X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fpr2%2Fclen.ma;h=b518e4360fc92acbc7ab326af7b7a3ea91486495;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=0b256622f96dc1332a23bc5f1517d8915e9bd6d3;hpb=40e0eed5dc1a5f56f089f491841a5077723b891a;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma
index 0b256622f..b518e4360 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/clen.ma
@@ -18,230 +18,134 @@ include "basic_1/pr2/props.ma".
 
 include "basic_1/clen/getl.ma".
 
-theorem pr2_gen_ctail:
+lemma pr2_gen_ctail:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
 (t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: 
 T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 
 (clen c) u t t2)))))))))
 \def
  \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(let TMP_1 \def (CTail k u c) 
-in (let TMP_2 \def (\lambda (c0: C).(pr2 c0 t1 t2)) in (let TMP_10 \def 
-(\lambda (_: C).(let TMP_3 \def (pr2 c t1 t2) in (let TMP_5 \def (\lambda (_: 
-T).(let TMP_4 \def (Bind Abbr) in (eq K k TMP_4))) in (let TMP_6 \def 
-(\lambda (t: T).(pr0 t1 t)) in (let TMP_8 \def (\lambda (t: T).(let TMP_7 
-\def (clen c) in (subst0 TMP_7 u t t2))) in (let TMP_9 \def (ex3 T TMP_5 
-TMP_6 TMP_8) in (or TMP_3 TMP_9))))))) in (let TMP_126 \def (\lambda (y: 
-C).(\lambda (H0: (pr2 y t1 t2)).(let TMP_18 \def (\lambda (c0: C).(\lambda 
-(t: T).(\lambda (t0: T).((eq C c0 (CTail k u c)) \to (let TMP_11 \def (pr2 c 
-t t0) in (let TMP_13 \def (\lambda (_: T).(let TMP_12 \def (Bind Abbr) in (eq 
-K k TMP_12))) in (let TMP_14 \def (\lambda (t3: T).(pr0 t t3)) in (let TMP_16 
-\def (\lambda (t3: T).(let TMP_15 \def (clen c) in (subst0 TMP_15 u t3 t0))) 
-in (let TMP_17 \def (ex3 T TMP_13 TMP_14 TMP_16) in (or TMP_11 
-TMP_17)))))))))) in (let TMP_27 \def (\lambda (c0: C).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k 
-u c))).(let TMP_19 \def (pr2 c t3 t4) in (let TMP_21 \def (\lambda (_: 
-T).(let TMP_20 \def (Bind Abbr) in (eq K k TMP_20))) in (let TMP_22 \def 
-(\lambda (t: T).(pr0 t3 t)) in (let TMP_24 \def (\lambda (t: T).(let TMP_23 
-\def (clen c) in (subst0 TMP_23 u t t4))) in (let TMP_25 \def (ex3 T TMP_21 
-TMP_22 TMP_24) in (let TMP_26 \def (pr2_free c t3 t4 H1) in (or_introl TMP_19 
-TMP_25 TMP_26)))))))))))) in (let TMP_125 \def (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind 
-Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 
-t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 t)).(\lambda (H4: (eq C c0 
-(CTail k u c))).(let TMP_30 \def (\lambda (c1: C).(let TMP_28 \def (Bind 
-Abbr) in (let TMP_29 \def (CHead d TMP_28 u0) in (getl i c1 TMP_29)))) in 
-(let TMP_31 \def (CTail k u c) in (let H5 \def (eq_ind C c0 TMP_30 H1 TMP_31 
-H4) in (let H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x 
-in (let TMP_33 \def (\lambda (e: C).(let TMP_32 \def (CTail k u e) in (eq C d 
-TMP_32))) in (let TMP_36 \def (\lambda (e: C).(let TMP_34 \def (Bind Abbr) in 
-(let TMP_35 \def (CHead e TMP_34 u0) in (getl i c TMP_35)))) in (let TMP_37 
-\def (ex2 C TMP_33 TMP_36) in (let TMP_39 \def (\lambda (_: nat).(let TMP_38 
-\def (clen c) in (eq nat i TMP_38))) in (let TMP_41 \def (\lambda (_: 
-nat).(let TMP_40 \def (Bind Abbr) in (eq K k TMP_40))) in (let TMP_42 \def 
-(\lambda (_: nat).(eq T u u0)) in (let TMP_44 \def (\lambda (n: nat).(let 
-TMP_43 \def (CSort n) in (eq C d TMP_43))) in (let TMP_45 \def (ex4 nat 
-TMP_39 TMP_41 TMP_42 TMP_44) in (let TMP_46 \def (pr2 c t3 t) in (let TMP_48 
-\def (\lambda (_: T).(let TMP_47 \def (Bind Abbr) in (eq K k TMP_47))) in 
-(let TMP_49 \def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_51 \def (\lambda 
-(t0: T).(let TMP_50 \def (clen c) in (subst0 TMP_50 u t0 t))) in (let TMP_52 
-\def (ex3 T TMP_48 TMP_49 TMP_51) in (let TMP_53 \def (or TMP_46 TMP_52) in 
-(let TMP_76 \def (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d (CTail k u e))) 
-(\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0))))).(let TMP_55 \def 
-(\lambda (e: C).(let TMP_54 \def (CTail k u e) in (eq C d TMP_54))) in (let 
-TMP_58 \def (\lambda (e: C).(let TMP_56 \def (Bind Abbr) in (let TMP_57 \def 
-(CHead e TMP_56 u0) in (getl i c TMP_57)))) in (let TMP_59 \def (pr2 c t3 t) 
