0 d w TMP_77 TMP_79) 
+in (let TMP_81 \def (S O) in (let TMP_82 \def (lift TMP_81 d x) in (let 
+TMP_83 \def (S O) in (let TMP_84 \def (lift TMP_83 d x) in (let TMP_85 \def 
+(eq T TMP_82 TMP_84) in (let TMP_86 \def (S O) in (let TMP_87 \def (lift 
+TMP_86 d x) in (let TMP_88 \def (refl_equal T TMP_87) in (let TMP_89 \def 
+(or_intror TMP_80 TMP_85 TMP_88) in (let TMP_90 \def (ex_intro T TMP_75 x 
+TMP_89) in (eq_ind_r T TMP_56 TMP_64 TMP_90 t H1)))))))))))))))))))))) in 
+(ex_ind T TMP_46 TMP_54 TMP_91 H0)))))) in (or_ind TMP_4 TMP_8 TMP_16 TMP_43 
+TMP_92 H)))))))))))).
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/defs.ma
index a493a7ac2c31531d70f1ad0e48a595fbed800df5..fca79e9d90ea98e38524413b78989aa45070e969 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/defs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/defs.ma
@@ -14,7 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/lift/defs.ma".
+include "basic_1/lift/defs.ma".
 
 inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def
 | subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma
index 165555fe2fed927839b9c759928eb01a56c48076..12a582906a7148e860e64ada6962f2cfde0c7bf8 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/fwd.ma
@@ -14,9 +14,28 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/subst0/defs.ma".
+include "basic_1/subst0/defs.ma".
 
-include "Basic-1/lift/props.ma".
+include "basic_1/lift/fwd.ma".
+
+let rec subst0_ind (P: (nat \to (T \to (T \to (T \to Prop))))) (f: (\forall 
+(v: T).(\forall (i: nat).(P i v (TLRef i) (lift (S i) O v))))) (f0: (\forall 
+(v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: nat).((subst0 i v u1 
+u2) \to ((P i v u1 u2) \to (\forall (t: T).(\forall (k: K).(P i v (THead k u1 
+t) (THead k u2 t))))))))))) (f1: (\forall (k: K).(\forall (v: T).(\forall 
+(t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to ((P 
+(s k i) v t1 t2) \to (\forall (u: T).(P i v (THead k u t1) (THead k u 
+t2))))))))))) (f2: (\forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall 
+(i: nat).((subst0 i v u1 u2) \to ((P i v u1 u2) \to (\forall (k: K).(\forall 
+(t1: T).(\forall (t2: T).((subst0 (s k i) v t1 t2) \to ((P (s k i) v t1 t2) 
+\to (P i v (THead k u1 t1) (THead k u2 t2)))))))))))))) (n: nat) (t: T) (t0: 
+T) (t1: T) (s0: subst0 n t t0 t1) on s0: P n t t0 t1 \def match s0 with 
+[(subst0_lref v i) \Rightarrow (f v i) | (subst0_fst v u2 u1 i s1 t2 k) 
+\Rightarrow (f0 v u2 u1 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 s1) t2 k) | 
+(subst0_snd k v t2 t3 i s1 u) \Rightarrow (f1 k v t2 t3 i s1 ((subst0_ind P f 
+f0 f1 f2) (s k i) v t3 t2 s1) u) | (subst0_both v u1 u2 i s1 k t2 t3 s2) 
+\Rightarrow (f2 v u1 u2 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 s1) k t2 t3 
+s2 ((subst0_ind P f f0 f1 f2) (s k i) v t2 t3 s2))].
 
 theorem subst0_gen_sort:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
@@ -28,33 +47,29 @@ i v (TSort n) x) \to (\forall (P: Prop).P)))))
 (H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda 
 (t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_: 
 T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def 
-(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2))))) 
-(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
-nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) 
-\to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t) 
-(TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee 
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
-_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in 
-(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1 
-t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda 
+(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(TSort n) H1) in (False_ind P H2))))) (\lambda (v0: T).(\lambda (u2: 
+T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 
+u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (t: T).(\lambda (k: 
+K).(\lambda (H3: (eq T (THead k u1 t) (TSort n))).(let H4 \def (eq_ind T 
+(THead k u1 t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
+H3) in (False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda 
+(t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 
+t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda 
 (H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1) 
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in 
+(False_ind P H4))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T 
+u1 (TSort n)) \to P))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TSort 
+n)) \to P))).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let H6 \def 
+(eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TSort n) H3) in (False_ind P H4))))))))))) (\lambda (v0: 
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 
-i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k: 
-K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1 
-t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (H5: (eq T (THead k 
-u1 t1) (TSort n))).(let H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) H)))))).
-(* COMMENTS
-Initial nodes: 445
-END *)
+True])) I (TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) 
+H)))))).
 
 theorem subst0_gen_lref:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
@@ -67,17 +82,16 @@ i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v))))
 nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n)) 
 \to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0: 
 T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def 
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with 
-[(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _) 
-\Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0: 
-nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj 
-(eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n) 
-(refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2: 
-T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 
-u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 
-(lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T 
-(THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda 
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+(f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i0 | 
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef 
+n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(land (eq nat n n0) (eq T (lift 
+(S n0) O v0) (lift (S n) O v0)))) (conj (eq nat n n) (eq T (lift (S n) O v0) 
+(lift (S n) O v0)) (refl_equal nat n) (refl_equal T (lift (S n) O v0))) i0 
+H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) 
+\to (land (eq nat n i0) (eq T u2 (lift (S n) O v0)))))).(\lambda (t: 
+T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t) (TLRef n))).(let H4 
+\def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee with [(TSort _) 
 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
 True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2 
 t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0: 
@@ -85,23 +99,19 @@ T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0
 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s 
 k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T 
 (THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda 
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u 
-t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1: 
-T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 
-u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 
-(lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef 
-n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda 
-(H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead k u1 t1) 
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-True])) I (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2 
-t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 779
-END *)
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in (False_ind (land 
+(eq nat n i0) (eq T (THead k u t2) (lift (S n) O v0))) H4))))))))))) (\lambda 
+(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: 
+(subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n 
+i0) (eq T u2 (lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 
+(TLRef n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O 
+v0)))))).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind 
+T (THead k u1 t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
+(TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2 t2) (lift 
+(S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
 
 theorem subst0_gen_head:
  \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall 
@@ -129,30 +139,28 @@ T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_:
 T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1 
 t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef 
 i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
-(THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S 
-i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T 
-(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2: 
-T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
-T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) 
-v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: 
-T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq 
-T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 
-t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T 
-u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0: 
-K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \def 
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead k u1 t1) H1) in (False_ind (or3 
+(ex2 T (\lambda (u2: T).(eq T (lift (S i0) O v0) (THead k u2 t1))) (\lambda 
+(u2: T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t2: T).(eq T (lift (S i0) O 
+v0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))) H2))))) (\lambda (v0: 
+T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 
+i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead k u1 t1)) \to (or3 (ex2 T 
+(\lambda (u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 
+u3))) (ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: 
+T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 
+u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 
+t2)))))))).(\lambda (t: T).(\lambda (k0: K).(\lambda (H3: (eq T (THead k0 u0 
+t) (THead k u1 t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e with 
 [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
 \Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _) 
-\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H6 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef 
+_) \Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead k0 u0 t) (THead k 
+u1 t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e with 
 [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) 
 \Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T 
 u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
@@ -192,44 +200,42 @@ T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2)))
 T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 
 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
 t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 
-t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda 
-(_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead 
-k1 _ _) \Rightarrow k1])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H5 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) 
-\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H6 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
-\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u 
-u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T 
-(\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2: 
-T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2) 
-(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0 
-(\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: 
-T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) 
-(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 
-(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
-t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s 
-k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1: 
-K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 
-(THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T 
-(\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s 
-k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
-(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 
-t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s 
-k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
-(\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2: 
-T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2) 
-(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T 
+t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) 
+(THead k0 u t0) (THead k u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | 
+(THead _ t _) \Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let 
+H6 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 
+| (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u t0) 
+(THead k u1 t1) H3) in (\lambda (H7: (eq T u u1)).(\lambda (H8: (eq K k0 
+k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T (\lambda (u2: T).(eq T (THead 
+k0 t t2) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T 
+(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u1 t3))) (\lambda (t3: 
+T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead k0 t t2) (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) 
+v0 t1 t3)))))) (let H9 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead k u1 
+t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda 
+(u2: T).(subst0 (s k0 i0) v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead 
+k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H2 t1 H6) in (let H10 \def 
+(eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t t2)) H1 t1 H6) in (let 
+H11 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1 (THead k u1 t1)) \to (or3 
+(ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 
+(s k1 i0) v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) 
+(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 (s k1 i0) v0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s 
+k (s k1 i0)) v0 t1 t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda 
+(k1: K).(subst0 (s k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: 
+K).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) 
+(\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead 
+k1 u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda 
+(_: T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T 
 (\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: 
 T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) 
 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
@@ -254,58 +260,54 @@ v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) 
 (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
 t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6 
-\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
-with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
-\Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H7 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
+\def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k0 | 
+(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t0) 
+(THead k u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e 
+with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) 
 \Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
-[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
-\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T 
-u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t: 
-T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead 
-k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda 
-(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) 
-v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
-t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4 
-t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t 
-t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1 
-(THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1))) 
-(\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T 
-t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))) 
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) 
-(\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in 
-(let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12 
-k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T 
-(THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) 
-(ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda 
-(t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda 
-(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k 
-i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead 
-k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) 
-(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2 
+(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef 
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t0) (THead k 
+u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k0 k)).(let 
+H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 
+T (\lambda (u3: T).(eq T t2 (THead k u3 t1))) (\lambda (u3: T).(subst0 (s k0 
+i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda 
+(t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(subst0 (s k0 i0) v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s 
+k (s k0 i0)) v0 t1 t3))))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda 
+(t: T).(subst0 (s k0 i0) v0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 
+(\lambda (k1: K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: 
+T).(eq T t2 (THead k u3 t1))) (\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) 
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 
+(s k (s k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq 
+T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 
+u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 
+t3))))))) H11 k H10) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 
+(s k1 i0) v0 t1 t2)) H12 k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
+(\lambda (u3: T).(eq T (THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: 
+T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) 
 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
-(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0 
-(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda 
-(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 
-v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3))) 
-(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: 
-T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: 
-T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda 
-(t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda 
-(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k 
-i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0 
-H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))).
-(* COMMENTS
-Initial nodes: 4255
-END *)
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H15 \def (eq_ind T 
+u0 (\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: 
+T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T 
+(\lambda (t3: T).(eq T u2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) 
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 
+\def (eq_ind T u0 (\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in 
+(or3_intro2 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead 
+k u2 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k 
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda 
+(_: T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k 
+u2 t2)) H16 H14)))) k0 H10)))))))) H7)) H6)))))))))))))) i v y x H0))) 
+H))))))).
 
 theorem subst0_gen_lift_lt:
  \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
@@ -489,9 +491,6 @@ t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S
 i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k 
 (lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i 
 H2))))))))))))) t1)).
-(* COMMENTS
-Initial nodes: 5157
-END *)
 
 theorem subst0_gen_lift_false:
  \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall 
@@ -551,21 +550,18 @@ T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u
 (lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k 
 (lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1) 
 x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h)) 
-(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i 
-(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind 
-T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: 
-T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7: 
-(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) 
-t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d 
-t0) (lift h (s k d) t1) x i H4))))))))))))))))) t).
-(* COMMENTS
-Initial nodes: 1621
-END *)
+(\lambda (n: nat).(lt (s k i) n)) (s_lt k i (plus d h) H2) (plus (s k d) h) 
+(s_plus k d h)) H7 P)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
+(s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
+(s k i) u (lift h (s k d) t1) t2))) P (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7: (subst0 i u (lift h d 
+t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) t1) x1)).(H u x0 h d 
+i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d t0) (lift h (s k d) 
+t1) x i H4))))))))))))))))) t).
 
 theorem subst0_gen_lift_ge:
  \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
@@ -614,7 +610,7 @@ u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0:
 nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift 
 (plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n 
 h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d 
-(plus O (S n)) (le_plus_plus O O d (S n) (le_n O) (le_S d n H1))) (le_O_n 
+(plus O (S n)) (le_plus_plus O O d (S n) (le_O_n O) (le_S d n H1))) (le_O_n 
 d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i 
 H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k: 
 K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall 
@@ -722,7 +718,195 @@ x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind
 nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i 
 H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u 
 (lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)).
-(* COMMENTS
-Initial nodes: 4191
-END *)
+
+theorem subst0_gen_lift_rev_ge:
+ \forall (t1: T).(\forall (v: T).(\forall (u2: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst0 i v t1 (lift h d u2)) \to ((le (plus d h) 
+i) \to (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: 
+T).(eq T t1 (lift h d u1)))))))))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (v: T).(\forall (u2: 
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i v t (lift 
+h d u2)) \to ((le (plus d h) i) \to (ex2 T (\lambda (u1: T).(subst0 (minus i 
+h) v u1 u2)) (\lambda (u1: T).(eq T t (lift h d u1)))))))))))) (\lambda (n: 
+nat).(\lambda (v: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (subst0 i v (TSort n) (lift h d 
+u2))).(\lambda (_: (le (plus d h) i)).(subst0_gen_sort v (lift h d u2) i n H 
+(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T 
+(TSort n) (lift h d u1))))))))))))) (\lambda (n: nat).(\lambda (v: 
+T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H: (subst0 i v (TLRef n) (lift h d u2))).(\lambda (H0: (le 
+(plus d h) i)).(land_ind (eq nat n i) (eq T (lift h d u2) (lift (S n) O v)) 
+(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T 
+(TLRef n) (lift h d u1)))) (\lambda (H1: (eq nat n i)).(\lambda (H2: (eq T 
+(lift h d u2) (lift (S n) O v))).(let H3 \def (eq_ind_r nat i (\lambda (n0: 
+nat).(le (plus d h) n0)) H0 n H1) in (eq_ind nat n (\lambda (n0: nat).(ex2 T 
+(\lambda (u1: T).(subst0 (minus n0 h) v u1 u2)) (\lambda (u1: T).(eq T (TLRef 
+n) (lift h d u1))))) (eq_ind_r nat (plus (minus n h) h) (\lambda (n0: 
+nat).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 u2)) (\lambda (u1: 
+T).(eq T (TLRef n0) (lift h d u1))))) (eq_ind T (lift h d (TLRef (minus n 
+h))) (\lambda (t: T).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 u2)) 
+(\lambda (u1: T).(eq T t (lift h d u1))))) (let H4 \def (eq_ind nat n 
+(\lambda (n0: nat).(eq T (lift h d u2) (lift (S n0) O v))) H2 (plus h (minus 
+n h)) (le_plus_minus h n (le_trans h (plus d h) n (le_plus_r d h) H3))) in 
+(let H5 \def (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(eq T 
+(lift h d u2) (lift n0 O v))) H4 (plus h (S (minus n h))) (plus_n_Sm h (minus 
+n h))) in (let H6 \def (eq_ind_r T (lift (plus h (S (minus n h))) O v) 
+(\lambda (t: T).(eq T (lift h d u2) t)) H5 (lift h d (lift (S (minus n h)) O 
+v)) (lift_free v (S (minus n h)) h O d (le_S d (minus n h) (le_minus d n h 
+H3)) (le_O_n d))) in (eq_ind_r T (lift (S (minus n h)) O v) (\lambda (t: 
+T).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 t)) (\lambda (u1: T).(eq 
+T (lift h d (TLRef (minus n h))) (lift h d u1))))) (ex_intro2 T (\lambda (u1: 
+T).(subst0 (minus n h) v u1 (lift (S (minus n h)) O v))) (\lambda (u1: T).(eq 
+T (lift h d (TLRef (minus n h))) (lift h d u1))) (TLRef (minus n h)) 
+(subst0_lref v (minus n h)) (refl_equal T (lift h d (TLRef (minus n h))))) u2 
+(lift_inj u2 (lift (S (minus n h)) O v) h d H6))))) (TLRef (plus (minus n h) 
+h)) (lift_lref_ge (minus n h) h d (le_minus d n h H3))) n (le_plus_minus_sym 
+h n (le_trans h (plus d h) n (le_plus_r d h) H3))) i H1)))) (subst0_gen_lref 
+v (lift h d u2) i n H)))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: 
+((\forall (v: T).(\forall (u2: T).(\forall (i: nat).(\forall (h: 
+nat).(\forall (d: nat).((subst0 i v t (lift h d u2)) \to ((le (plus d h) i) 
+\to (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: 
+T).(eq T t (lift h d u1))))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
+(v: T).(\forall (u2: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: 
+nat).((subst0 i v t0 (lift h d u2)) \to ((le (plus d h) i) \to (ex2 T 
+(\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T t0 
+(lift h d u1))))))))))))).(\lambda (v: T).(\lambda (u2: T).(\lambda (i: 
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (subst0 i v (THead k t 
+t0) (lift h d u2))).(\lambda (H2: (le (plus d h) i)).(or3_ind (ex2 T (\lambda 
+(u3: T).(eq T (lift h d u2) (THead k u3 t0))) (\lambda (u3: T).(subst0 i v t 
+u3))) (ex2 T (\lambda (t2: T).(eq T (lift h d u2) (THead k t t2))) (\lambda 
+(t2: T).(subst0 (s k i) v t0 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T (lift h d u2) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(subst0 i v t u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t0 
+t2)))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: 
+T).(eq T (THead k t t0) (lift h d u1)))) (\lambda (H3: (ex2 T (\lambda (u3: 
+T).(eq T (lift h d u2) (THead k u3 t0))) (\lambda (u3: T).(subst0 i v t 
+u3)))).(ex2_ind T (\lambda (u3: T).(eq T (lift h d u2) (THead k u3 t0))) 
+(\lambda (u3: T).(subst0 i v t u3)) (ex2 T (\lambda (u1: T).(subst0 (minus i 
+h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda 
+(x: T).(\lambda (H4: (eq T (lift h d u2) (THead k x t0))).(\lambda (H5: 
+(subst0 i v t x)).(let H6 \def (sym_eq T (lift h d u2) (THead k x t0) H4) in 
+(let H_x \def (lift_gen_head k x t0 u2 h d H6) in (let H7 \def H_x in 
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T x (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t0 (lift h (s k d) z)))) (ex2 T (\lambda (u1: 
+T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift 
+h d u1)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T u2 (THead k 
+x0 x1))).(\lambda (H9: (eq T x (lift h d x0))).(\lambda (H10: (eq T t0 (lift 
+h (s k d) x1))).(let H11 \def (eq_ind T x (\lambda (t2: T).(subst0 i v t t2)) 
+H5 (lift h d x0) H9) in (eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T 
+(\lambda (u1: T).(subst0 (minus i h) v u1 t2)) (\lambda (u1: T).(eq T (THead 
+k t t0) (lift h d u1))))) (eq_ind_r T (lift h (s k d) x1) (\lambda (t2: 
+T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) 
+(\lambda (u1: T).(eq T (THead k t t2) (lift h d u1))))) (let H_x0 \def (H v 
+x0 i h d H11 H2) in (let H12 \def H_x0 in (ex2_ind T (\lambda (u1: T).(subst0 
+(minus i h) v u1 x0)) (\lambda (u1: T).(eq T t (lift h d u1))) (ex2 T 
+(\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: 
+T).(eq T (THead k t (lift h (s k d) x1)) (lift h d u1)))) (\lambda (x2: 
+T).(\lambda (H13: (subst0 (minus i h) v x2 x0)).(\lambda (H14: (eq T t (lift 
+h d x2))).(eq_ind_r T (lift h d x2) (\lambda (t2: T).(ex2 T (\lambda (u1: 
+T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T (THead k 
+t2 (lift h (s k d) x1)) (lift h d u1))))) (eq_ind T (lift h d (THead k x2 
+x1)) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead 
+k x0 x1))) (\lambda (u1: T).(eq T t2 (lift h d u1))))) (ex_intro2 T (\lambda 
+(u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T 
+(lift h d (THead k x2 x1)) (lift h d u1))) (THead k x2 x1) (subst0_fst v x0 
+x2 (minus i h) H13 x1 k) (refl_equal T (lift h d (THead k x2 x1)))) (THead k 
+(lift h d x2) (lift h (s k d) x1)) (lift_head k x2 x1 h d)) t H14)))) H12))) 
+t0 H10) u2 H8))))))) H7))))))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq 
+T (lift h d u2) (THead k t t2))) (\lambda (t2: T).(subst0 (s k i) v t0 
+t2)))).(ex2_ind T (\lambda (t2: T).(eq T (lift h d u2) (THead k t t2))) 
+(\lambda (t2: T).(subst0 (s k i) v t0 t2)) (ex2 T (\lambda (u1: T).(subst0 
+(minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) 
+(\lambda (x: T).(\lambda (H4: (eq T (lift h d u2) (THead k t x))).(\lambda 
+(H5: (subst0 (s k i) v t0 x)).(let H6 \def (sym_eq T (lift h d u2) (THead k t 
+x) H4) in (let H_x \def (lift_gen_head k t x u2 h d H6) in (let H7 \def H_x 
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T t (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T x (lift h (s k d) z)))) (ex2 T (\lambda (u1: 
+T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift 
+h d u1)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T u2 (THead k 
+x0 x1))).(\lambda (H9: (eq T t (lift h d x0))).(\lambda (H10: (eq T x (lift h 
+(s k d) x1))).(let H11 \def (eq_ind T x (\lambda (t2: T).(subst0 (s k i) v t0 
+t2)) H5 (lift h (s k d) x1) H10) in (eq_ind_r T (THead k x0 x1) (\lambda (t2: 
+T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 t2)) (\lambda (u1: T).(eq 
+T (THead k t t0) (lift h d u1))))) (eq_ind_r T (lift h d x0) (\lambda (t2: 
+T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) 
+(\lambda (u1: T).(eq T (THead k t2 t0) (lift h d u1))))) (let H_y \def (H0 v 
+x1 (s k i) h (s k d) H11) in (let H12 \def (eq_ind_r nat (plus (s k d) h) 
+(\lambda (n: nat).((le n (s k i)) \to (ex2 T (\lambda (u1: T).(subst0 (minus 
+(s k i) h) v u1 x1)) (\lambda (u1: T).(eq T t0 (lift h (s k d) u1)))))) H_y 
+(s k (plus d h)) (s_plus k d h)) in (let H13 \def (eq_ind_r nat (minus (s k 
+i) h) (\lambda (n: nat).((le (s k (plus d h)) (s k i)) \to (ex2 T (\lambda 
+(u1: T).(subst0 n v u1 x1)) (\lambda (u1: T).(eq T t0 (lift h (s k d) 
+u1)))))) H12 (s k (minus i h)) (s_minus k i h (le_trans h (plus d h) i 
+(le_plus_r d h) H2))) in (let H14 \def (H13 (s_le k (plus d h) i H2)) in 
+(ex2_ind T (\lambda (u1: T).(subst0 (s k (minus i h)) v u1 x1)) (\lambda (u1: 
+T).(eq T t0 (lift h (s k d) u1))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) 
+v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T (THead k (lift h d x0) t0) 
+(lift h d u1)))) (\lambda (x2: T).(\lambda (H15: (subst0 (s k (minus i h)) v 
+x2 x1)).(\lambda (H16: (eq T t0 (lift h (s k d) x2))).(eq_ind_r T (lift h (s 
+k d) x2) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
+(THead k x0 x1))) (\lambda (u1: T).(eq T (THead k (lift h d x0) t2) (lift h d 
+u1))))) (eq_ind T (lift h d (THead k x0 x2)) (\lambda (t2: T).(ex2 T (\lambda 
+(u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T t2 
+(lift h d u1))))) (ex_intro2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
+(THead k x0 x1))) (\lambda (u1: T).(eq T (lift h d (THead k x0 x2)) (lift h d 
+u1))) (THead k x0 x2) (subst0_snd k v x1 x2 (minus i h) H15 x0) (refl_equal T 
+(lift h d (THead k x0 x2)))) (THead k (lift h d x0) (lift h (s k d) x2)) 
+(lift_head k x0 x2 h d)) t0 H16)))) H14))))) t H9) u2 H8))))))) H7))))))) 
+H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (lift h 
+d u2) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v t u3))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t0 t2))))).(ex3_2_ind T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T (lift h d u2) (THead k u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i v t u3))) (\lambda (_: T).(\lambda 
+(t2: T).(subst0 (s k i) v t0 t2))) (ex2 T (\lambda (u1: T).(subst0 (minus i 
+h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H4: (eq T (lift h d u2) (THead k x0 
+x1))).(\lambda (H5: (subst0 i v t x0)).(\lambda (H6: (subst0 (s k i) v t0 
+x1)).(let H7 \def (sym_eq T (lift h d u2) (THead k x0 x1) H4) in (let H_x 
+\def (lift_gen_head k x0 x1 u2 h d H7) in (let H8 \def H_x in (ex3_2_ind T T 
+(\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z)))) (\lambda (y: 
+T).(\lambda (_: T).(eq T x0 (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T x1 (lift h (s k d) z)))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) 
+v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T u2 (THead k x2 x3))).(\lambda 
+(H10: (eq T x0 (lift h d x2))).(\lambda (H11: (eq T x1 (lift h (s k d) 
+x3))).(let H12 \def (eq_ind T x1 (\lambda (t2: T).(subst0 (s k i) v t0 t2)) 
+H6 (lift h (s k d) x3) H11) in (let H13 \def (eq_ind T x0 (\lambda (t2: 
+T).(subst0 i v t t2)) H5 (lift h d x2) H10) in (eq_ind_r T (THead k x2 x3) 
+(\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 t2)) 
+(\lambda (u1: T).(eq T (THead k t t0) (lift h d u1))))) (let H_x0 \def (H v 
+x2 i h d H13 H2) in (let H14 \def H_x0 in (ex2_ind T (\lambda (u1: T).(subst0 
+(minus i h) v u1 x2)) (\lambda (u1: T).(eq T t (lift h d u1))) (ex2 T 
+(\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x2 x3))) (\lambda (u1: 
+T).(eq T (THead k t t0) (lift h d u1)))) (\lambda (x: T).(\lambda (H15: 
+(subst0 (minus i h) v x x2)).(\lambda (H16: (eq T t (lift h d x))).(eq_ind_r 
+T (lift h d x) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v 
+u1 (THead k x2 x3))) (\lambda (u1: T).(eq T (THead k t2 t0) (lift h d u1))))) 
+(let H_y \def (H0 v x3 (s k i) h (s k d) H12) in (let H17 \def (eq_ind_r nat 
+(plus (s k d) h) (\lambda (n: nat).((le n (s k i)) \to (ex2 T (\lambda (u1: 
+T).(subst0 (minus (s k i) h) v u1 x3)) (\lambda (u1: T).(eq T t0 (lift h (s k 
+d) u1)))))) H_y (s k (plus d h)) (s_plus k d h)) in (let H18 \def (eq_ind_r 
+nat (minus (s k i) h) (\lambda (n: nat).((le (s k (plus d h)) (s k i)) \to 
+(ex2 T (\lambda (u1: T).(subst0 n v u1 x3)) (\lambda (u1: T).(eq T t0 (lift h 
+(s k d) u1)))))) H17 (s k (minus i h)) (s_minus k i h (le_trans h (plus d h) 
+i (le_plus_r d h) H2))) in (let H19 \def (H18 (s_le k (plus d h) i H2)) in 
+(ex2_ind T (\lambda (u1: T).(subst0 (s k (minus i h)) v u1 x3)) (\lambda (u1: 
+T).(eq T t0 (lift h (s k d) u1))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) 
+v u1 (THead k x2 x3))) (\lambda (u1: T).(eq T (THead k (lift h d x) t0) (lift 
+h d u1)))) (\lambda (x4: T).(\lambda (H20: (subst0 (s k (minus i h)) v x4 
+x3)).(\lambda (H21: (eq T t0 (lift h (s k d) x4))).(eq_ind_r T (lift h (s k 
+d) x4) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
+(THead k x2 x3))) (\lambda (u1: T).(eq T (THead k (lift h d x) t2) (lift h d 
+u1))))) (eq_ind T (lift h d (THead k x x4)) (\lambda (t2: T).(ex2 T (\lambda 
+(u1: T).(subst0 (minus i h) v u1 (THead k x2 x3))) (\lambda (u1: T).(eq T t2 
+(lift h d u1))))) (ex_intro2 T (\lambda (u1: T).(subst0 (minus i h) v u1 
+(THead k x2 x3))) (\lambda (u1: T).(eq T (lift h d (THead k x x4)) (lift h d 
+u1))) (THead k x x4) (subst0_both v x x2 (minus i h) H15 k x4 x3 H20) 
+(refl_equal T (lift h d (THead k x x4)))) (THead k (lift h d x) (lift h (s k 
+d) x4)) (lift_head k x x4 h d)) t0 H21)))) H19))))) t H16)))) H14))) u2 
+H9)))))))) H8))))))))) H3)) (subst0_gen_head k v t t0 (lift h d u2) i 
+H1)))))))))))))) t1).
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma
index 5da05fa2ad5e5a13e50f88a54e536139e3271600..04300ba9b17940db2d474d76edb124b2ad637d46 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/props.ma
@@ -14,69 +14,93 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/subst0/fwd.ma".
+include "basic_1/subst0/fwd.ma".
 
 theorem subst0_refl:
  \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to 
 (\forall (P: Prop).P))))
 \def
- \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: 
-nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n: 
-nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort 
-n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda 
-(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef 
-n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O 
-u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O 
-u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1 
-P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P: 
-Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u 
-t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0 
-d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T 
-(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2: 
-T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead 
-k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda 
-(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: 
-T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T 
-(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0 
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) 
-(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T 
-(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t2 _) 
-\Rightarrow t2])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def 
-(eq_ind_r T x (\lambda (t2: T).(subst0 d u t0 t2)) H4 t0 H5) in (H d H6 
-P)))))) H2)) (\lambda (H2: (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) 
-(THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2)))).(ex2_ind T 
-(\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2: 
-T).(subst0 (s k d) u t1 t2)) P (\lambda (x: T).(\lambda (H3: (eq T (THead k 
-t0 t1) (THead k t0 x))).(\lambda (H4: (subst0 (s k d) u t1 x)).(let H5 \def 
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+ \lambda (u: T).(\lambda (t: T).(let TMP_1 \def (\lambda (t0: T).(\forall (d: 
+nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) in (let TMP_3 \def 
+(\lambda (n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort 
+n))).(\lambda (P: Prop).(let TMP_2 \def (TSort n) in (subst0_gen_sort u TMP_2 
+d n H P)))))) in (let TMP_17 \def (\lambda (n: nat).(\lambda (d: 
+nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef n))).(\lambda (P: Prop).(let 
+TMP_4 \def (eq nat n d) in (let TMP_5 \def (TLRef n) in (let TMP_6 \def (S n) 
+in (let TMP_7 \def (lift TMP_6 O u) in (let TMP_8 \def (eq T TMP_5 TMP_7) in 
+(let TMP_14 \def (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) 
+(lift (S n) O u))).(let TMP_9 \def (S n) in (let TMP_10 \def (le_O_n n) in 
+(let TMP_11 \def (S n) in (let TMP_12 \def (plus O TMP_11) in (let TMP_13 
+\def (le_n TMP_12) in (lift_gen_lref_false TMP_9 O n TMP_10 TMP_13 u H1 
+P)))))))) in (let TMP_15 \def (TLRef n) in (let TMP_16 \def (subst0_gen_lref 
+u TMP_15 d n H) in (land_ind TMP_4 TMP_8 P TMP_14 TMP_16))))))))))))) in (let 
+TMP_79 \def (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (d: 
+nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P))))).(\lambda (t1: 
+T).(\lambda (H0: ((\forall (d: nat).((subst0 d u t1 t1) \to (\forall (P: 
+Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0 d u (THead k t0 t1) 
+(THead k t0 t1))).(\lambda (P: Prop).(let TMP_20 \def (\lambda (u2: T).(let 
+TMP_18 \def (THead k t0 t1) in (let TMP_19 \def (THead k u2 t1) in (eq T 
+TMP_18 TMP_19)))) in (let TMP_21 \def (\lambda (u2: T).(subst0 d u t0 u2)) in 
+(let TMP_22 \def (ex2 T TMP_20 TMP_21) in (let TMP_25 \def (\lambda (t2: 
+T).(let TMP_23 \def (THead k t0 t1) in (let TMP_24 \def (THead k t0 t2) in 
+(eq T TMP_23 TMP_24)))) in (let TMP_27 \def (\lambda (t2: T).(let TMP_26 \def 
+(s k d) in (subst0 TMP_26 u t1 t2))) in (let TMP_28 \def (ex2 T TMP_25 
+TMP_27) in (let TMP_31 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_29 
+\def (THead k t0 t1) in (let TMP_30 \def (THead k u2 t2) in (eq T TMP_29 
+TMP_30))))) in (let TMP_32 \def (\lambda (u2: T).(\lambda (_: T).(subst0 d u 
+t0 u2))) in (let TMP_34 \def (\lambda (_: T).(\lambda (t2: T).(let TMP_33 
+\def (s k d) in (subst0 TMP_33 u t1 t2)))) in (let TMP_35 \def (ex3_2 T T 
+TMP_31 TMP_32 TMP_34) in (let TMP_45 \def (\lambda (H2: (ex2 T (\lambda (u2: 
+T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0 
+u2)))).(let TMP_38 \def (\lambda (u2: T).(let TMP_36 \def (THead k t0 t1) in 
+(let TMP_37 \def (THead k u2 t1) in (eq T TMP_36 TMP_37)))) in (let TMP_39 
+\def (\lambda (u2: T).(subst0 d u t0 u2)) in (let TMP_44 \def (\lambda (x: 
+T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 
+d u t0 x)).(let TMP_40 \def (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t2 _) \Rightarrow t2])) 
+in (let TMP_41 \def (THead k t0 t1) in (let TMP_42 \def (THead k x t1) in 
+(let H5 \def (f_equal T T TMP_40 TMP_41 TMP_42 H3) in (let TMP_43 \def 
+(\lambda (t2: T).(subst0 d u t0 t2)) in (let H6 \def (eq_ind_r T x TMP_43 H4 
+t0 H5) in (H d H6 P)))))))))) in (ex2_ind T TMP_38 TMP_39 P TMP_44 H2))))) in 
+(let TMP_58 \def (\lambda (H2: (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) 
+(THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2)))).(let TMP_48 
+\def (\lambda (t2: T).(let TMP_46 \def (THead k t0 t1) in (let TMP_47 \def 
+(THead k t0 t2) in (eq T TMP_46 TMP_47)))) in (let TMP_50 \def (\lambda (t2: 
+T).(let TMP_49 \def (s k d) in (subst0 TMP_49 u t1 t2))) in (let TMP_57 \def 
+(\lambda (x: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k t0 x))).(\lambda 
+(H4: (subst0 (s k d) u t1 x)).(let TMP_51 \def (\lambda (e: T).(match e with 
 [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) 
-\Rightarrow t2])) (THead k t0 t1) (THead k t0 x) H3) in (let H6 \def 
-(eq_ind_r T x (\lambda (t2: T).(subst0 (s k d) u t1 t2)) H4 t1 H5) in (H0 (s 
-k d) H6 P)))))) H2)) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+\Rightarrow t2])) in (let TMP_52 \def (THead k t0 t1) in (let TMP_53 \def 
+(THead k t0 x) in (let H5 \def (f_equal T T TMP_51 TMP_52 TMP_53 H3) in (let 
+TMP_55 \def (\lambda (t2: T).(let TMP_54 \def (s k d) in (subst0 TMP_54 u t1 
+t2))) in (let H6 \def (eq_ind_r T x TMP_55 H4 t1 H5) in (let TMP_56 \def (s k 
+d) in (H0 TMP_56 H6 P))))))))))) in (ex2_ind T TMP_48 TMP_50 P TMP_57 H2))))) 
+in (let TMP_76 \def (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
 T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
 T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 
-t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 
-t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2))) 
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0: 
+t2))))).(let TMP_61 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_59 \def 
+(THead k t0 t1) in (let TMP_60 \def (THead k u2 t2) in (eq T TMP_59 
+TMP_60))))) in (let TMP_62 \def (\lambda (u2: T).(\lambda (_: T).(subst0 d u 
+t0 u2))) in (let TMP_64 \def (\lambda (_: T).(\lambda (t2: T).(let TMP_63 
+\def (s k d) in (subst0 TMP_63 u t1 t2)))) in (let TMP_75 \def (\lambda (x0: 
 T).(\lambda (x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0 
 x1))).(\lambda (H4: (subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1 
-x1)).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
-(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead 
-_ t2 _) \Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in ((let H7 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t2) 
-\Rightarrow t2])) (THead k t0 t1) (THead k x0 x1) H3) in (\lambda (H8: (eq T 
-t0 x0)).(let H9 \def (eq_ind_r T x1 (\lambda (t2: T).(subst0 (s k d) u t1 
-t2)) H5 t1 H7) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: T).(subst0 d u 
-t0 t2)) H4 t0 H8) in (H d H10 P))))) H6))))))) H2)) (subst0_gen_head k u t0 
-t1 (THead k t0 t1) d H1)))))))))) t)).
-(* COMMENTS
-Initial nodes: 1119
-END *)
+x1)).(let TMP_65 \def (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 
+| (TLRef _) \Rightarrow t0 | (THead _ t2 _) \Rightarrow t2])) in (let TMP_66 
+\def (THead k t0 t1) in (let TMP_67 \def (THead k x0 x1) in (let H6 \def 
+(f_equal T T TMP_65 TMP_66 TMP_67 H3) in (let TMP_68 \def (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | 
+(THead _ _ t2) \Rightarrow t2])) in (let TMP_69 \def (THead k t0 t1) in (let 
+TMP_70 \def (THead k x0 x1) in (let H7 \def (f_equal T T TMP_68 TMP_69 TMP_70 
+H3) in (let TMP_74 \def (\lambda (H8: (eq T t0 x0)).(let TMP_72 \def (\lambda 
+(t2: T).(let TMP_71 \def (s k d) in (subst0 TMP_71 u t1 t2))) in (let H9 \def 
+(eq_ind_r T x1 TMP_72 H5 t1 H7) in (let TMP_73 \def (\lambda (t2: T).(subst0 
+d u t0 t2)) in (let H10 \def (eq_ind_r T x0 TMP_73 H4 t0 H8) in (H d H10 
+P)))))) in (TMP_74 H6))))))))))))))) in (ex3_2_ind T T TMP_61 TMP_62 TMP_64 P 
+TMP_75 H2)))))) in (let TMP_77 \def (THead k t0 t1) in (let TMP_78 \def 
+(subst0_gen_head k u t0 t1 TMP_77 d H1) in (or3_ind TMP_22 TMP_28 TMP_35 P 
+TMP_45 TMP_58 TMP_76 TMP_78)))))))))))))))))))))))) in (T_ind TMP_1 TMP_3 
+TMP_17 TMP_79 t)))))).
 
 theorem subst0_lift_lt:
  \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
@@ -84,72 +108,163 @@ i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i
 (lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: 
+(H: (subst0 i u t1 t2)).(let TMP_6 \def (\lambda (n: nat).(\lambda (t: 
 T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall 
-(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d 
-t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda 
-(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t: 
-T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v)))) 
-(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0))) 
-(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O 
-v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t: 
-T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S 
-i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d 
-v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r 
-(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda 
-(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: 
-(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall 
-(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d 
-u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt 
-i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) 
-t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d 
-(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t)) 
-(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d 
-u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h 
-d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k 
-u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h 
-d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: 
-((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) 
-(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda 
-(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3 
-\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d0: nat).((lt (s 
-k i0) d0) \to (\forall (h0: nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) 
-(lift h0 d0 t3) (lift h0 d0 t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r 
-T (THead k (lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0 
-(lift h (minus d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead 
-k (lift h d u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h 
-(minus d (S i0)) v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind 
-nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v) 
-(THead k (lift h d u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s 
-k d) t0)))) (subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s 
-k d) t0) (lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d 
-u0)) (minus d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0)) 
-(lift_head k u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h 
-d)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(h: nat).(let TMP_1 \def (S n) in (let TMP_2 \def (minus d TMP_1) in (let 
+TMP_3 \def (lift h TMP_2 t) in (let TMP_4 \def (lift h d t0) in (let TMP_5 
+\def (lift h d t3) in (subst0 n TMP_3 TMP_4 TMP_5))))))))))))) in (let TMP_59 
+\def (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (lt 
+i0 d)).(\lambda (h: nat).(let TMP_7 \def (TLRef i0) in (let TMP_14 \def 
+(\lambda (t: T).(let TMP_8 \def (S i0) in (let TMP_9 \def (minus d TMP_8) in 
+(let TMP_10 \def (lift h TMP_9 v) in (let TMP_11 \def (S i0) in (let TMP_12 
+\def (lift TMP_11 O v) in (let TMP_13 \def (lift h d TMP_12) in (subst0 i0 
+TMP_10 t TMP_13)))))))) in (let TMP_15 \def (S i0) in (let w \def (minus d 
+TMP_15) in (let TMP_16 \def (S i0) in (let TMP_17 \def (S i0) in (let TMP_18 
+\def (minus d TMP_17) in (let TMP_19 \def (plus TMP_16 TMP_18) in (let TMP_25 
+\def (\lambda (n: nat).(let TMP_20 \def (lift h w v) in (let TMP_21 \def 
+(TLRef i0) in (let TMP_22 \def (S i0) in (let TMP_23 \def (lift TMP_22 O v) 
+in (let TMP_24 \def (lift h n TMP_23) in (subst0 i0 TMP_20 TMP_21 
+TMP_24))))))) in (let TMP_26 \def (S i0) in (let TMP_27 \def (S i0) in (let 
+TMP_28 \def (minus d TMP_27) in (let TMP_29 \def (lift h TMP_28 v) in (let 
+TMP_30 \def (lift TMP_26 O TMP_29) in (let TMP_33 \def (\lambda (t: T).(let 
+TMP_31 \def (lift h w v) in (let TMP_32 \def (TLRef i0) in (subst0 i0 TMP_31 
+TMP_32 t)))) in (let TMP_34 \def (S i0) in (let TMP_35 \def (minus d TMP_34) 
+in (let TMP_36 \def (lift h TMP_35 v) in (let TMP_37 \def (subst0_lref TMP_36 
+i0) in (let TMP_38 \def (S i0) in (let TMP_39 \def (S i0) in (let TMP_40 \def 
+(minus d TMP_39) in (let TMP_41 \def (plus TMP_38 TMP_40) in (let TMP_42 \def 
+(S i0) in (let TMP_43 \def (lift TMP_42 O v) in (let TMP_44 \def (lift h 
+TMP_41 TMP_43) in (let TMP_45 \def (S i0) in (let TMP_46 \def (S i0) in (let 
+TMP_47 \def (minus d TMP_46) in (let TMP_48 \def (S i0) in (let TMP_49 \def 
+(minus d TMP_48) in (let TMP_50 \def (le_O_n TMP_49) in (let TMP_51 \def 
+(lift_d v h TMP_45 TMP_47 O TMP_50) in (let TMP_52 \def (eq_ind_r T TMP_30 
+TMP_33 TMP_37 TMP_44 TMP_51) in (let TMP_53 \def (S i0) in (let TMP_54 \def 
+(le_plus_minus_r TMP_53 d H0) in (let TMP_55 \def (eq_ind nat TMP_19 TMP_25 
+TMP_52 d TMP_54) in (let TMP_56 \def (TLRef i0) in (let TMP_57 \def (lift h d 
+TMP_56) in (let TMP_58 \def (lift_lref_lt i0 h d H0) in (eq_ind_r T TMP_7 
+TMP_14 TMP_55 TMP_57 TMP_58)))))))))))))))))))))))))))))))))))))))))))))) in 
+(let TMP_98 \def (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda 
 (i0: nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: 
 nat).((lt i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v) 
-(lift h d u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda 
-(t3: T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d: 
+(lift h d u1) (lift h d u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda 
+(d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let TMP_60 \def (lift h 
+d u1) in (let TMP_61 \def (s k d) in (let TMP_62 \def (lift h TMP_61 t) in 
+(let TMP_63 \def (THead k TMP_60 TMP_62) in (let TMP_69 \def (\lambda (t0: 
+T).(let TMP_64 \def (S i0) in (let TMP_65 \def (minus d TMP_64) in (let 
+TMP_66 \def (lift h TMP_65 v) in (let TMP_67 \def (THead k u2 t) in (let 
+TMP_68 \def (lift h d TMP_67) in (subst0 i0 TMP_66 t0 TMP_68))))))) in (let 
+TMP_70 \def (lift h d u2) in (let TMP_71 \def (s k d) in (let TMP_72 \def 
+(lift h TMP_71 t) in (let TMP_73 \def (THead k TMP_70 TMP_72) in (let TMP_81 
+\def (\lambda (t0: T).(let TMP_74 \def (S i0) in (let TMP_75 \def (minus d 
+TMP_74) in (let TMP_76 \def (lift h TMP_75 v) in (let TMP_77 \def (lift h d 
+u1) in (let TMP_78 \def (s k d) in (let TMP_79 \def (lift h TMP_78 t) in (let 
+TMP_80 \def (THead k TMP_77 TMP_79) in (subst0 i0 TMP_76 TMP_80 t0))))))))) 
+in (let TMP_82 \def (S i0) in (let TMP_83 \def (minus d TMP_82) in (let 
+TMP_84 \def (lift h TMP_83 v) in (let TMP_85 \def (lift h d u2) in (let 
+TMP_86 \def (lift h d u1) in (let TMP_87 \def (H1 d H2 h) in (let TMP_88 \def 
+(s k d) in (let TMP_89 \def (lift h TMP_88 t) in (let TMP_90 \def (subst0_fst 
+TMP_84 TMP_85 TMP_86 i0 TMP_87 TMP_89 k) in (let TMP_91 \def (THead k u2 t) 
+in (let TMP_92 \def (lift h d TMP_91) in (let TMP_93 \def (lift_head k u2 t h 
+d) in (let TMP_94 \def (eq_ind_r T TMP_73 TMP_81 TMP_90 TMP_92 TMP_93) in 
+(let TMP_95 \def (THead k u1 t) in (let TMP_96 \def (lift h d TMP_95) in (let 
+TMP_97 \def (lift_head k u1 t h d) in (eq_ind_r T TMP_63 TMP_69 TMP_94 TMP_96 
+TMP_97)))))))))))))))))))))))))))))))))))))) in (let TMP_172 \def (\lambda 
+(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d: 
+nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d 
+(S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda (u0: T).(\lambda 
+(d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let TMP_99 \def (s k i0) 
+in (let TMP_100 \def (S TMP_99) in (let TMP_106 \def (\lambda (n: 
+nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0: nat).(let TMP_101 
+\def (s k i0) in (let TMP_102 \def (minus d0 n) in (let TMP_103 \def (lift h0 
+TMP_102 v) in (let TMP_104 \def (lift h0 d0 t3) in (let TMP_105 \def (lift h0 
+d0 t0) in (subst0 TMP_101 TMP_103 TMP_104 TMP_105)))))))))) in (let TMP_107 
+\def (S i0) in (let TMP_108 \def (s k TMP_107) in (let TMP_109 \def (s_S k 
+i0) in (let H3 \def (eq_ind_r nat TMP_100 TMP_106 H1 TMP_108 TMP_109) in (let 
+TMP_110 \def (lift h d u0) in (let TMP_111 \def (s k d) in (let TMP_112 \def 
+(lift h TMP_111 t3) in (let TMP_113 \def (THead k TMP_110 TMP_112) in (let 
+TMP_119 \def (\lambda (t: T).(let TMP_114 \def (S i0) in (let TMP_115 \def 
+(minus d TMP_114) in (let TMP_116 \def (lift h TMP_115 v) in (let TMP_117 
+\def (THead k u0 t0) in (let TMP_118 \def (lift h d TMP_117) in (subst0 i0 
+TMP_116 t TMP_118))))))) in (let TMP_120 \def (lift h d u0) in (let TMP_121 
+\def (s k d) in (let TMP_122 \def (lift h TMP_121 t0) in (let TMP_123 \def 
+(THead k TMP_120 TMP_122) in (let TMP_131 \def (\lambda (t: T).(let TMP_124 
+\def (S i0) in (let TMP_125 \def (minus d TMP_124) in (let TMP_126 \def (lift 
+h TMP_125 v) in (let TMP_127 \def (lift h d u0) in (let TMP_128 \def (s k d) 
+in (let TMP_129 \def (lift h TMP_128 t3) in (let TMP_130 \def (THead k 
+TMP_127 TMP_129) in (subst0 i0 TMP_126 TMP_130 t))))))))) in (let TMP_132 
+\def (s k d) in (let TMP_133 \def (S i0) in (let TMP_134 \def (s k TMP_133) 
+in (let TMP_135 \def (minus TMP_132 TMP_134) in (let TMP_145 \def (\lambda 
+(n: nat).(let TMP_136 \def (lift h n v) in (let TMP_137 \def (lift h d u0) in 
+(let TMP_138 \def (s k d) in (let TMP_139 \def (lift h TMP_138 t3) in (let 
+TMP_140 \def (THead k TMP_137 TMP_139) in (let TMP_141 \def (lift h d u0) in 
+(let TMP_142 \def (s k d) in (let TMP_143 \def (lift h TMP_142 t0) in (let 
+TMP_144 \def (THead k TMP_141 TMP_143) in (subst0 i0 TMP_136 TMP_140 
+TMP_144))))))))))) in (let TMP_146 \def (s k d) in (let TMP_147 \def (S i0) 
+in (let TMP_148 \def (s k TMP_147) in (let TMP_149 \def (minus TMP_146 
+TMP_148) in (let TMP_150 \def (lift h TMP_149 v) in (let TMP_151 \def (s k d) 
+in (let TMP_152 \def (lift h TMP_151 t0) in (let TMP_153 \def (s k d) in (let 
+TMP_154 \def (lift h TMP_153 t3) in (let TMP_155 \def (s k d) in (let TMP_156 
+\def (s_lt k i0 d H2) in (let TMP_157 \def (H3 TMP_155 TMP_156 h) in (let 
+TMP_158 \def (lift h d u0) in (let TMP_159 \def (subst0_snd k TMP_150 TMP_152 
+TMP_154 i0 TMP_157 TMP_158) in (let TMP_160 \def (S i0) in (let TMP_161 \def 
+(minus d TMP_160) in (let TMP_162 \def (S i0) in (let TMP_163 \def (minus_s_s 
+k d TMP_162) in (let TMP_164 \def (eq_ind nat TMP_135 TMP_145 TMP_159 TMP_161 
+TMP_163) in (let TMP_165 \def (THead k u0 t0) in (let TMP_166 \def (lift h d 
+TMP_165) in (let TMP_167 \def (lift_head k u0 t0 h d) in (let TMP_168 \def 
+(eq_ind_r T TMP_123 TMP_131 TMP_164 TMP_166 TMP_167) in (let TMP_169 \def 
+(THead k u0 t3) in (let TMP_170 \def (lift h d TMP_169) in (let TMP_171 \def 
+(lift_head k u0 t3 h d) in (eq_ind_r T TMP_113 TMP_119 TMP_168 TMP_170 
+TMP_171)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (let 
+TMP_243 \def (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt 
+i0 d) \to (\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d 
+u1) (lift h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d: 
 nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d 
 (S (s k i0))) v) (lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda 
-(H4: (lt i0 d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0)) 
-(\lambda (n: nat).(\forall (d0: nat).((lt (s k i0) d0) \to (\forall (h0: 
-nat).(subst0 (s k i0) (lift h0 (minus d0 n) v) (lift h0 d0 t0) (lift h0 d0 
-t3)))))) H3 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k (lift h d u1) 
-(lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t 
-(lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k 
-d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift 
-h d u1) (lift h (s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v) 
-(lift h d u1) (lift h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k 
-d) t3) (eq_ind nat (minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s 
-k i0) (lift h n v) (lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d) 
-(s_lt k i0 d H4) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d 
-(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) 
-(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))).
-(* COMMENTS
-Initial nodes: 1805
-END *)
+(H4: (lt i0 d)).(\lambda (h: nat).(let TMP_173 \def (s k i0) in (let TMP_174 
+\def (S TMP_173) in (let TMP_180 \def (\lambda (n: nat).(\forall (d0: 
+nat).((lt (s k i0) d0) \to (\forall (h0: nat).(let TMP_175 \def (s k i0) in 
+(let TMP_176 \def (minus d0 n) in (let TMP_177 \def (lift h0 TMP_176 v) in 
+(let TMP_178 \def (lift h0 d0 t0) in (let TMP_179 \def (lift h0 d0 t3) in 
+(subst0 TMP_175 TMP_177 TMP_178 TMP_179)))))))))) in (let TMP_181 \def (S i0) 
+in (let TMP_182 \def (s k TMP_181) in (let TMP_183 \def (s_S k i0) in (let H5 
+\def (eq_ind_r nat TMP_174 TMP_180 H3 TMP_182 TMP_183) in (let TMP_184 \def 
+(lift h d u1) in (let TMP_185 \def (s k d) in (let TMP_186 \def (lift h 
+TMP_185 t0) in (let TMP_187 \def (THead k TMP_184 TMP_186) in (let TMP_193 
+\def (\lambda (t: T).(let TMP_188 \def (S i0) in (let TMP_189 \def (minus d 
+TMP_188) in (let TMP_190 \def (lift h TMP_189 v) in (let TMP_191 \def (THead 
+k u2 t3) in (let TMP_192 \def (lift h d TMP_191) in (subst0 i0 TMP_190 t 
+TMP_192))))))) in (let TMP_194 \def (lift h d u2) in (let TMP_195 \def (s k 
+d) in (let TMP_196 \def (lift h TMP_195 t3) in (let TMP_197 \def (THead k 
+TMP_194 TMP_196) in (let TMP_205 \def (\lambda (t: T).(let TMP_198 \def (S 
+i0) in (let TMP_199 \def (minus d TMP_198) in (let TMP_200 \def (lift h 
+TMP_199 v) in (let TMP_201 \def (lift h d u1) in (let TMP_202 \def (s k d) in 
+(let TMP_203 \def (lift h TMP_202 t0) in (let TMP_204 \def (THead k TMP_201 
+TMP_203) in (subst0 i0 TMP_200 TMP_204 t))))))))) in (let TMP_206 \def (S i0) 
+in (let TMP_207 \def (minus d TMP_206) in (let TMP_208 \def (lift h TMP_207 
+v) in (let TMP_209 \def (lift h d u1) in (let TMP_210 \def (lift h d u2) in 
+(let TMP_211 \def (H1 d H4 h) in (let TMP_212 \def (s k d) in (let TMP_213 
+\def (lift h TMP_212 t0) in (let TMP_214 \def (s k d) in (let TMP_215 \def 
+(lift h TMP_214 t3) in (let TMP_216 \def (s k d) in (let TMP_217 \def (S i0) 
+in (let TMP_218 \def (s k TMP_217) in (let TMP_219 \def (minus TMP_216 
+TMP_218) in (let TMP_226 \def (\lambda (n: nat).(let TMP_220 \def (s k i0) in 
+(let TMP_221 \def (lift h n v) in (let TMP_222 \def (s k d) in (let TMP_223 
+\def (lift h TMP_222 t0) in (let TMP_224 \def (s k d) in (let TMP_225 \def 
+(lift h TMP_224 t3) in (subst0 TMP_220 TMP_221 TMP_223 TMP_225)))))))) in 
+(let TMP_227 \def (s k d) in (let TMP_228 \def (s_lt k i0 d H4) in (let 
+TMP_229 \def (H5 TMP_227 TMP_228 h) in (let TMP_230 \def (S i0) in (let 
+TMP_231 \def (minus d TMP_230) in (let TMP_232 \def (S i0) in (let TMP_233 
+\def (minus_s_s k d TMP_232) in (let TMP_234 \def (eq_ind nat TMP_219 TMP_226 
+TMP_229 TMP_231 TMP_233) in (let TMP_235 \def (subst0_both TMP_208 TMP_209 
+TMP_210 i0 TMP_211 k TMP_213 TMP_215 TMP_234) in (let TMP_236 \def (THead k 
+u2 t3) in (let TMP_237 \def (lift h d TMP_236) in (let TMP_238 \def 
+(lift_head k u2 t3 h d) in (let TMP_239 \def (eq_ind_r T TMP_197 TMP_205 
+TMP_235 TMP_237 TMP_238) in (let TMP_240 \def (THead k u1 t0) in (let TMP_241 
+\def (lift h d TMP_240) in (let TMP_242 \def (lift_head k u1 t0 h d) in 
+(eq_ind_r T TMP_187 TMP_193 TMP_239 TMP_241 
+TMP_242))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in 
+(subst0_ind TMP_6 TMP_59 TMP_98 TMP_172 TMP_243 i u t1 t2 H)))))))))).
 
 theorem subst0_lift_ge:
  \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall 
@@ -157,60 +272,124 @@ theorem subst0_lift_ge:
 (plus i h) u (lift h d t1) (lift h d t2)))))))))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: 
+(h: nat).(\lambda (H: (subst0 i u t1 t2)).(let TMP_4 \def (\lambda (n: 
 nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le 
-d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v: 
-T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T 
-(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift 
-(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0 
-(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda 
-(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r 
-nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0)) 
-O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S 
-i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d 
-(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d 
-H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
-nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le 
-d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t: 
-T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T 
-(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 
-h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift 
-h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1) 
-(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0 
-h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2 
-t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda 
-(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: 
-nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d: 
-nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d 
-t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3 
-\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d0: 
-nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t3) (lift h d0 t0))))) H1 
-(s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0) 
-(lift h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead 
-k u0 t0)))) (eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda 
-(t: T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t)) 
-(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k 
-d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k 
-u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d))))))))))))) 
-(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda 
+d n) \to (let TMP_1 \def (plus n h) in (let TMP_2 \def (lift h d t0) in (let 
+TMP_3 \def (lift h d t3) in (subst0 TMP_1 t TMP_2 TMP_3)))))))))) in (let 
+TMP_52 \def (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda 
+(H0: (le d i0)).(let TMP_5 \def (plus i0 h) in (let TMP_6 \def (TLRef TMP_5) 
+in (let TMP_11 \def (\lambda (t: T).(let TMP_7 \def (plus i0 h) in (let TMP_8 
+\def (S i0) in (let TMP_9 \def (lift TMP_8 O v) in (let TMP_10 \def (lift h d 
+TMP_9) in (subst0 TMP_7 v t TMP_10)))))) in (let TMP_12 \def (S i0) in (let 
+TMP_13 \def (plus h TMP_12) in (let TMP_14 \def (lift TMP_13 O v) in (let 
+TMP_18 \def (\lambda (t: T).(let TMP_15 \def (plus i0 h) in (let TMP_16 \def 
+(plus i0 h) in (let TMP_17 \def (TLRef TMP_16) in (subst0 TMP_15 v TMP_17 
+t))))) in (let TMP_19 \def (plus h i0) in (let TMP_20 \def (S TMP_19) in (let 
+TMP_25 \def (\lambda (n: nat).(let TMP_21 \def (plus i0 h) in (let TMP_22 
+\def (plus i0 h) in (let TMP_23 \def (TLRef TMP_22) in (let TMP_24 \def (lift 
+n O v) in (subst0 TMP_21 v TMP_23 TMP_24)))))) in (let TMP_26 \def (plus h 
+i0) in (let TMP_31 \def (\lambda (n: nat).(let TMP_27 \def (TLRef n) in (let 
+TMP_28 \def (plus h i0) in (let TMP_29 \def (S TMP_28) in (let TMP_30 \def 
+(lift TMP_29 O v) in (subst0 n v TMP_27 TMP_30)))))) in (let TMP_32 \def 
+(plus h i0) in (let TMP_33 \def (subst0_lref v TMP_32) in (let TMP_34 \def 
+(plus i0 h) in (let TMP_35 \def (plus_sym i0 h) in (let TMP_36 \def (eq_ind_r 
+nat TMP_26 TMP_31 TMP_33 TMP_34 TMP_35) in (let TMP_37 \def (S i0) in (let 
+TMP_38 \def (plus h TMP_37) in (let TMP_39 \def (plus_n_Sm h i0) in (let 
+TMP_40 \def (eq_ind nat TMP_20 TMP_25 TMP_36 TMP_38 TMP_39) in (let TMP_41 
+\def (S i0) in (let TMP_42 \def (lift TMP_41 O v) in (let TMP_43 \def (lift h 
+d TMP_42) in (let TMP_44 \def (S i0) in (let TMP_45 \def (le_S d i0 H0) in 
+(let TMP_46 \def (le_O_n d) in (let TMP_47 \def (lift_free v TMP_44 h O d 
+TMP_45 TMP_46) in (let TMP_48 \def (eq_ind_r T TMP_14 TMP_18 TMP_40 TMP_43 
+TMP_47) in (let TMP_49 \def (TLRef i0) in (let TMP_50 \def (lift h d TMP_49) 
+in (let TMP_51 \def (lift_lref_ge i0 h d H0) in (eq_ind_r T TMP_6 TMP_11 
+TMP_48 TMP_50 TMP_51))))))))))))))))))))))))))))))))))))) in (let TMP_85 \def 
+(\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda 
 (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to 
-(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k: 
+(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t: 
+T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let TMP_53 
+\def (lift h d u1) in (let TMP_54 \def (s k d) in (let TMP_55 \def (lift h 
+TMP_54 t) in (let TMP_56 \def (THead k TMP_53 TMP_55) in (let TMP_60 \def 
+(\lambda (t0: T).(let TMP_57 \def (plus i0 h) in (let TMP_58 \def (THead k u2 
+t) in (let TMP_59 \def (lift h d TMP_58) in (subst0 TMP_57 v t0 TMP_59))))) 
+in (let TMP_61 \def (lift h d u2) in (let TMP_62 \def (s k d) in (let TMP_63 
+\def (lift h TMP_62 t) in (let TMP_64 \def (THead k TMP_61 TMP_63) in (let 
+TMP_70 \def (\lambda (t0: T).(let TMP_65 \def (plus i0 h) in (let TMP_66 \def 
+(lift h d u1) in (let TMP_67 \def (s k d) in (let TMP_68 \def (lift h TMP_67 
+t) in (let TMP_69 \def (THead k TMP_66 TMP_68) in (subst0 TMP_65 v TMP_69 
+t0))))))) in (let TMP_71 \def (lift h d u2) in (let TMP_72 \def (lift h d u1) 
+in (let TMP_73 \def (plus i0 h) in (let TMP_74 \def (H1 d H2) in (let TMP_75 
+\def (s k d) in (let TMP_76 \def (lift h TMP_75 t) in (let TMP_77 \def 
+(subst0_fst v TMP_71 TMP_72 TMP_73 TMP_74 TMP_76 k) in (let TMP_78 \def 
+(THead k u2 t) in (let TMP_79 \def (lift h d TMP_78) in (let TMP_80 \def 
+(lift_head k u2 t h d) in (let TMP_81 \def (eq_ind_r T TMP_64 TMP_70 TMP_77 
+TMP_79 TMP_80) in (let TMP_82 \def (THead k u1 t) in (let TMP_83 \def (lift h 
+d TMP_82) in (let TMP_84 \def (lift_head k u1 t h d) in (eq_ind_r T TMP_56 
+TMP_60 TMP_81 TMP_83 TMP_84))))))))))))))))))))))))))))))))))) in (let 
+TMP_129 \def (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: 
+((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d 
+t3) (lift h d t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d 
+i0)).(let TMP_86 \def (s k i0) in (let TMP_87 \def (plus TMP_86 h) in (let 
+TMP_90 \def (\lambda (n: nat).(\forall (d0: nat).((le d0 (s k i0)) \to (let 
+TMP_88 \def (lift h d0 t3) in (let TMP_89 \def (lift h d0 t0) in (subst0 n v 
+TMP_88 TMP_89)))))) in (let TMP_91 \def (plus i0 h) in (let TMP_92 \def (s k 
+TMP_91) in (let TMP_93 \def (s_plus k i0 h) in (let H3 \def (eq_ind_r nat 
+TMP_87 TMP_90 H1 TMP_92 TMP_93) in (let TMP_94 \def (lift h d u0) in (let 
+TMP_95 \def (s k d) in (let TMP_96 \def (lift h TMP_95 t3) in (let TMP_97 
+\def (THead k TMP_94 TMP_96) in (let TMP_101 \def (\lambda (t: T).(let TMP_98 
+\def (plus i0 h) in (let TMP_99 \def (THead k u0 t0) in (let TMP_100 \def 
+(lift h d TMP_99) in (subst0 TMP_98 v t TMP_100))))) in (let TMP_102 \def 
+(lift h d u0) in (let TMP_103 \def (s k d) in (let TMP_104 \def (lift h 
+TMP_103 t0) in (let TMP_105 \def (THead k TMP_102 TMP_104) in (let TMP_111 
+\def (\lambda (t: T).(let TMP_106 \def (plus i0 h) in (let TMP_107 \def (lift 
+h d u0) in (let TMP_108 \def (s k d) in (let TMP_109 \def (lift h TMP_108 t3) 
+in (let TMP_110 \def (THead k TMP_107 TMP_109) in (subst0 TMP_106 v TMP_110 
+t))))))) in (let TMP_112 \def (s k d) in (let TMP_113 \def (lift h TMP_112 
+t0) in (let TMP_114 \def (s k d) in (let TMP_115 \def (lift h TMP_114 t3) in 
+(let TMP_116 \def (plus i0 h) in (let TMP_117 \def (s k d) in (let TMP_118 
+\def (s_le k d i0 H2) in (let TMP_119 \def (H3 TMP_117 TMP_118) in (let 
+TMP_120 \def (lift h d u0) in (let TMP_121 \def (subst0_snd k v TMP_113 
+TMP_115 TMP_116 TMP_119 TMP_120) in (let TMP_122 \def (THead k u0 t0) in (let 
+TMP_123 \def (lift h d TMP_122) in (let TMP_124 \def (lift_head k u0 t0 h d) 
+in (let TMP_125 \def (eq_ind_r T TMP_105 TMP_111 TMP_121 TMP_123 TMP_124) in 
+(let TMP_126 \def (THead k u0 t3) in (let TMP_127 \def (lift h d TMP_126) in 
+(let TMP_128 \def (lift_head k u0 t3 h d) in (eq_ind_r T TMP_97 TMP_101 
+TMP_125 TMP_127 TMP_128))))))))))))))))))))))))))))))))))))))))))))) in (let 
+TMP_175 \def (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le 
+d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k: 
 K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0 
 t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k 
 i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le 
-d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: 
-nat).(\forall (d0: nat).((le d0 (s k i0)) \to (subst0 n v (lift h d0 t0) 
-(lift h d0 t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T 
-(THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0 
-h) v t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift 
-h (s k d) t3)) (\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1) 
-(lift h (s k d) t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0 
-h) (H1 d H4) k (lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d 
-i0 H4))) (lift h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead 
-k u1 t0)) (lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 1449
-END *)
+d i0)).(let TMP_130 \def (s k i0) in (let TMP_131 \def (plus TMP_130 h) in 
+(let TMP_134 \def (\lambda (n: nat).(\forall (d0: nat).((le d0 (s k i0)) \to 
+(let TMP_132 \def (lift h d0 t0) in (let TMP_133 \def (lift h d0 t3) in 
+(subst0 n v TMP_132 TMP_133)))))) in (let TMP_135 \def (plus i0 h) in (let 
+TMP_136 \def (s k TMP_135) in (let TMP_137 \def (s_plus k i0 h) in (let H5 
+\def (eq_ind_r nat TMP_131 TMP_134 H3 TMP_136 TMP_137) in (let TMP_138 \def 
+(lift h d u1) in (let TMP_139 \def (s k d) in (let TMP_140 \def (lift h 
+TMP_139 t0) in (let TMP_141 \def (THead k TMP_138 TMP_140) in (let TMP_145 
+\def (\lambda (t: T).(let TMP_142 \def (plus i0 h) in (let TMP_143 \def 
+(THead k u2 t3) in (let TMP_144 \def (lift h d TMP_143) in (subst0 TMP_142 v 
+t TMP_144))))) in (let TMP_146 \def (lift h d u2) in (let TMP_147 \def (s k 
+d) in (let TMP_148 \def (lift h TMP_147 t3) in (let TMP_149 \def (THead k 
+TMP_146 TMP_148) in (let TMP_155 \def (\lambda (t: T).(let TMP_150 \def (plus 
+i0 h) in (let TMP_151 \def (lift h d u1) in (let TMP_152 \def (s k d) in (let 
+TMP_153 \def (lift h TMP_152 t0) in (let TMP_154 \def (THead k TMP_151 
+TMP_153) in (subst0 TMP_150 v TMP_154 t))))))) in (let TMP_156 \def (lift h d 
+u1) in (let TMP_157 \def (lift h d u2) in (let TMP_158 \def (plus i0 h) in 
+(let TMP_159 \def (H1 d H4) in (let TMP_160 \def (s k d) in (let TMP_161 \def 
+(lift h TMP_160 t0) in (let TMP_162 \def (s k d) in (let TMP_163 \def (lift h 
+TMP_162 t3) in (let TMP_164 \def (s k d) in (let TMP_165 \def (s_le k d i0 
+H4) in (let TMP_166 \def (H5 TMP_164 TMP_165) in (let TMP_167 \def 
+(subst0_both v TMP_156 TMP_157 TMP_158 TMP_159 k TMP_161 TMP_163 TMP_166) in 
+(let TMP_168 \def (THead k u2 t3) in (let TMP_169 \def (lift h d TMP_168) in 
+(let TMP_170 \def (lift_head k u2 t3 h d) in (let TMP_171 \def (eq_ind_r T 
+TMP_149 TMP_155 TMP_167 TMP_169 TMP_170) in (let TMP_172 \def (THead k u1 t0) 
+in (let TMP_173 \def (lift h d TMP_172) in (let TMP_174 \def (lift_head k u1 
+t0 h d) in (eq_ind_r T TMP_141 TMP_145 TMP_171 TMP_173 
+TMP_174)))))))))))))))))))))))))))))))))))))))))))))))))) in (subst0_ind 
+TMP_4 TMP_52 TMP_85 TMP_129 TMP_175 i u t1 t2 H))))))))))).
 
 theorem subst0_lift_ge_S:
  \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
@@ -218,14 +397,22 @@ i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d
 t1) (lift (S O) d t2))))))))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat 
-(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d 
-t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O) 
-i) (\lambda (n: nat).(eq nat n (S i))) (refl_equal nat (S i)) (plus i (S O)) 
-(plus_sym i (S O)))))))))).
-(* COMMENTS
-Initial nodes: 137
-END *)
+(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(let TMP_1 
+\def (S O) in (let TMP_2 \def (plus i TMP_1) in (let TMP_7 \def (\lambda (n: 
+nat).(let TMP_3 \def (S O) in (let TMP_4 \def (lift TMP_3 d t1) in (let TMP_5 
+\def (S O) in (let TMP_6 \def (lift TMP_5 d t2) in (subst0 n u TMP_4 
+TMP_6)))))) in (let TMP_8 \def (S O) in (let TMP_9 \def (subst0_lift_ge t1 t2 
+u i TMP_8 H d H0) in (let TMP_10 \def (S i) in (let TMP_11 \def (S O) in (let 
+TMP_12 \def (plus TMP_11 i) in (let TMP_14 \def (\lambda (n: nat).(let TMP_13 
+\def (S i) in (eq nat n TMP_13))) in (let TMP_15 \def (S O) in (let TMP_16 
+\def (plus TMP_15 i) in (let TMP_17 \def (S i) in (let TMP_18 \def (S i) in 
+(let TMP_19 \def (le_n TMP_18) in (let TMP_20 \def (S O) in (let TMP_21 \def 
+(plus TMP_20 i) in (let TMP_22 \def (le_n TMP_21) in (let TMP_23 \def 
+(le_antisym TMP_16 TMP_17 TMP_19 TMP_22) in (let TMP_24 \def (S O) in (let 
+TMP_25 \def (plus i TMP_24) in (let TMP_26 \def (S O) in (let TMP_27 \def 
+(plus_sym i TMP_26) in (let TMP_28 \def (eq_ind_r nat TMP_12 TMP_14 TMP_23 
+TMP_25 TMP_27) in (eq_ind nat TMP_2 TMP_7 TMP_9 TMP_10 
+TMP_28)))))))))))))))))))))))))))))).
 
 theorem subst0_lift_ge_s:
  \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 
@@ -235,7 +422,4 @@ i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s
  \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda 
 (H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda 
 (_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))).
-(* COMMENTS
-Initial nodes: 43
-END *)
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/subst0.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/subst0.ma
index 66c167d2b117015bdc09de1b10c02c545404c7fd..99caee518484e50d5af25e2d9c3bf65cddf9c053 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/subst0.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/subst0.ma
@@ -14,7 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/subst0/props.ma".
+include "basic_1/subst0/props.ma".
+
+include "basic_1/s/fwd.ma".
 
 theorem subst0_subst0:
  \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 
@@ -91,9 +93,6 @@ T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u
 t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5) 
 (subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4))))) 
 (H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))).
-(* COMMENTS
-Initial nodes: 1613
-END *)
 
 theorem subst0_subst0_back:
  \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 
@@ -170,9 +169,6 @@ t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0
 x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i)) 
 H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2 
 t1 t2 H))))).
-(* COMMENTS
-Initial nodes: 1613
-END *)
 
 theorem subst0_trans:
  \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0 
@@ -285,9 +281,6 @@ i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))
 T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0 
 x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0 
 H4))))))))))))))) i v t1 t2 H))))).
-(* COMMENTS
-Initial nodes: 2555
-END *)
 
 theorem subst0_confluence_neq:
  \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: 
@@ -309,70 +302,68 @@ t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda
 nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda 
 (t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda 
 (t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2)) in 
-False return (\lambda (_: False).(ex2 T (\lambda (t: T).(subst0 i2 u2 (lift 
-(S i) O v) t)) (\lambda (t: T).(subst0 i v (lift (S i) O u2) t)))) with []) 
-in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) (\lambda (v: 
-T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (subst0 
-i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: T).(\forall (i2: 
-nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: 
-T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda 
-(t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: T).(\lambda (i2: 
-nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) t2)).(\lambda (H3: (not (eq 
-nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t))) 
-(\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda (t3: T).(eq T t2 
-(THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex3_2 T T 
-(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: 
-T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead 
-k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T 
-(\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 
-u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: 
-T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) 
-t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq 
-T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k 
-x t) (\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) 
-t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) (ex2_ind T (\lambda (t3: 
-T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i v x t3)) (ex2 T (\lambda 
-(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead 
-k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda 
-(H8: (subst0 i v x x0)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k 
-u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t) 
-(subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2 
-H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead 
-k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3)))).(ex2_ind T (\lambda 
+False with []) in H5) t2 H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) 
+(\lambda (v: T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
+(H0: (subst0 i v u0 u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: 
+T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 
+T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 
+t)))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: 
+T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) 
+t2)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq 
+T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda 
 (t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t 
-t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: 
-T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0 
-x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x) 
-(\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4)) 
-(\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0 
-i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3)) 
-(THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k)) 
-t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq 
-T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 
-u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t 
-t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 
+t3))) (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 
 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3: 
+T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: 
 T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead k x0 
-x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: (subst0 (s k i2) u3 t 
-x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 T (\lambda (t4: 
+(\lambda (H4: (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: 
+T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 
+t))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 
+i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: 
+T).(\lambda (H5: (eq T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 
+x)).(eq_ind_r T (THead k x t) (\lambda (t3: T).(ex2 T (\lambda (t4: 
 T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) 
 (ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i 
-v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda 
-(t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: T).(\lambda (H8: 
-(subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 x)).(ex_intro2 T (\lambda 
+v x t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda 
+(t3: T).(subst0 i v (THead k x t) t3))) (\lambda (x0: T).(\lambda (H7: 
+(subst0 i2 u3 u2 x0)).(\lambda (H8: (subst0 i v x x0)).(ex_intro2 T (\lambda 
 (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead 
-k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x i2 H8 k t x1 H7) 
-(subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 H5)))))) H4)) 
-(subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: K).(\lambda 
-(v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H0: 
-(subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: T).(\forall (u2: 
-T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq nat (s k i) i2)) 
-\to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: T).(subst0 (s k 
-i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (u2: 
-T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3) 
+k x t) t3)) (THead k x0 t) (subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x 
+i H8 t k))))) (H1 x u3 i2 H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T 
+(\lambda (t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) 
+u3 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u0 t3))) (\lambda 
+(t3: T).(subst0 (s k i2) u3 t t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 
+(THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: 
+T).(\lambda (H5: (eq T t2 (THead k u0 x))).(\lambda (H6: (subst0 (s k i2) u3 
+t x)).(eq_ind_r T (THead k u0 x) (\lambda (t3: T).(ex2 T (\lambda (t4: 
+T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) 
+(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: 
+T).(subst0 i v (THead k u0 x) t3)) (THead k u2 x) (subst0_snd k u3 x t i2 H6 
+u2) (subst0_fst v u2 u0 i H0 x k)) t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T 
+(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: 
+T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s k i2) u3 t t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda 
+(t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 
+i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) 
+(ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: 
+T).(subst0 i v t2 t3))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T 
+t2 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: 
+(subst0 (s k i2) u3 t x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 
+T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 
+i v t3 t4)))) (ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: 
+T).(subst0 i v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) 
+t3)) (\lambda (t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: 
+T).(\lambda (H8: (subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 
+x)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda 
+(t3: T).(subst0 i v (THead k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x 
+i2 H8 k t x1 H7) (subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 
+H5)))))) H4)) (subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: 
+K).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: 
+nat).(\lambda (H0: (subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: 
+T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq 
+nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: 
+T).(subst0 (s k i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda 
+(u2: T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3) 
 t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq 
 T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda 
 (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3 
@@ -511,9 +502,6 @@ i2) u3 t3 x2)).(\lambda (_: (subst0 (s k i) v x1 x2)).(\lambda (H14: (eq nat
 (H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H12)))))))))) (H1 x0 u3 i2 
 H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2 t4 i2 
 H4)))))))))))))))))) i1 u1 t0 t1 H))))).
-(* COMMENTS
-Initial nodes: 5375
-END *)
 
 theorem subst0_confluence_eq:
  \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 
@@ -1368,9 +1356,6 @@ k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda
 k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2 
 i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5)) 
 (subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))).
-(* COMMENTS
-Initial nodes: 25595
-END *)
 
 theorem subst0_confluence_lift:
  \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 
@@ -1401,7 +1386,4 @@ t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i)
 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) 
 i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2))) 
 (subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))).
-(* COMMENTS
-Initial nodes: 703
-END *)
 

diff --git a/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma b/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma
index c8e8420bb49b18b157d8b16a13bc85683e0256d2..7fe9f6ef23c9c742dfd06968d508d21c8481d9ce 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/subst0/tlt.ma
@@ -14,11 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/subst0/defs.ma".
+include "basic_1/subst0/fwd.ma".
 
-include "Basic-1/lift/props.ma".
-
-include "Basic-1/lift/tlt.ma".
+include "basic_1/lift/tlt.ma".
 
 theorem subst0_weight_le:
  \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
@@ -27,200 +25,381 @@ nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
 d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t))))))))))
 \def
  \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda 
-(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda 
-(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1)))))))))) 
-(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: 
-((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda 
-(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift 
-(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O 
-v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2: 
-T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 
-u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+(H: (subst0 d u t z)).(let TMP_3 \def (\lambda (n: nat).(\lambda (t0: 
+T).(\lambda (t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall 
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt 
+(weight_map f (lift (S n) O t0)) (g n)) \to (let TMP_1 \def (weight_map f t2) 
+in (let TMP_2 \def (weight_map g t1) in (le TMP_1 TMP_2))))))))))) in (let 
+TMP_33 \def (\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H1: (lt (weight_map f (lift (S i) O v)) (g i))).(let 
+TMP_4 \def (S i) in (let TMP_5 \def (lift TMP_4 O v) in (let TMP_6 \def 
+(weight_map f TMP_5) in (let TMP_7 \def (TLRef i) in (let TMP_8 \def 
+(weight_map g TMP_7) in (let TMP_9 \def (S i) in (let TMP_10 \def (lift TMP_9 
+O v) in (let TMP_11 \def (weight_map f TMP_10) in (let TMP_12 \def (S TMP_11) 
+in (let TMP_13 \def (TLRef i) in (let TMP_14 \def (weight_map g TMP_13) in 
+(let TMP_15 \def (S TMP_14) in (let TMP_16 \def (S i) in (let TMP_17 \def 
+(lift TMP_16 O v) in (let TMP_18 \def (weight_map f TMP_17) in (let TMP_19 
+\def (S TMP_18) in (let TMP_20 \def (S TMP_19) in (let TMP_21 \def (TLRef i) 
+in (let TMP_22 \def (weight_map g TMP_21) in (let TMP_23 \def (S TMP_22) in 
+(let TMP_24 \def (S i) in (let TMP_25 \def (lift TMP_24 O v) in (let TMP_26 
+\def (weight_map f TMP_25) in (let TMP_27 \def (S TMP_26) in (let TMP_28 \def 
+(TLRef i) in (let TMP_29 \def (weight_map g TMP_28) in (let TMP_30 \def 
+(le_n_S TMP_27 TMP_29 H1) in (let TMP_31 \def (le_S TMP_20 TMP_23 TMP_30) in 
+(let TMP_32 \def (le_S_n TMP_12 TMP_15 TMP_31) in (le_S_n TMP_6 TMP_8 
+TMP_32)))))))))))))))))))))))))))))))))))) in (let TMP_146 \def (\lambda (v: 
+T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i 
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
 i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda 
-(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead 
-k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind 
+(t0: T).(\lambda (k: K).(let TMP_38 \def (\lambda (k0: K).(\forall (f: ((nat 
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g 
+m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (let TMP_34 \def 
+(THead k0 u2 t0) in (let TMP_35 \def (weight_map f TMP_34) in (let TMP_36 
+\def (THead k0 u1 t0) in (let TMP_37 \def (weight_map g TMP_36) in (le TMP_35 
+TMP_37)))))))))) in (let TMP_131 \def (\lambda (b: B).(let TMP_45 \def 
 (\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g 
-(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: 
-((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g 
-m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S 
-(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus 
-(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) 
-(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
-(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f 
-g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map 
-g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S 
-(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 
-H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt 
-(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) 
-(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) 
-t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
-t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd 
-g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda (f: 
+i) O v)) (g i)) \to (let TMP_39 \def (Bind b0) in (let TMP_40 \def (THead 
+TMP_39 u2 t0) in (let TMP_41 \def (weight_map f TMP_40) in (let TMP_42 \def 
+(Bind b0) in (let TMP_43 \def (THead TMP_42 u1 t0) in (let TMP_44 \def 
+(weight_map g TMP_43) in (le TMP_41 TMP_44)))))))))))) in (let TMP_86 \def 
+(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: 
+((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift 
+(S i) O v)) (g i))).(let TMP_46 \def (weight_map f u2) in (let TMP_47 \def 
+(weight_map f u2) in (let TMP_48 \def (S TMP_47) in (let TMP_49 \def (wadd f 
+TMP_48) in (let TMP_50 \def (weight_map TMP_49 t0) in (let TMP_51 \def (plus 
+TMP_46 TMP_50) in (let TMP_52 \def (weight_map g u1) in (let TMP_53 \def 
+(weight_map g u1) in (let TMP_54 \def (S TMP_53) in (let TMP_55 \def (wadd g 
+TMP_54) in (let TMP_56 \def (weight_map TMP_55 t0) in (let TMP_57 \def (plus 
+TMP_52 TMP_56) in (let TMP_58 \def (weight_map f u2) in (let TMP_59 \def 
+(weight_map g u1) in (let TMP_60 \def (weight_map f u2) in (let TMP_61 \def 
+(S TMP_60) in (let TMP_62 \def (wadd f TMP_61) in (let TMP_63 \def 
+(weight_map TMP_62 t0) in (let TMP_64 \def (weight_map g u1) in (let TMP_65 
+\def (S TMP_64) in (let TMP_66 \def (wadd g TMP_65) in (let TMP_67 \def 
+(weight_map TMP_66 t0) in (let TMP_68 \def (H1 f g H2 H3) in (let TMP_69 \def 
+(weight_map f u2) in (let TMP_70 \def (S TMP_69) in (let TMP_71 \def (wadd f 
+TMP_70) in (let TMP_72 \def (weight_map g u1) in (let TMP_73 \def (S TMP_72) 
+in (let TMP_74 \def (wadd g TMP_73) in (let TMP_83 \def (\lambda (n: 
+nat).(let TMP_75 \def (weight_map f u2) in (let TMP_76 \def (S TMP_75) in 
+(let TMP_77 \def (weight_map g u1) in (let TMP_78 \def (S TMP_77) in (let 
+TMP_79 \def (weight_map f u2) in (let TMP_80 \def (weight_map g u1) in (let 
+TMP_81 \def (H1 f g H2 H3) in (let TMP_82 \def (le_n_S TMP_79 TMP_80 TMP_81) 
+in (wadd_le f g H2 TMP_76 TMP_78 TMP_82 n)))))))))) in (let TMP_84 \def 
+(weight_le t0 TMP_71 TMP_74 TMP_83) in (let TMP_85 \def (le_plus_plus TMP_58 
+TMP_59 TMP_63 TMP_67 TMP_68 TMP_84) in (le_n_S TMP_51 TMP_57 
+TMP_85))))))))))))))))))))))))))))))))))))) in (let TMP_108 \def (\lambda (f: 
 ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
 nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus 
-(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2) 
-(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f 
-g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g 
-H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 
-m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g 
-u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1) 
-(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g 
-H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: 
-T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 
-t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+i))).(let TMP_87 \def (weight_map f u2) in (let TMP_88 \def (wadd f O) in 
+(let TMP_89 \def (weight_map TMP_88 t0) in (let TMP_90 \def (plus TMP_87 
+TMP_89) in (let TMP_91 \def (weight_map g u1) in (let TMP_92 \def (wadd g O) 
+in (let TMP_93 \def (weight_map TMP_92 t0) in (let TMP_94 \def (plus TMP_91 
+TMP_93) in (let TMP_95 \def (weight_map f u2) in (let TMP_96 \def (weight_map 
+g u1) in (let TMP_97 \def (wadd f O) in (let TMP_98 \def (weight_map TMP_97 
+t0) in (let TMP_99 \def (wadd g O) in (let TMP_100 \def (weight_map TMP_99 
+t0) in (let TMP_101 \def (H1 f g H2 H3) in (let TMP_102 \def (wadd f O) in 
+(let TMP_103 \def (wadd g O) in (let TMP_105 \def (\lambda (n: nat).(let 
+TMP_104 \def (le_O_n O) in (wadd_le f g H2 O O TMP_104 n))) in (let TMP_106 
+\def (weight_le t0 TMP_102 TMP_103 TMP_105) in (let TMP_107 \def 
+(le_plus_plus TMP_95 TMP_96 TMP_98 TMP_100 TMP_101 TMP_106) in (le_n_S TMP_90 
+TMP_94 TMP_107))))))))))))))))))))))))) in (let TMP_130 \def (\lambda (f: 
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(let TMP_109 \def (weight_map f u2) in (let TMP_110 \def (wadd f O) in 
+(let TMP_111 \def (weight_map TMP_110 t0) in (let TMP_112 \def (plus TMP_109 
+TMP_111) in (let TMP_113 \def (weight_map g u1) in (let TMP_114 \def (wadd g 
+O) in (let TMP_115 \def (weight_map TMP_114 t0) in (let TMP_116 \def (plus 
+TMP_113 TMP_115) in (let TMP_117 \def (weight_map f u2) in (let TMP_118 \def 
+(weight_map g u1) in (let TMP_119 \def (wadd f O) in (let TMP_120 \def 
+(weight_map TMP_119 t0) in (let TMP_121 \def (wadd g O) in (let TMP_122 \def 
+(weight_map TMP_121 t0) in (let TMP_123 \def (H1 f g H2 H3) in (let TMP_124 
+\def (wadd f O) in (let TMP_125 \def (wadd g O) in (let TMP_127 \def (\lambda 
+(n: nat).(let TMP_126 \def (le_O_n O) in (wadd_le f g H2 O O TMP_126 n))) in 
+(let TMP_128 \def (weight_le t0 TMP_124 TMP_125 TMP_127) in (let TMP_129 \def 
+(le_plus_plus TMP_117 TMP_118 TMP_120 TMP_122 TMP_123 TMP_128) in (le_n_S 
+TMP_112 TMP_116 TMP_129))))))))))))))))))))))))) in (B_ind TMP_45 TMP_86 
+TMP_108 TMP_130 b)))))) in (let TMP_145 \def (\lambda (_: F).(\lambda (f0: 
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
+nat).(le (f0 m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) 
+(g i))).(let TMP_132 \def (weight_map f0 u2) in (let TMP_133 \def (weight_map 
+f0 t0) in (let TMP_134 \def (plus TMP_132 TMP_133) in (let TMP_135 \def 
+(weight_map g u1) in (let TMP_136 \def (weight_map g t0) in (let TMP_137 \def 
+(plus TMP_135 TMP_136) in (let TMP_138 \def (weight_map f0 u2) in (let 
+TMP_139 \def (weight_map g u1) in (let TMP_140 \def (weight_map f0 t0) in 
+(let TMP_141 \def (weight_map g t0) in (let TMP_142 \def (H1 f0 g H2 H3) in 
+(let TMP_143 \def (weight_le t0 f0 g H2) in (let TMP_144 \def (le_plus_plus 
+TMP_138 TMP_139 TMP_140 TMP_141 TMP_142 TMP_143) in (le_n_S TMP_134 TMP_137 
+TMP_144))))))))))))))))))) in (K_ind TMP_38 TMP_131 TMP_145 k)))))))))))) in 
+(let TMP_302 \def (\lambda (k: K).(let TMP_151 \def (\lambda (k0: K).(\forall 
+(v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) 
+v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
 (s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f t2) (weight_map g 
 t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: 
 ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
-f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead k0 u0 t2)) 
-(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda 
-(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: 
-nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) 
-\to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: 
+f (lift (S i) O v)) (g i)) \to (let TMP_147 \def (THead k0 u0 t2) in (let 
+TMP_148 \def (weight_map f TMP_147) in (let TMP_149 \def (THead k0 u0 t1) in 
+(let TMP_150 \def (weight_map g TMP_149) in (le TMP_148 
+TMP_150))))))))))))))))) in (let TMP_287 \def (\lambda (b: B).(let TMP_158 
+\def (\lambda (b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: 
+T).(\forall (i: nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind 
+b0) i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: 
 T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
 nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to 
-(le (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 
-t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 
-(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: 
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le 
+(let TMP_152 \def (Bind b0) in (let TMP_153 \def (THead TMP_152 u0 t2) in 
+(let TMP_154 \def (weight_map f TMP_153) in (let TMP_155 \def (Bind b0) in 
+(let TMP_156 \def (THead TMP_155 u0 t1) in (let TMP_157 \def (weight_map g 
+TMP_156) in (le TMP_154 TMP_157))))))))))))))))))) in (let TMP_216 \def 
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f 
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(let 
+TMP_159 \def (weight_map f u0) in (let TMP_160 \def (weight_map f u0) in (let 
+TMP_161 \def (S TMP_160) in (let TMP_162 \def (wadd f TMP_161) in (let 
+TMP_163 \def (weight_map TMP_162 t2) in (let TMP_164 \def (plus TMP_159 
+TMP_163) in (let TMP_165 \def (weight_map g u0) in (let TMP_166 \def 
+(weight_map g u0) in (let TMP_167 \def (S TMP_166) in (let TMP_168 \def (wadd 
+g TMP_167) in (let TMP_169 \def (weight_map TMP_168 t1) in (let TMP_170 \def 
+(plus TMP_165 TMP_169) in (let TMP_171 \def (weight_map f u0) in (let TMP_172 
+\def (weight_map g u0) in (let TMP_173 \def (weight_map f u0) in (let TMP_174 
+\def (S TMP_173) in (let TMP_175 \def (wadd f TMP_174) in (let TMP_176 \def 
+(weight_map TMP_175 t2) in (let TMP_177 \def (weight_map g u0) in (let 
+TMP_178 \def (S TMP_177) in (let TMP_179 \def (wadd g TMP_178) in (let 
+TMP_180 \def (weight_map TMP_179 t1) in (let TMP_181 \def (weight_le u0 f g 
+H2) in (let TMP_182 \def (weight_map f u0) in (let TMP_183 \def (S TMP_182) 
+in (let TMP_184 \def (wadd f TMP_183) in (let TMP_185 \def (weight_map g u0) 
+in (let TMP_186 \def (S TMP_185) in (let TMP_187 \def (wadd g TMP_186) in 
+(let TMP_196 \def (\lambda (m: nat).(let TMP_188 \def (weight_map f u0) in 
+(let TMP_189 \def (S TMP_188) in (let TMP_190 \def (weight_map g u0) in (let 
+TMP_191 \def (S TMP_190) in (let TMP_192 \def (weight_map f u0) in (let 
+TMP_193 \def (weight_map g u0) in (let TMP_194 \def (weight_le u0 f g H2) in 
+(let TMP_195 \def (le_n_S TMP_192 TMP_193 TMP_194) in (wadd_le f g H2 TMP_189 
+TMP_191 TMP_195 m)))))))))) in (let TMP_197 \def (S i) in (let TMP_198 \def 
+(lift TMP_197 O v) in (let TMP_199 \def (weight_map f TMP_198) in (let 
+TMP_201 \def (\lambda (n: nat).(let TMP_200 \def (g i) in (lt n TMP_200))) in 
+(let TMP_202 \def (weight_map f u0) in (let TMP_203 \def (S TMP_202) in (let 
+TMP_204 \def (wadd f TMP_203) in (let TMP_205 \def (S i) in (let TMP_206 \def 
+(S TMP_205) in (let TMP_207 \def (lift TMP_206 O v) in (let TMP_208 \def 
+(weight_map TMP_204 TMP_207) in (let TMP_209 \def (weight_map f u0) in (let 
+TMP_210 \def (S TMP_209) in (let TMP_211 \def (S i) in (let TMP_212 \def 
+(lift_weight_add_O TMP_210 v TMP_211 f) in (let TMP_213 \def (eq_ind nat 
+TMP_199 TMP_201 H3 TMP_208 TMP_212) in (let TMP_214 \def (H1 TMP_184 TMP_187 
+TMP_196 TMP_213) in (let TMP_215 \def (le_plus_plus TMP_171 TMP_172 TMP_176 
+TMP_180 TMP_181 TMP_214) in (le_n_S TMP_164 TMP_170 
+TMP_215)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (let 
+TMP_251 \def (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: 
+nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat 
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g 
+m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le 
 (weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: 
 ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
 nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f 
-u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) 
-t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S 
-(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) 
-(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S 
-(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) 
-(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le 
-u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
-nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S 
-i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) 
+i))).(let TMP_217 \def (weight_map f u0) in (let TMP_218 \def (wadd f O) in 
+(let TMP_219 \def (weight_map TMP_218 t2) in (let TMP_220 \def (plus TMP_217 
+TMP_219) in (let TMP_221 \def (weight_map g u0) in (let TMP_222 \def (wadd g 
+O) in (let TMP_223 \def (weight_map TMP_222 t1) in (let TMP_224 \def (plus 
+TMP_221 TMP_223) in (let TMP_225 \def (weight_map f u0) in (let TMP_226 \def 
+(weight_map g u0) in (let TMP_227 \def (wadd f O) in (let TMP_228 \def 
+(weight_map TMP_227 t2) in (let TMP_229 \def (wadd g O) in (let TMP_230 \def 
+(weight_map TMP_229 t1) in (let TMP_231 \def (weight_le u0 f g H2) in (let 
+TMP_232 \def (wadd f O) in (let TMP_233 \def (wadd g O) in (let TMP_235 \def 
+(\lambda (m: nat).(let TMP_234 \def (le_O_n O) in (wadd_le f g H2 O O TMP_234 
+m))) in (let TMP_236 \def (S i) in (let TMP_237 \def (lift TMP_236 O v) in 
+(let TMP_238 \def (weight_map f TMP_237) in (let TMP_240 \def (\lambda (n: 
+nat).(let TMP_239 \def (g i) in (lt n TMP_239))) in (let TMP_241 \def (wadd f 
+O) in (let TMP_242 \def (S i) in (let TMP_243 \def (S TMP_242) in (let 
+TMP_244 \def (lift TMP_243 O v) in (let TMP_245 \def (weight_map TMP_241 
+TMP_244) in (let TMP_246 \def (S i) in (let TMP_247 \def (lift_weight_add_O O 
+v TMP_246 f) in (let TMP_248 \def (eq_ind nat TMP_238 TMP_240 H3 TMP_245 
+TMP_247) in (let TMP_249 \def (H1 TMP_232 TMP_233 TMP_235 TMP_248) in (let 
+TMP_250 \def (le_plus_plus TMP_225 TMP_226 TMP_228 TMP_230 TMP_231 TMP_249) 
+in (le_n_S TMP_220 TMP_224 
+TMP_250)))))))))))))))))))))))))))))))))))))))))))) in (let TMP_286 \def 
 (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
 (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to 
 nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
 \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f 
 t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to 
 nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
-m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus 
-(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0) 
-(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) 
-(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f 
-g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda 
-(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) 
-(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 
-t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(let 
+TMP_252 \def (weight_map f u0) in (let TMP_253 \def (wadd f O) in (let 
+TMP_254 \def (weight_map TMP_253 t2) in (let TMP_255 \def (plus TMP_252 
+TMP_254) in (let TMP_256 \def (weight_map g u0) in (let TMP_257 \def (wadd g 
+O) in (let TMP_258 \def (weight_map TMP_257 t1) in (let TMP_259 \def (plus 
+TMP_256 TMP_258) in (let TMP_260 \def (weight_map f u0) in (let TMP_261 \def 
+(weight_map g u0) in (let TMP_262 \def (wadd f O) in (let TMP_263 \def 
+(weight_map TMP_262 t2) in (let TMP_264 \def (wadd g O) in (let TMP_265 \def 
+(weight_map TMP_264 t1) in (let TMP_266 \def (weight_le u0 f g H2) in (let 
+TMP_267 \def (wadd f O) in (let TMP_268 \def (wadd g O) in (let TMP_270 \def 
+(\lambda (m: nat).(let TMP_269 \def (le_O_n O) in (wadd_le f g H2 O O TMP_269 
+m))) in (let TMP_271 \def (S i) in (let TMP_272 \def (lift TMP_271 O v) in 
+(let TMP_273 \def (weight_map f TMP_272) in (let TMP_275 \def (\lambda (n: 
+nat).(let TMP_274 \def (g i) in (lt n TMP_274))) in (let TMP_276 \def (wadd f 
+O) in (let TMP_277 \def (S i) in (let TMP_278 \def (S TMP_277) in (let 
+TMP_279 \def (lift TMP_278 O v) in (let TMP_280 \def (weight_map TMP_276 
+TMP_279) in (let TMP_281 \def (S i) in (let TMP_282 \def (lift_weight_add_O O 
+v TMP_281 f) in (let TMP_283 \def (eq_ind nat TMP_273 TMP_275 H3 TMP_280 
+TMP_282) in (let TMP_284 \def (H1 TMP_267 TMP_268 TMP_270 TMP_283) in (let 
+TMP_285 \def (le_plus_plus TMP_260 TMP_261 TMP_263 TMP_265 TMP_266 TMP_284) 
+in (le_n_S TMP_255 TMP_259 
+TMP_285)))))))))))))))))))))))))))))))))))))))))))) in (B_ind TMP_158 TMP_216 
+TMP_251 TMP_286 b)))))) in (let TMP_301 \def (\lambda (_: F).(\lambda (v: 
+T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i 
+v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat 
+\to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 
+(lift (S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g 
+t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
+(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(let TMP_288 \def 
+(weight_map f0 u0) in (let TMP_289 \def (weight_map f0 t2) in (let TMP_290 
+\def (plus TMP_288 TMP_289) in (let TMP_291 \def (weight_map g u0) in (let 
+TMP_292 \def (weight_map g t1) in (let TMP_293 \def (plus TMP_291 TMP_292) in 
+(let TMP_294 \def (weight_map f0 u0) in (let TMP_295 \def (weight_map g u0) 
+in (let TMP_296 \def (weight_map f0 t2) in (let TMP_297 \def (weight_map g 
+t1) in (let TMP_298 \def (weight_le u0 f0 g H2) in (let TMP_299 \def (H1 f0 g 
+H2 H3) in (let TMP_300 \def (le_plus_plus TMP_294 TMP_295 TMP_296 TMP_297 
+TMP_298 TMP_299) in (le_n_S TMP_290 TMP_293 TMP_300)))))))))))))))))))))))))) 
+in (K_ind TMP_151 TMP_287 TMP_301 k))))) in (let TMP_458 \def (\lambda (v: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (_: (subst0 i 
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g 
-t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
-\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: 
-(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) 
-(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) 
-t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O) 
-t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g 
-O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map 
-f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O) 
-(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) b)) 
-(\lambda (_: F).(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 
-(i: nat).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat 
-\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g 
-m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0 
-t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 
-m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g 
-u0) (weight_map g t1)) (le_plus_plus (weight_map f0 u0) (weight_map g u0) 
-(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 
-H3))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall 
-(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f 
-m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le 
-(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda 
-(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to 
-(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s 
-k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f: 
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
-(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map 
-f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: 
-B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s 
-(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat 
-\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f 
-(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f 
+i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda 
+(k: K).(let TMP_307 \def (\lambda (k0: K).(\forall (t1: T).(\forall (t2: 
+T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall 
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt 
+(weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (le (weight_map f 
 t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: 
 ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
-f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2)) 
-(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: 
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le 
-(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f 
-m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f 
-u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) 
-t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
-(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f 
-g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1))) 
-(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1)) 
-(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat 
-(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 
-(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) 
-(lift_weight_add_O (S (weight_map f u2)) v (S i) f))))))))))))) (\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: 
+f (lift (S i) O v)) (g i)) \to (let TMP_303 \def (THead k0 u2 t2) in (let 
+TMP_304 \def (weight_map f TMP_303) in (let TMP_305 \def (THead k0 u1 t1) in 
+(let TMP_306 \def (weight_map g TMP_305) in (le TMP_304 TMP_306)))))))))))))) 
+in (let TMP_443 \def (\lambda (b: B).(let TMP_314 \def (\lambda (b0: 
+B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v t1 t2) \to 
+(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s (Bind b0) i)) O 
+v)) (g (s (Bind b0) i))) \to (le (weight_map f t2) (weight_map g t1))))))) 
+\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
+(m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) 
+\to (let TMP_308 \def (Bind b0) in (let TMP_309 \def (THead TMP_308 u2 t2) in 
+(let TMP_310 \def (weight_map f TMP_309) in (let TMP_311 \def (Bind b0) in 
+(let TMP_312 \def (THead TMP_311 u1 t1) in (let TMP_313 \def (weight_map g 
+TMP_312) in (le TMP_310 TMP_313)))))))))))))))) in (let TMP_372 \def (\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: 
 ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
 nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S 
 i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat 
 \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le 
 (f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus 
-(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2) 
-(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f 
-g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O 
-(le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
-nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) 
-(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f 
-t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
-\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: 
-(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2) 
-(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) 
-t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
-t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) 
-(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f 
-(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) 
-(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b)) 
-(\lambda (_: F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 
-t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
-(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g 
-t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5: 
-(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2) 
-(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus 
-(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 
-f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))).
-(* COMMENTS
-Initial nodes: 4101
-END *)
+i))).(let TMP_315 \def (weight_map f u2) in (let TMP_316 \def (weight_map f 
+u2) in (let TMP_317 \def (S TMP_316) in (let TMP_318 \def (wadd f TMP_317) in 
+(let TMP_319 \def (weight_map TMP_318 t2) in (let TMP_320 \def (plus TMP_315 
+TMP_319) in (let TMP_321 \def (weight_map g u1) in (let TMP_322 \def 
+(weight_map g u1) in (let TMP_323 \def (S TMP_322) in (let TMP_324 \def (wadd 
+g TMP_323) in (let TMP_325 \def (weight_map TMP_324 t1) in (let TMP_326 \def 
+(plus TMP_321 TMP_325) in (let TMP_327 \def (weight_map f u2) in (let TMP_328 
+\def (weight_map g u1) in (let TMP_329 \def (weight_map f u2) in (let TMP_330 
+\def (S TMP_329) in (let TMP_331 \def (wadd f TMP_330) in (let TMP_332 \def 
+(weight_map TMP_331 t2) in (let TMP_333 \def (weight_map g u1) in (let 
+TMP_334 \def (S TMP_333) in (let TMP_335 \def (wadd g TMP_334) in (let 
+TMP_336 \def (weight_map TMP_335 t1) in (let TMP_337 \def (H1 f g H4 H5) in 
+(let TMP_338 \def (weight_map f u2) in (let TMP_339 \def (S TMP_338) in (let 
+TMP_340 \def (wadd f TMP_339) in (let TMP_341 \def (weight_map g u1) in (let 
+TMP_342 \def (S TMP_341) in (let TMP_343 \def (wadd g TMP_342) in (let 
+TMP_352 \def (\lambda (m: nat).(let TMP_344 \def (weight_map f u2) in (let 
+TMP_345 \def (S TMP_344) in (let TMP_346 \def (weight_map g u1) in (let 
+TMP_347 \def (S TMP_346) in (let TMP_348 \def (weight_map f u2) in (let 
+TMP_349 \def (weight_map g u1) in (let TMP_350 \def (H1 f g H4 H5) in (let 
+TMP_351 \def (le_n_S TMP_348 TMP_349 TMP_350) in (wadd_le f g H4 TMP_345 
+TMP_347 TMP_351 m)))))))))) in (let TMP_353 \def (S i) in (let TMP_354 \def 
+(lift TMP_353 O v) in (let TMP_355 \def (weight_map f TMP_354) in (let 
+TMP_357 \def (\lambda (n: nat).(let TMP_356 \def (g i) in (lt n TMP_356))) in 
+(let TMP_358 \def (weight_map f u2) in (let TMP_359 \def (S TMP_358) in (let 
+TMP_360 \def (wadd f TMP_359) in (let TMP_361 \def (S i) in (let TMP_362 \def 
+(S TMP_361) in (let TMP_363 \def (lift TMP_362 O v) in (let TMP_364 \def 
+(weight_map TMP_360 TMP_363) in (let TMP_365 \def (weight_map f u2) in (let 
+TMP_366 \def (S TMP_365) in (let TMP_367 \def (S i) in (let TMP_368 \def 
+(lift_weight_add_O TMP_366 v TMP_367 f) in (let TMP_369 \def (eq_ind nat 
+TMP_355 TMP_357 H5 TMP_364 TMP_368) in (let TMP_370 \def (H3 TMP_340 TMP_343 
+TMP_352 TMP_369) in (let TMP_371 \def (le_plus_plus TMP_327 TMP_328 TMP_332 
+TMP_336 TMP_337 TMP_370) in (le_n_S TMP_320 TMP_326 
+TMP_371))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (let 
+TMP_407 \def (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v 
+t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g 
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt 
+(weight_map f (lift (S i) O v)) (g i))).(let TMP_373 \def (weight_map f u2) 
+in (let TMP_374 \def (wadd f O) in (let TMP_375 \def (weight_map TMP_374 t2) 
+in (let TMP_376 \def (plus TMP_373 TMP_375) in (let TMP_377 \def (weight_map 
+g u1) in (let TMP_378 \def (wadd g O) in (let TMP_379 \def (weight_map 
+TMP_378 t1) in (let TMP_380 \def (plus TMP_377 TMP_379) in (let TMP_381 \def 
+(weight_map f u2) in (let TMP_382 \def (weight_map g u1) in (let TMP_383 \def 
+(wadd f O) in (let TMP_384 \def (weight_map TMP_383 t2) in (let TMP_385 \def 
+(wadd g O) in (let TMP_386 \def (weight_map TMP_385 t1) in (let TMP_387 \def 
+(H1 f g H4 H5) in (let TMP_388 \def (wadd f O) in (let TMP_389 \def (wadd g 
+O) in (let TMP_391 \def (\lambda (m: nat).(let TMP_390 \def (le_O_n O) in 
+(wadd_le f g H4 O O TMP_390 m))) in (let TMP_392 \def (S i) in (let TMP_393 
+\def (lift TMP_392 O v) in (let TMP_394 \def (weight_map f TMP_393) in (let 
+TMP_396 \def (\lambda (n: nat).(let TMP_395 \def (g i) in (lt n TMP_395))) in 
+(let TMP_397 \def (wadd f O) in (let TMP_398 \def (S i) in (let TMP_399 \def 
+(S TMP_398) in (let TMP_400 \def (lift TMP_399 O v) in (let TMP_401 \def 
+(weight_map TMP_397 TMP_400) in (let TMP_402 \def (S i) in (let TMP_403 \def 
+(lift_weight_add_O O v TMP_402 f) in (let TMP_404 \def (eq_ind nat TMP_394 
+TMP_396 H5 TMP_401 TMP_403) in (let TMP_405 \def (H3 TMP_388 TMP_389 TMP_391 
+TMP_404) in (let TMP_406 \def (le_plus_plus TMP_381 TMP_382 TMP_384 TMP_386 
+TMP_387 TMP_405) in (le_n_S TMP_376 TMP_380 
+TMP_406))))))))))))))))))))))))))))))))))))))))) in (let TMP_442 \def 
+(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 
+t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g 
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt 
+(weight_map f (lift (S i) O v)) (g i))).(let TMP_408 \def (weight_map f u2) 
+in (let TMP_409 \def (wadd f O) in (let TMP_410 \def (weight_map TMP_409 t2) 
+in (let TMP_411 \def (plus TMP_408 TMP_410) in (let TMP_412 \def (weight_map 
+g u1) in (let TMP_413 \def (wadd g O) in (let TMP_414 \def (weight_map 
+TMP_413 t1) in (let TMP_415 \def (plus TMP_412 TMP_414) in (let TMP_416 \def 
+(weight_map f u2) in (let TMP_417 \def (weight_map g u1) in (let TMP_418 \def 
+(wadd f O) in (let TMP_419 \def (weight_map TMP_418 t2) in (let TMP_420 \def 
+(wadd g O) in (let TMP_421 \def (weight_map TMP_420 t1) in (let TMP_422 \def 
+(H1 f g H4 H5) in (let TMP_423 \def (wadd f O) in (let TMP_424 \def (wadd g 
+O) in (let TMP_426 \def (\lambda (m: nat).(let TMP_425 \def (le_O_n O) in 
+(wadd_le f g H4 O O TMP_425 m))) in (let TMP_427 \def (S i) in (let TMP_428 
+\def (lift TMP_427 O v) in (let TMP_429 \def (weight_map f TMP_428) in (let 
+TMP_431 \def (\lambda (n: nat).(let TMP_430 \def (g i) in (lt n TMP_430))) in 
+(let TMP_432 \def (wadd f O) in (let TMP_433 \def (S i) in (let TMP_434 \def 
+(S TMP_433) in (let TMP_435 \def (lift TMP_434 O v) in (let TMP_436 \def 
+(weight_map TMP_432 TMP_435) in (let TMP_437 \def (S i) in (let TMP_438 \def 
+(lift_weight_add_O O v TMP_437 f) in (let TMP_439 \def (eq_ind nat TMP_429 
+TMP_431 H5 TMP_436 TMP_438) in (let TMP_440 \def (H3 TMP_423 TMP_424 TMP_426 
+TMP_439) in (let TMP_441 \def (le_plus_plus TMP_416 TMP_417 TMP_419 TMP_421 
+TMP_422 TMP_440) in (le_n_S TMP_411 TMP_415 
+TMP_441))))))))))))))))))))))))))))))))))))))))) in (B_ind TMP_314 TMP_372 
+TMP_407 TMP_442 b)))))) in (let TMP_457 \def (\lambda (_: F).(\lambda (t1: 
+T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall 
+(f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le 
+(f0 m) (g m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le 
+(weight_map f0 t2) (weight_map g t1)))))))).(\lambda (f0: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 
+m) (g m))))).(\lambda (H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(let 
+TMP_444 \def (weight_map f0 u2) in (let TMP_445 \def (weight_map f0 t2) in 
+(let TMP_446 \def (plus TMP_444 TMP_445) in (let TMP_447 \def (weight_map g 
+u1) in (let TMP_448 \def (weight_map g t1) in (let TMP_449 \def (plus TMP_447 
+TMP_448) in (let TMP_450 \def (weight_map f0 u2) in (let TMP_451 \def 
+(weight_map g u1) in (let TMP_452 \def (weight_map f0 t2) in (let TMP_453 
+\def (weight_map g t1) in (let TMP_454 \def (H1 f0 g H4 H5) in (let TMP_455 
+\def (H3 f0 g H4 H5) in (let TMP_456 \def (le_plus_plus TMP_450 TMP_451 
+TMP_452 TMP_453 TMP_454 TMP_455) in (le_n_S TMP_446 TMP_449 
+TMP_456))))))))))))))))))))))) in (K_ind TMP_307 TMP_443 TMP_457 k))))))))))) 
+in (subst0_ind TMP_3 TMP_33 TMP_146 TMP_302 TMP_458 d u t z H)))))))))).
 
 theorem subst0_weight_lt:
  \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d 
@@ -229,240 +408,447 @@ nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
 d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t))))))))))
 \def
  \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda 
-(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda 
-(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1)))))))))) 
-(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g: 
-((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda 
-(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v: 
-T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i 
-v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda 
-(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead 
-k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind 
-(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g 
-(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: 
-((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g 
-m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S 
-(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus 
-(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0)) 
-(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S 
-(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f 
-g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map 
-g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S 
-(weight_map g u1)) (lt_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2 
-H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
-nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt 
-(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) 
-(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O) 
-t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f 
-O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) 
-(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd 
-f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f: 
+(H: (subst0 d u t z)).(let TMP_3 \def (\lambda (n: nat).(\lambda (t0: 
+T).(\lambda (t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall 
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt 
+(weight_map f (lift (S n) O t0)) (g n)) \to (let TMP_1 \def (weight_map f t2) 
+in (let TMP_2 \def (weight_map g t1) in (lt TMP_1 TMP_2))))))))))) in (let 
+TMP_4 \def (\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H1: (lt (weight_map f (lift (S i) O v)) (g 
+i))).H1)))))) in (let TMP_129 \def (\lambda (v: T).(\lambda (u2: T).(\lambda 
+(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: 
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to 
+(lt (weight_map f u2) (weight_map g u1)))))))).(\lambda (t0: T).(\lambda (k: 
+K).(let TMP_9 \def (\lambda (k0: K).(\forall (f: ((nat \to nat))).(\forall 
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt 
+(weight_map f (lift (S i) O v)) (g i)) \to (let TMP_5 \def (THead k0 u2 t0) 
+in (let TMP_6 \def (weight_map f TMP_5) in (let TMP_7 \def (THead k0 u1 t0) 
+in (let TMP_8 \def (weight_map g TMP_7) in (lt TMP_6 TMP_8)))))))))) in (let 
+TMP_114 \def (\lambda (b: B).(let TMP_16 \def (\lambda (b0: B).(\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (let TMP_10 \def 
+(Bind b0) in (let TMP_11 \def (THead TMP_10 u2 t0) in (let TMP_12 \def 
+(weight_map f TMP_11) in (let TMP_13 \def (Bind b0) in (let TMP_14 \def 
+(THead TMP_13 u1 t0) in (let TMP_15 \def (weight_map g TMP_14) in (lt TMP_12 
+TMP_15)))))))))))) in (let TMP_57 \def (\lambda (f: ((nat \to nat))).(\lambda 
+(g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g 
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(let TMP_17 
+\def (weight_map f u2) in (let TMP_18 \def (weight_map f u2) in (let TMP_19 
+\def (S TMP_18) in (let TMP_20 \def (wadd f TMP_19) in (let TMP_21 \def 
+(weight_map TMP_20 t0) in (let TMP_22 \def (plus TMP_17 TMP_21) in (let 
+TMP_23 \def (weight_map g u1) in (let TMP_24 \def (weight_map g u1) in (let 
+TMP_25 \def (S TMP_24) in (let TMP_26 \def (wadd g TMP_25) in (let TMP_27 
+\def (weight_map TMP_26 t0) in (let TMP_28 \def (plus TMP_23 TMP_27) in (let 
+TMP_29 \def (weight_map f u2) in (let TMP_30 \def (weight_map g u1) in (let 
+TMP_31 \def (weight_map f u2) in (let TMP_32 \def (S TMP_31) in (let TMP_33 
+\def (wadd f TMP_32) in (let TMP_34 \def (weight_map TMP_33 t0) in (let 
+TMP_35 \def (weight_map g u1) in (let TMP_36 \def (S TMP_35) in (let TMP_37 
+\def (wadd g TMP_36) in (let TMP_38 \def (weight_map TMP_37 t0) in (let 
+TMP_39 \def (H1 f g H2 H3) in (let TMP_40 \def (weight_map f u2) in (let 
+TMP_41 \def (S TMP_40) in (let TMP_42 \def (wadd f TMP_41) in (let TMP_43 
+\def (weight_map g u1) in (let TMP_44 \def (S TMP_43) in (let TMP_45 \def 
+(wadd g TMP_44) in (let TMP_54 \def (\lambda (n: nat).(let TMP_46 \def 
+(weight_map f u2) in (let TMP_47 \def (S TMP_46) in (let TMP_48 \def 
+(weight_map g u1) in (let TMP_49 \def (S TMP_48) in (let TMP_50 \def 
+(weight_map f u2) in (let TMP_51 \def (weight_map g u1) in (let TMP_52 \def 
+(H1 f g H2 H3) in (let TMP_53 \def (lt_n_S TMP_50 TMP_51 TMP_52) in (wadd_lt 
+f g H2 TMP_47 TMP_49 TMP_53 n)))))))))) in (let TMP_55 \def (weight_le t0 
+TMP_42 TMP_45 TMP_54) in (let TMP_56 \def (lt_le_plus_plus TMP_29 TMP_30 
+TMP_34 TMP_38 TMP_39 TMP_55) in (lt_n_S TMP_22 TMP_28 
+TMP_56))))))))))))))))))))))))))))))))))))) in (let TMP_85 \def (\lambda (f: 
 ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
 nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus 
-(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f 
-u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) 
-(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n 
-(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O 
-O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to 
+i))).(let TMP_58 \def (weight_map f u2) in (let TMP_59 \def (wadd f O) in 
+(let TMP_60 \def (weight_map TMP_59 t0) in (let TMP_61 \def (plus TMP_58 
+TMP_60) in (let TMP_62 \def (weight_map g u1) in (let TMP_63 \def (wadd g O) 
+in (let TMP_64 \def (weight_map TMP_63 t0) in (let TMP_65 \def (plus TMP_62 
+TMP_64) in (let TMP_66 \def (weight_map f u2) in (let TMP_67 \def (weight_map 
+g u1) in (let TMP_68 \def (wadd f O) in (let TMP_69 \def (weight_map TMP_68 
+t0) in (let TMP_70 \def (wadd g O) in (let TMP_71 \def (weight_map TMP_70 t0) 
+in (let TMP_72 \def (H1 f g H2 H3) in (let TMP_73 \def (wadd f O) in (let 
+TMP_74 \def (wadd g O) in (let TMP_82 \def (\lambda (n: nat).(let TMP_75 \def 
+(wadd f O n) in (let TMP_76 \def (wadd g O n) in (let TMP_77 \def (wadd f O 
+n) in (let TMP_78 \def (wadd g O n) in (let TMP_79 \def (le_O_n O) in (let 
+TMP_80 \def (wadd_le f g H2 O O TMP_79 n) in (let TMP_81 \def (le_n_S TMP_77 
+TMP_78 TMP_80) in (le_S_n TMP_75 TMP_76 TMP_81))))))))) in (let TMP_83 \def 
+(weight_le t0 TMP_73 TMP_74 TMP_82) in (let TMP_84 \def (lt_le_plus_plus 
+TMP_66 TMP_67 TMP_69 TMP_71 TMP_72 TMP_83) in (lt_n_S TMP_61 TMP_65 
+TMP_84))))))))))))))))))))))))) in (let TMP_113 \def (\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(let 
+TMP_86 \def (weight_map f u2) in (let TMP_87 \def (wadd f O) in (let TMP_88 
+\def (weight_map TMP_87 t0) in (let TMP_89 \def (plus TMP_86 TMP_88) in (let 
+TMP_90 \def (weight_map g u1) in (let TMP_91 \def (wadd g O) in (let TMP_92 
+\def (weight_map TMP_91 t0) in (let TMP_93 \def (plus TMP_90 TMP_92) in (let 
+TMP_94 \def (weight_map f u2) in (let TMP_95 \def (weight_map g u1) in (let 
+TMP_96 \def (wadd f O) in (let TMP_97 \def (weight_map TMP_96 t0) in (let 
+TMP_98 \def (wadd g O) in (let TMP_99 \def (weight_map TMP_98 t0) in (let 
+TMP_100 \def (H1 f g H2 H3) in (let TMP_101 \def (wadd f O) in (let TMP_102 
+\def (wadd g O) in (let TMP_110 \def (\lambda (n: nat).(let TMP_103 \def 
+(wadd f O n) in (let TMP_104 \def (wadd g O n) in (let TMP_105 \def (wadd f O 
+n) in (let TMP_106 \def (wadd g O n) in (let TMP_107 \def (le_O_n O) in (let 
+TMP_108 \def (wadd_le f g H2 O O TMP_107 n) in (let TMP_109 \def (le_n_S 
+TMP_105 TMP_106 TMP_108) in (le_S_n TMP_103 TMP_104 TMP_109))))))))) in (let 
+TMP_111 \def (weight_le t0 TMP_101 TMP_102 TMP_110) in (let TMP_112 \def 
+(lt_le_plus_plus TMP_94 TMP_95 TMP_97 TMP_99 TMP_100 TMP_111) in (lt_n_S 
+TMP_89 TMP_93 TMP_112))))))))))))))))))))))))) in (B_ind TMP_16 TMP_57 TMP_85 
+TMP_113 b)))))) in (let TMP_128 \def (\lambda (_: F).(\lambda (f0: ((nat \to 
 nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 
-m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g 
-u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1) 
-(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g 
-H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: 
-T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 
-t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(let 
+TMP_115 \def (weight_map f0 u2) in (let TMP_116 \def (weight_map f0 t0) in 
+(let TMP_117 \def (plus TMP_115 TMP_116) in (let TMP_118 \def (weight_map g 
+u1) in (let TMP_119 \def (weight_map g t0) in (let TMP_120 \def (plus TMP_118 
+TMP_119) in (let TMP_121 \def (weight_map f0 u2) in (let TMP_122 \def 
+(weight_map g u1) in (let TMP_123 \def (weight_map f0 t0) in (let TMP_124 
+\def (weight_map g t0) in (let TMP_125 \def (H1 f0 g H2 H3) in (let TMP_126 
+\def (weight_le t0 f0 g H2) in (let TMP_127 \def (lt_le_plus_plus TMP_121 
+TMP_122 TMP_123 TMP_124 TMP_125 TMP_126) in (lt_n_S TMP_117 TMP_120 
+TMP_127))))))))))))))))))) in (K_ind TMP_9 TMP_114 TMP_128 k)))))))))))) in 
+(let TMP_285 \def (\lambda (k: K).(let TMP_134 \def (\lambda (k0: K).(\forall 
+(v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) 
+v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
 (s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g 
 t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to nat))).(\forall (g: 
 ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
-f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead k0 u0 t2)) 
-(weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda 
-(b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: 
-nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) 
-\to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: 
+f (lift (S i) O v)) (g i)) \to (let TMP_130 \def (THead k0 u0 t2) in (let 
+TMP_131 \def (weight_map f TMP_130) in (let TMP_132 \def (THead k0 u0 t1) in 
+(let TMP_133 \def (weight_map g TMP_132) in (lt TMP_131 
+TMP_133))))))))))))))))) in (let TMP_270 \def (\lambda (b: B).(let TMP_141 
+\def (\lambda (b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1: 
+T).(\forall (i: nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f: 
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
+(g m)))) \to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind 
+b0) i))) \to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: 
 T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
 nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to 
-(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 
-t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 
-(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: 
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt 
+(let TMP_135 \def (Bind b0) in (let TMP_136 \def (THead TMP_135 u0 t2) in 
+(let TMP_137 \def (weight_map f TMP_136) in (let TMP_138 \def (Bind b0) in 
+(let TMP_139 \def (THead TMP_138 u0 t1) in (let TMP_140 \def (weight_map g 
+TMP_139) in (lt TMP_137 TMP_140))))))))))))))))))) in (let TMP_199 \def 
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to 
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f 
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to 
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(let 
+TMP_142 \def (weight_map f u0) in (let TMP_143 \def (weight_map f u0) in (let 
+TMP_144 \def (S TMP_143) in (let TMP_145 \def (wadd f TMP_144) in (let 
+TMP_146 \def (weight_map TMP_145 t2) in (let TMP_147 \def (plus TMP_142 
+TMP_146) in (let TMP_148 \def (weight_map g u0) in (let TMP_149 \def 
+(weight_map g u0) in (let TMP_150 \def (S TMP_149) in (let TMP_151 \def (wadd 
+g TMP_150) in (let TMP_152 \def (weight_map TMP_151 t1) in (let TMP_153 \def 
+(plus TMP_148 TMP_152) in (let TMP_154 \def (weight_map f u0) in (let TMP_155 
+\def (weight_map g u0) in (let TMP_156 \def (weight_map f u0) in (let TMP_157 
+\def (S TMP_156) in (let TMP_158 \def (wadd f TMP_157) in (let TMP_159 \def 
+(weight_map TMP_158 t2) in (let TMP_160 \def (weight_map g u0) in (let 
+TMP_161 \def (S TMP_160) in (let TMP_162 \def (wadd g TMP_161) in (let 
+TMP_163 \def (weight_map TMP_162 t1) in (let TMP_164 \def (weight_le u0 f g 
+H2) in (let TMP_165 \def (weight_map f u0) in (let TMP_166 \def (S TMP_165) 
+in (let TMP_167 \def (wadd f TMP_166) in (let TMP_168 \def (weight_map g u0) 
+in (let TMP_169 \def (S TMP_168) in (let TMP_170 \def (wadd g TMP_169) in 
+(let TMP_179 \def (\lambda (m: nat).(let TMP_171 \def (weight_map f u0) in 
+(let TMP_172 \def (S TMP_171) in (let TMP_173 \def (weight_map g u0) in (let 
+TMP_174 \def (S TMP_173) in (let TMP_175 \def (weight_map f u0) in (let 
+TMP_176 \def (weight_map g u0) in (let TMP_177 \def (weight_le u0 f g H2) in 
+(let TMP_178 \def (le_n_S TMP_175 TMP_176 TMP_177) in (wadd_le f g H2 TMP_172 
+TMP_174 TMP_178 m)))))))))) in (let TMP_180 \def (S i) in (let TMP_181 \def 
+(lift TMP_180 O v) in (let TMP_182 \def (weight_map f TMP_181) in (let 
+TMP_184 \def (\lambda (n: nat).(let TMP_183 \def (g i) in (lt n TMP_183))) in 
+(let TMP_185 \def (weight_map f u0) in (let TMP_186 \def (S TMP_185) in (let 
+TMP_187 \def (wadd f TMP_186) in (let TMP_188 \def (S i) in (let TMP_189 \def 
+(S TMP_188) in (let TMP_190 \def (lift TMP_189 O v) in (let TMP_191 \def 
+(weight_map TMP_187 TMP_190) in (let TMP_192 \def (weight_map f u0) in (let 
+TMP_193 \def (S TMP_192) in (let TMP_194 \def (S i) in (let TMP_195 \def 
+(lift_weight_add_O TMP_193 v TMP_194 f) in (let TMP_196 \def (eq_ind nat 
+TMP_182 TMP_184 H3 TMP_191 TMP_195) in (let TMP_197 \def (H1 TMP_167 TMP_170 
+TMP_179 TMP_196) in (let TMP_198 \def (le_lt_plus_plus TMP_154 TMP_155 
+TMP_159 TMP_163 TMP_164 TMP_197) in (lt_n_S TMP_147 TMP_153 
+TMP_198)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (let 
+TMP_234 \def (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: 
+nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat 
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g 
+m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt 
 (weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: 
 ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: 
 nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f 
-u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0))) 
-t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f 
-(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1) 
-(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S 
-(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0)) 
-(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le 
-u0 f g H2)) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: 
-nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f u0))) (lift (S (S 
-i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i) f)))))))))))))))) 
+i))).(let TMP_200 \def (weight_map f u0) in (let TMP_201 \def (wadd f O) in 
+(let TMP_202 \def (weight_map TMP_201 t2) in (let TMP_203 \def (plus TMP_200 
+TMP_202) in (let TMP_204 \def (weight_map g u0) in (let TMP_205 \def (wadd g 
+O) in (let TMP_206 \def (weight_map TMP_205 t1) in (let TMP_207 \def (plus 
+TMP_204 TMP_206) in (let TMP_208 \def (weight_map f u0) in (let TMP_209 \def 
+(weight_map g u0) in (let TMP_210 \def (wadd f O) in (let TMP_211 \def 
+(weight_map TMP_210 t2) in (let TMP_212 \def (wadd g O) in (let TMP_213 \def 
+(weight_map TMP_212 t1) in (let TMP_214 \def (weight_le u0 f g H2) in (let 
+TMP_215 \def (wadd f O) in (let TMP_216 \def (wadd g O) in (let TMP_218 \def 
+(\lambda (m: nat).(let TMP_217 \def (le_O_n O) in (wadd_le f g H2 O O TMP_217 
+m))) in (let TMP_219 \def (S i) in (let TMP_220 \def (lift TMP_219 O v) in 
+(let TMP_221 \def (weight_map f TMP_220) in (let TMP_223 \def (\lambda (n: 
+nat).(let TMP_222 \def (g i) in (lt n TMP_222))) in (let TMP_224 \def (wadd f 
+O) in (let TMP_225 \def (S i) in (let TMP_226 \def (S TMP_225) in (let 
+TMP_227 \def (lift TMP_226 O v) in (let TMP_228 \def (weight_map TMP_224 
+TMP_227) in (let TMP_229 \def (S i) in (let TMP_230 \def (lift_weight_add_O O 
+v TMP_229 f) in (let TMP_231 \def (eq_ind nat TMP_221 TMP_223 H3 TMP_228 
+TMP_230) in (let TMP_232 \def (H1 TMP_215 TMP_216 TMP_218 TMP_231) in (let 
+TMP_233 \def (le_lt_plus_plus TMP_208 TMP_209 TMP_211 TMP_213 TMP_214 
+TMP_232) in (lt_n_S TMP_203 TMP_207 
+TMP_233)))))))))))))))))))))))))))))))))))))))))))) in (let TMP_269 \def 
 (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda 
 (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to 
 nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
 \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f 
 t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to 
 nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f 
-m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus 
-(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f 
-u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) 
-(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f 
-g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda 
-(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v)) 
-(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 
-t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g 
-t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat 
-\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: 
-(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0) 
-(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O) 
-t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f 
-O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd 
-g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat 
-(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 
-(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i) 
-f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2: 
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1 
-t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
-nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
-(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g 
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(let 
+TMP_235 \def (weight_map f u0) in (let TMP_236 \def (wadd f O) in (let 
+TMP_237 \def (weight_map TMP_236 t2) in (let TMP_238 \def (plus TMP_235 
+TMP_237) in (let TMP_239 \def (weight_map g u0) in (let TMP_240 \def (wadd g 
+O) in (let TMP_241 \def (weight_map TMP_240 t1) in (let TMP_242 \def (plus 
+TMP_239 TMP_241) in (let TMP_243 \def (weight_map f u0) in (let TMP_244 \def 
+(weight_map g u0) in (let TMP_245 \def (wadd f O) in (let TMP_246 \def 
+(weight_map TMP_245 t2) in (let TMP_247 \def (wadd g O) in (let TMP_248 \def 
+(weight_map TMP_247 t1) in (let TMP_249 \def (weight_le u0 f g H2) in (let 
+TMP_250 \def (wadd f O) in (let TMP_251 \def (wadd g O) in (let TMP_253 \def 
+(\lambda (m: nat).(let TMP_252 \def (le_O_n O) in (wadd_le f g H2 O O TMP_252 
+m))) in (let TMP_254 \def (S i) in (let TMP_255 \def (lift TMP_254 O v) in 
+(let TMP_256 \def (weight_map f TMP_255) in (let TMP_258 \def (\lambda (n: 
+nat).(let TMP_257 \def (g i) in (lt n TMP_257))) in (let TMP_259 \def (wadd f 
+O) in (let TMP_260 \def (S i) in (let TMP_261 \def (S TMP_260) in (let 
+TMP_262 \def (lift TMP_261 O v) in (let TMP_263 \def (weight_map TMP_259 
+TMP_262) in (let TMP_264 \def (S i) in (let TMP_265 \def (lift_weight_add_O O 
+v TMP_264 f) in (let TMP_266 \def (eq_ind nat TMP_256 TMP_258 H3 TMP_263 
+TMP_265) in (let TMP_267 \def (H1 TMP_250 TMP_251 TMP_253 TMP_266) in (let 
+TMP_268 \def (le_lt_plus_plus TMP_243 TMP_244 TMP_246 TMP_248 TMP_249 
+TMP_267) in (lt_n_S TMP_238 TMP_242 
+TMP_268)))))))))))))))))))))))))))))))))))))))))))) in (B_ind TMP_141 TMP_199 
+TMP_234 TMP_269 b)))))) in (let TMP_284 \def (\lambda (_: F).(\lambda (v: 
+T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i 
+v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat 
+\to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 
+(lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g 
 t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
 \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
-(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map 
-f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1)) 
-(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) 
-(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) 
-(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda 
-(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall 
+(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(let TMP_271 \def 
+(weight_map f0 u0) in (let TMP_272 \def (weight_map f0 t2) in (let TMP_273 
+\def (plus TMP_271 TMP_272) in (let TMP_274 \def (weight_map g u0) in (let 
+TMP_275 \def (weight_map g t1) in (let TMP_276 \def (plus TMP_274 TMP_275) in 
+(let TMP_277 \def (weight_map f0 u0) in (let TMP_278 \def (weight_map g u0) 
+in (let TMP_279 \def (weight_map f0 t2) in (let TMP_280 \def (weight_map g 
+t1) in (let TMP_281 \def (weight_le u0 f0 g H2) in (let TMP_282 \def (H1 f0 g 
+H2 H3) in (let TMP_283 \def (le_lt_plus_plus TMP_277 TMP_278 TMP_279 TMP_280 
+TMP_281 TMP_282) in (lt_n_S TMP_273 TMP_276 TMP_283)))))))))))))))))))))))))) 
+in (K_ind TMP_134 TMP_270 TMP_284 k))))) in (let TMP_454 \def (\lambda (v: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (_: (subst0 i 
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
+i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda 
+(k: K).(let TMP_290 \def (\lambda (k0: K).(\forall (t1: T).(\forall (t2: 
+T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall 
 (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt 
-(weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map 
-g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1: 
-T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt 
-(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) 
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead 
-k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind 
-(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v 
-t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
+(weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f 
+t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: 
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map 
+f (lift (S i) O v)) (g i)) \to (let TMP_286 \def (THead k0 u2 t2) in (let 
+TMP_287 \def (weight_map f TMP_286) in (let TMP_288 \def (THead k0 u1 t1) in 
+(let TMP_289 \def (weight_map g TMP_288) in (lt TMP_287 TMP_289)))))))))))))) 
+in (let TMP_439 \def (\lambda (b: B).(let TMP_297 \def (\lambda (b0: 
+B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v t1 t2) \to 
+(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s (Bind b0) i)) O 
+v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2) (weight_map g t1))))))) 
+\to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall 
+(m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) 
+\to (let TMP_291 \def (Bind b0) in (let TMP_292 \def (THead TMP_291 u2 t2) in 
+(let TMP_293 \def (weight_map f TMP_292) in (let TMP_294 \def (Bind b0) in 
+(let TMP_295 \def (THead TMP_294 u1 t1) in (let TMP_296 \def (weight_map g 
+TMP_295) in (lt TMP_293 TMP_296)))))))))))))))) in (let TMP_356 \def (\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: 
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S 
+i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat 
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le 
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
+i))).(let TMP_298 \def (weight_map f u2) in (let TMP_299 \def (weight_map f 
+u2) in (let TMP_300 \def (S TMP_299) in (let TMP_301 \def (wadd f TMP_300) in 
+(let TMP_302 \def (weight_map TMP_301 t2) in (let TMP_303 \def (plus TMP_298 
+TMP_302) in (let TMP_304 \def (weight_map g u1) in (let TMP_305 \def 
+(weight_map g u1) in (let TMP_306 \def (S TMP_305) in (let TMP_307 \def (wadd 
+g TMP_306) in (let TMP_308 \def (weight_map TMP_307 t1) in (let TMP_309 \def 
+(plus TMP_304 TMP_308) in (let TMP_310 \def (weight_map f u2) in (let TMP_311 
+\def (weight_map g u1) in (let TMP_312 \def (weight_map f u2) in (let TMP_313 
+\def (S TMP_312) in (let TMP_314 \def (wadd f TMP_313) in (let TMP_315 \def 
+(weight_map TMP_314 t2) in (let TMP_316 \def (weight_map g u1) in (let 
+TMP_317 \def (S TMP_316) in (let TMP_318 \def (wadd g TMP_317) in (let 
+TMP_319 \def (weight_map TMP_318 t1) in (let TMP_320 \def (H1 f g H4 H5) in 
+(let TMP_321 \def (S i) in (let TMP_322 \def (weight_map f u2) in (let 
+TMP_323 \def (S TMP_322) in (let TMP_324 \def (wadd f TMP_323) in (let 
+TMP_325 \def (weight_map g u1) in (let TMP_326 \def (S TMP_325) in (let 
+TMP_327 \def (wadd g TMP_326) in (let TMP_336 \def (\lambda (m: nat).(let 
+TMP_328 \def (weight_map f u2) in (let TMP_329 \def (S TMP_328) in (let 
+TMP_330 \def (weight_map g u1) in (let TMP_331 \def (S TMP_330) in (let 
+TMP_332 \def (weight_map f u2) in (let TMP_333 \def (weight_map g u1) in (let 
+TMP_334 \def (H1 f g H4 H5) in (let TMP_335 \def (lt_n_S TMP_332 TMP_333 
+TMP_334) in (wadd_lt f g H4 TMP_329 TMP_331 TMP_335 m)))))))))) in (let 
+TMP_337 \def (S i) in (let TMP_338 \def (lift TMP_337 O v) in (let TMP_339 
+\def (weight_map f TMP_338) in (let TMP_341 \def (\lambda (n: nat).(let 
+TMP_340 \def (g i) in (lt n TMP_340))) in (let TMP_342 \def (weight_map f u2) 
+in (let TMP_343 \def (S TMP_342) in (let TMP_344 \def (wadd f TMP_343) in 
+(let TMP_345 \def (S i) in (let TMP_346 \def (S TMP_345) in (let TMP_347 \def 
+(lift TMP_346 O v) in (let TMP_348 \def (weight_map TMP_344 TMP_347) in (let 
+TMP_349 \def (weight_map f u2) in (let TMP_350 \def (S TMP_349) in (let 
+TMP_351 \def (S i) in (let TMP_352 \def (lift_weight_add_O TMP_350 v TMP_351 
+f) in (let TMP_353 \def (eq_ind nat TMP_339 TMP_341 H5 TMP_348 TMP_352) in 
+(let TMP_354 \def (subst0_weight_le v t1 t2 TMP_321 H2 TMP_324 TMP_327 
+TMP_336 TMP_353) in (let TMP_355 \def (lt_le_plus_plus TMP_310 TMP_311 
+TMP_315 TMP_319 TMP_320 TMP_354) in (lt_n_S TMP_303 TMP_309 
+TMP_355)))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (let 
+TMP_397 \def (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v 
+t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
-(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (lt (weight_map f t2) 
-(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat 
-\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f 
-(lift (S i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t2)) 
-(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H2: (subst0 (S i) v t1 t2)).(\lambda (_: ((\forall (f: 
-((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) 
-(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt 
-(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to 
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f 
-m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f 
-u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) 
-t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f 
-(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 
-f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2))) 
-(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S 
-(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2) 
-(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O 
-v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f 
-u2))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i) 
-f))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v 
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g 
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt 
+(weight_map f (lift (S i) O v)) (g i))).(let TMP_357 \def (weight_map f u2) 
+in (let TMP_358 \def (wadd f O) in (let TMP_359 \def (weight_map TMP_358 t2) 
+in (let TMP_360 \def (plus TMP_357 TMP_359) in (let TMP_361 \def (weight_map 
+g u1) in (let TMP_362 \def (wadd g O) in (let TMP_363 \def (weight_map 
+TMP_362 t1) in (let TMP_364 \def (plus TMP_361 TMP_363) in (let TMP_365 \def 
+(weight_map f u2) in (let TMP_366 \def (weight_map g u1) in (let TMP_367 \def 
+(wadd f O) in (let TMP_368 \def (weight_map TMP_367 t2) in (let TMP_369 \def 
+(wadd g O) in (let TMP_370 \def (weight_map TMP_369 t1) in (let TMP_371 \def 
+(H1 f g H4 H5) in (let TMP_372 \def (wadd f O) in (let TMP_373 \def (wadd g 
+O) in (let TMP_381 \def (\lambda (m: nat).(let TMP_374 \def (wadd f O m) in 
+(let TMP_375 \def (wadd g O m) in (let TMP_376 \def (wadd f O m) in (let 
+TMP_377 \def (wadd g O m) in (let TMP_378 \def (le_O_n O) in (let TMP_379 
+\def (wadd_le f g H4 O O TMP_378 m) in (let TMP_380 \def (le_n_S TMP_376 
+TMP_377 TMP_379) in (le_S_n TMP_374 TMP_375 TMP_380))))))))) in (let TMP_382 
+\def (S i) in (let TMP_383 \def (lift TMP_382 O v) in (let TMP_384 \def 
+(weight_map f TMP_383) in (let TMP_386 \def (\lambda (n: nat).(let TMP_385 
+\def (g i) in (lt n TMP_385))) in (let TMP_387 \def (wadd f O) in (let 
+TMP_388 \def (S i) in (let TMP_389 \def (S TMP_388) in (let TMP_390 \def 
+(lift TMP_389 O v) in (let TMP_391 \def (weight_map TMP_387 TMP_390) in (let 
+TMP_392 \def (S i) in (let TMP_393 \def (lift_weight_add_O O v TMP_392 f) in 
+(let TMP_394 \def (eq_ind nat TMP_384 TMP_386 H5 TMP_391 TMP_393) in (let 
+TMP_395 \def (H3 TMP_372 TMP_373 TMP_381 TMP_394) in (let TMP_396 \def 
+(lt_plus_plus TMP_365 TMP_366 TMP_368 TMP_370 TMP_371 TMP_395) in (lt_n_S 
+TMP_360 TMP_364 TMP_396))))))))))))))))))))))))))))))))))))))))) in (let 
+TMP_438 \def (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v 
 t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to 
 nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S 
 (S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g 
 t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to 
 nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt 
-(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2) 
-(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O) 
-t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O) 
-t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O) 
-(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m) 
-(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f 
-(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) 
-(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda 
-(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: 
-((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: 
-nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S 
-i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat 
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le 
-(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g 
-i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus 
-(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2) 
-(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f 
-g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m) 
-(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n O) 
-m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g 
-i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v 
-(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to 
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) 
-\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2) 
-(weight_map g t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat 
-\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda 
-(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map 
-f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) 
-(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) 
-(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t 
-z H))))).
-(* COMMENTS
-Initial nodes: 4207
-END *)
+(weight_map f (lift (S i) O v)) (g i))).(let TMP_398 \def (weight_map f u2) 
+in (let TMP_399 \def (wadd f O) in (let TMP_400 \def (weight_map TMP_399 t2) 
+in (let TMP_401 \def (plus TMP_398 TMP_400) in (let TMP_402 \def (weight_map 
+g u1) in (let TMP_403 \def (wadd g O) in (let TMP_404 \def (weight_map 
+TMP_403 t1) in (let TMP_405 \def (plus TMP_402 TMP_404) in (let TMP_406 \def 
+(weight_map f u2) in (let TMP_407 \def (weight_map g u1) in (let TMP_408 \def 
+(wadd f O) in (let TMP_409 \def (weight_map TMP_408 t2) in (let TMP_410 \def 
+(wadd g O) in (let TMP_411 \def (weight_map TMP_410 t1) in (let TMP_412 \def 
+(H1 f g H4 H5) in (let TMP_413 \def (wadd f O) in (let TMP_414 \def (wadd g 
+O) in (let TMP_422 \def (\lambda (m: nat).(let TMP_415 \def (wadd f O m) in 
+(let TMP_416 \def (wadd g O m) in (let TMP_417 \def (wadd f O m) in (let 
+TMP_418 \def (wadd g O m) in (let TMP_419 \def (le_O_n O) in (let TMP_420 
+\def (wadd_le f g H4 O O TMP_419 m) in (let TMP_421 \def (le_n_S TMP_417 
+TMP_418 TMP_420) in (le_S_n TMP_415 TMP_416 TMP_421))))))))) in (let TMP_423 
+\def (S i) in (let TMP_424 \def (lift TMP_423 O v) in (let TMP_425 \def 
+(weight_map f TMP_424) in (let TMP_427 \def (\lambda (n: nat).(let TMP_426 
+\def (g i) in (lt n TMP_426))) in (let TMP_428 \def (wadd f O) in (let 
+TMP_429 \def (S i) in (let TMP_430 \def (S TMP_429) in (let TMP_431 \def 
+(lift TMP_430 O v) in (let TMP_432 \def (weight_map TMP_428 TMP_431) in (let 
+TMP_433 \def (S i) in (let TMP_434 \def (lift_weight_add_O O v TMP_433 f) in 
+(let TMP_435 \def (eq_ind nat TMP_425 TMP_427 H5 TMP_432 TMP_434) in (let 
+TMP_436 \def (H3 TMP_413 TMP_414 TMP_422 TMP_435) in (let TMP_437 \def 
+(lt_plus_plus TMP_406 TMP_407 TMP_409 TMP_411 TMP_412 TMP_436) in (lt_n_S 
+TMP_401 TMP_405 TMP_437))))))))))))))))))))))))))))))))))))))))) in (B_ind 
+TMP_297 TMP_356 TMP_397 TMP_438 b)))))) in (let TMP_453 \def (\lambda (_: 
+F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 
+t2)).(\lambda (H3: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to 
+nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 
+(S i) O v)) (g i)) \to (lt (weight_map f0 t2) (weight_map g 
+t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to 
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5: 
+(lt (weight_map f0 (lift (S i) O v)) (g i))).(let TMP_440 \def (weight_map f0 
+u2) in (let TMP_441 \def (weight_map f0 t2) in (let TMP_442 \def (plus 
+TMP_440 TMP_441) in (let TMP_443 \def (weight_map g u1) in (let TMP_444 \def 
+(weight_map g t1) in (let TMP_445 \def (plus TMP_443 TMP_444) in (let TMP_446 
+\def (weight_map f0 u2) in (let TMP_447 \def (weight_map g u1) in (let 
+TMP_448 \def (weight_map f0 t2) in (let TMP_449 \def (weight_map g t1) in 
+(let TMP_450 \def (H1 f0 g H4 H5) in (let TMP_451 \def (H3 f0 g H4 H5) in 
+(let TMP_452 \def (lt_plus_plus TMP_446 TMP_447 TMP_448 TMP_449 TMP_450 
+TMP_451) in (lt_n_S TMP_442 TMP_445 TMP_452))))))))))))))))))))))) in (K_ind 
+TMP_290 TMP_439 TMP_453 k))))))))))) in (subst0_ind TMP_3 TMP_4 TMP_129 
+TMP_285 TMP_454 d u t z H)))))))))).
 
 theorem subst0_tlt_head:
  \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt 
 (THead (Bind Abbr) u z) (THead (Bind Abbr) u t)))))
 \def
  \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t 
-z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus 
-(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S 
-(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map 
-(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd 
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map 
-(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n 
-(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd 
-(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda 
-(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n 
-(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m))) 
-(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n: 
-nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda 
-(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_: 
-nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u) 
-(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda 
-(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda 
-(_: nat).O) u)) u O (\lambda (_: nat).O))))))))).
-(* COMMENTS
-Initial nodes: 347
-END *)
+z)).(let TMP_1 \def (\lambda (_: nat).O) in (let TMP_2 \def (weight_map TMP_1 
+u) in (let TMP_3 \def (\lambda (_: nat).O) in (let TMP_4 \def (\lambda (_: 
+nat).O) in (let TMP_5 \def (weight_map TMP_4 u) in (let TMP_6 \def (S TMP_5) 
+in (let TMP_7 \def (wadd TMP_3 TMP_6) in (let TMP_8 \def (weight_map TMP_7 z) 
+in (let TMP_9 \def (plus TMP_2 TMP_8) in (let TMP_10 \def (\lambda (_: 
+nat).O) in (let TMP_11 \def (weight_map TMP_10 u) in (let TMP_12 \def 
+(\lambda (_: nat).O) in (let TMP_13 \def (\lambda (_: nat).O) in (let TMP_14 
+\def (weight_map TMP_13 u) in (let TMP_15 \def (S TMP_14) in (let TMP_16 \def 
+(wadd TMP_12 TMP_15) in (let TMP_17 \def (weight_map TMP_16 t) in (let TMP_18 
+\def (plus TMP_11 TMP_17) in (let TMP_19 \def (\lambda (_: nat).O) in (let 
+TMP_20 \def (weight_map TMP_19 u) in (let TMP_21 \def (\lambda (_: nat).O) in 
+(let TMP_22 \def (weight_map TMP_21 u) in (let TMP_23 \def (\lambda (_: 
+nat).O) in (let TMP_24 \def (\lambda (_: nat).O) in (let TMP_25 \def 
+(weight_map TMP_24 u) in (let TMP_26 \def (S TMP_25) in (let TMP_27 \def 
+(wadd TMP_23 TMP_26) in (let TMP_28 \def (weight_map TMP_27 z) in (let TMP_29 
+\def (\lambda (_: nat).O) in (let TMP_30 \def (\lambda (_: nat).O) in (let 
+TMP_31 \def (weight_map TMP_30 u) in (let TMP_32 \def (S TMP_31) in (let 
+TMP_33 \def (wadd TMP_29 TMP_32) in (let TMP_34 \def (weight_map TMP_33 t) in 
+(let TMP_35 \def (\lambda (_: nat).O) in (let TMP_36 \def (weight_map TMP_35 
+u) in (let TMP_37 \def (le_n TMP_36) in (let TMP_38 \def (\lambda (_: nat).O) 
+in (let TMP_39 \def (\lambda (_: nat).O) in (let TMP_40 \def (weight_map 
+TMP_39 u) in (let TMP_41 \def (S TMP_40) in (let TMP_42 \def (wadd TMP_38 
+TMP_41) in (let TMP_43 \def (\lambda (_: nat).O) in (let TMP_44 \def (\lambda 
+(_: nat).O) in (let TMP_45 \def (weight_map TMP_44 u) in (let TMP_46 \def (S 
+TMP_45) in (let TMP_47 \def (wadd TMP_43 TMP_46) in (let TMP_53 \def (\lambda 
+(m: nat).(let TMP_48 \def (\lambda (_: nat).O) in (let TMP_49 \def (\lambda 
+(_: nat).O) in (let TMP_50 \def (weight_map TMP_49 u) in (let TMP_51 \def (S 
+TMP_50) in (let TMP_52 \def (wadd TMP_48 TMP_51 m) in (le_n TMP_52))))))) in 
+(let TMP_54 \def (\lambda (_: nat).O) in (let TMP_55 \def (lift O O u) in 
+(let TMP_56 \def (weight_map TMP_54 TMP_55) in (let TMP_60 \def (\lambda (n: 
+nat).(let TMP_57 \def (\lambda (_: nat).O) in (let TMP_58 \def (weight_map 
+TMP_57 u) in (let TMP_59 \def (S TMP_58) in (lt n TMP_59))))) in (let TMP_66 
+\def (\lambda (t0: T).(let TMP_61 \def (\lambda (_: nat).O) in (let TMP_62 
+\def (weight_map TMP_61 t0) in (let TMP_63 \def (\lambda (_: nat).O) in (let 
+TMP_64 \def (weight_map TMP_63 u) in (let TMP_65 \def (S TMP_64) in (lt 
+TMP_62 TMP_65))))))) in (let TMP_67 \def (\lambda (_: nat).O) in (let TMP_68 
+\def (weight_map TMP_67 u) in (let TMP_69 \def (S TMP_68) in (let TMP_70 \def 
+(le_n TMP_69) in (let TMP_71 \def (lift O O u) in (let TMP_72 \def (lift_r u 
+O) in (let TMP_73 \def (eq_ind_r T u TMP_66 TMP_70 TMP_71 TMP_72) in (let 
+TMP_74 \def (\lambda (_: nat).O) in (let TMP_75 \def (\lambda (_: nat).O) in 
+(let TMP_76 \def (weight_map TMP_75 u) in (let TMP_77 \def (S TMP_76) in (let 
+TMP_78 \def (wadd TMP_74 TMP_77) in (let TMP_79 \def (S O) in (let TMP_80 
+\def (lift TMP_79 O u) in (let TMP_81 \def (weight_map TMP_78 TMP_80) in (let 
+TMP_82 \def (\lambda (_: nat).O) in (let TMP_83 \def (weight_map TMP_82 u) in 
+(let TMP_84 \def (S TMP_83) in (let TMP_85 \def (\lambda (_: nat).O) in (let 
+TMP_86 \def (lift_weight_add_O TMP_84 u O TMP_85) in (let TMP_87 \def (eq_ind 
+nat TMP_56 TMP_60 TMP_73 TMP_81 TMP_86) in (let TMP_88 \def (subst0_weight_lt 
+u t z O H TMP_42 TMP_47 TMP_53 TMP_87) in (let TMP_89 \def (le_lt_plus_plus 
+TMP_20 TMP_22 TMP_28 TMP_34 TMP_37 TMP_88) in (lt_n_S TMP_9 TMP_18 
+TMP_89))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
+)))))))).
 
 theorem subst0_tlt:
  \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z 
 (THead (Bind Abbr) u t)))))
 \def
  \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t 
-z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx 
-(Bind Abbr) u z) (subst0_tlt_head u t z H))))).
-(* COMMENTS
-Initial nodes: 59
-END *)
+z)).(let TMP_1 \def (Bind Abbr) in (let TMP_2 \def (THead TMP_1 u z) in (let 
+TMP_3 \def (Bind Abbr) in (let TMP_4 \def (THead TMP_3 u t) in (let TMP_5 
+\def (Bind Abbr) in (let TMP_6 \def (tlt_head_dx TMP_5 u z) in (let TMP_7 
+\def (subst0_tlt_head u t z H) in (tlt_trans TMP_2 z TMP_4 TMP_6 
+TMP_7))))))))))).