-in (let TMP_61 \def (\lambda (_: T).(let TMP_60 \def (Bind Abbr) in (eq K k 
-TMP_60))) in (let TMP_62 \def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_64 
-\def (\lambda (t0: T).(let TMP_63 \def (clen c) in (subst0 TMP_63 u t0 t))) 
-in (let TMP_65 \def (ex3 T TMP_61 TMP_62 TMP_64) in (let TMP_66 \def (or 
-TMP_59 TMP_65) in (let TMP_75 \def (\lambda (x: C).(\lambda (_: (eq C d 
-(CTail k u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(let 
-TMP_67 \def (pr2 c t3 t) in (let TMP_69 \def (\lambda (_: T).(let TMP_68 \def 
-(Bind Abbr) in (eq K k TMP_68))) in (let TMP_70 \def (\lambda (t0: T).(pr0 t3 
-t0)) in (let TMP_72 \def (\lambda (t0: T).(let TMP_71 \def (clen c) in 
-(subst0 TMP_71 u t0 t))) in (let TMP_73 \def (ex3 T TMP_69 TMP_70 TMP_72) in 
-(let TMP_74 \def (pr2_delta c x u0 i H9 t3 t4 H2 t H3) in (or_introl TMP_67 
-TMP_73 TMP_74)))))))))) in (ex2_ind C TMP_55 TMP_58 TMP_66 TMP_75 
-H7))))))))))) in (let TMP_124 \def (\lambda (H7: (ex4 nat (\lambda (_: 
-nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda 
-(_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))))).(let TMP_78 
-\def (\lambda (_: nat).(let TMP_77 \def (clen c) in (eq nat i TMP_77))) in 
-(let TMP_80 \def (\lambda (_: nat).(let TMP_79 \def (Bind Abbr) in (eq K k 
-TMP_79))) in (let TMP_81 \def (\lambda (_: nat).(eq T u u0)) in (let TMP_83 
-\def (\lambda (n: nat).(let TMP_82 \def (CSort n) in (eq C d TMP_82))) in 
-(let TMP_84 \def (pr2 c t3 t) in (let TMP_86 \def (\lambda (_: T).(let TMP_85 
-\def (Bind Abbr) in (eq K k TMP_85))) in (let TMP_87 \def (\lambda (t0: 
-T).(pr0 t3 t0)) in (let TMP_89 \def (\lambda (t0: T).(let TMP_88 \def (clen 
-c) in (subst0 TMP_88 u t0 t))) in (let TMP_90 \def (ex3 T TMP_86 TMP_87 
-TMP_89) in (let TMP_91 \def (or TMP_84 TMP_90) in (let TMP_123 \def (\lambda 
-(x0: nat).(\lambda (H8: (eq nat i (clen c))).(\lambda (H9: (eq K k (Bind 
-Abbr))).(\lambda (H10: (eq T u u0)).(\lambda (_: (eq C d (CSort x0))).(let 
-TMP_92 \def (\lambda (n: nat).(subst0 n u0 t4 t)) in (let TMP_93 \def (clen 
-c) in (let H12 \def (eq_ind nat i TMP_92 H3 TMP_93 H8) in (let TMP_95 \def 
-(\lambda (t0: T).(let TMP_94 \def (clen c) in (subst0 TMP_94 t0 t4 t))) in 
-(let H13 \def (eq_ind_r T u0 TMP_95 H12 u H10) in (let TMP_96 \def (Bind 
-Abbr) in (let TMP_104 \def (\lambda (k0: K).(let TMP_97 \def (pr2 c t3 t) in 
-(let TMP_99 \def (\lambda (_: T).(let TMP_98 \def (Bind Abbr) in (eq K k0 
-TMP_98))) in (let TMP_100 \def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_102 
-\def (\lambda (t0: T).(let TMP_101 \def (clen c) in (subst0 TMP_101 u t0 t))) 
-in (let TMP_103 \def (ex3 T TMP_99 TMP_100 TMP_102) in (or TMP_97 
-TMP_103))))))) in (let TMP_105 \def (pr2 c t3 t) in (let TMP_108 \def 
-(\lambda (_: T).(let TMP_106 \def (Bind Abbr) in (let TMP_107 \def (Bind 
-Abbr) in (eq K TMP_106 TMP_107)))) in (let TMP_109 \def (\lambda (t0: T).(pr0 
-t3 t0)) in (let TMP_111 \def (\lambda (t0: T).(let TMP_110 \def (clen c) in 
-(subst0 TMP_110 u t0 t))) in (let TMP_112 \def (ex3 T TMP_108 TMP_109 
-TMP_111) in (let TMP_115 \def (\lambda (_: T).(let TMP_113 \def (Bind Abbr) 
-in (let TMP_114 \def (Bind Abbr) in (eq K TMP_113 TMP_114)))) in (let TMP_116 
-\def (\lambda (t0: T).(pr0 t3 t0)) in (let TMP_118 \def (\lambda (t0: T).(let 
-TMP_117 \def (clen c) in (subst0 TMP_117 u t0 t))) in (let TMP_119 \def (Bind 
-Abbr) in (let TMP_120 \def (refl_equal K TMP_119) in (let TMP_121 \def 
-(ex3_intro T TMP_115 TMP_116 TMP_118 t4 TMP_120 H2 H13) in (let TMP_122 \def 
-(or_intror TMP_105 TMP_112 TMP_121) in (eq_ind_r K TMP_96 TMP_104 TMP_122 k 
-H9))))))))))))))))))))))))) in (ex4_ind nat TMP_78 TMP_80 TMP_81 TMP_83 
-TMP_91 TMP_123 H7))))))))))))) in (or_ind TMP_37 TMP_45 TMP_53 TMP_76 TMP_124 
-H6))))))))))))))))))))))))))))))))) in (pr2_ind TMP_18 TMP_27 TMP_125 y t1 t2 
-H0)))))) in (insert_eq C TMP_1 TMP_2 TMP_10 TMP_126 H)))))))))).
+(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c) 
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T 
+(\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda 
+(t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 
+(CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind 
+Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 
+t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) 
+(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) 
+(\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1))))))) 
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda 
+(H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: 
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 
+t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda 
+(c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let 
+H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind 
+(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c 
+(CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) 
+(\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) 
+(\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: 
+T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: 
+T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d 
+(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) 
+u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: 
+C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_: 
+T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: 
+T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k 
+u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c 
+t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 
+t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4 
+H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen 
+c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) 
+(\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq 
+nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: 
+nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 
+T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) 
+(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8: 
+(eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T 
+u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda 
+(n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 
+(\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind 
+Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind 
+Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 
+t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind 
+Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 
+t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda 
+(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 
+(refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 
+t2 H0))) H)))))).
 
-theorem pr2_gen_cbind:
+lemma pr2_gen_cbind:
  \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
 (t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) 
 (THead (Bind b) v t2)))))))
 \def
  \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let TMP_1 \def (Bind 
-b) in (let TMP_2 \def (CHead c TMP_1 v) in (let TMP_3 \def (\lambda (c0: 
-C).(pr2 c0 t1 t2)) in (let TMP_8 \def (\lambda (_: C).(let TMP_4 \def (Bind 
-b) in (let TMP_5 \def (THead TMP_4 v t1) in (let TMP_6 \def (Bind b) in (let 
-TMP_7 \def (THead TMP_6 v t2) in (pr2 c TMP_5 TMP_7)))))) in (let TMP_119 
-\def (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(let TMP_13 \def (\lambda 
-(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v)) \to 
-(let TMP_9 \def (Bind b) in (let TMP_10 \def (THead TMP_9 v t) in (let TMP_11 
-\def (Bind b) in (let TMP_12 \def (THead TMP_11 v t0) in (pr2 c TMP_10 
-TMP_12))))))))) in (let TMP_21 \def (\lambda (c0: C).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c 
-(Bind b) v))).(let TMP_14 \def (Bind b) in (let TMP_15 \def (THead TMP_14 v 
-t3) in (let TMP_16 \def (Bind b) in (let TMP_17 \def (THead TMP_16 v t4) in 
-(let TMP_18 \def (pr0_refl v) in (let TMP_19 \def (Bind b) in (let TMP_20 
-\def (pr0_comp v v TMP_18 t3 t4 H1 TMP_19) in (pr2_free c TMP_15 TMP_17 
-TMP_20))))))))))))) in (let TMP_118 \def (\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind 
-Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 
-t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u t4 t)).(\lambda (H4: (eq C c0 
-(CHead c (Bind b) v))).(let TMP_24 \def (\lambda (c1: C).(let TMP_22 \def 
-(Bind Abbr) in (let TMP_23 \def (CHead d TMP_22 u) in (getl i c1 TMP_23)))) 
-in (let TMP_25 \def (Bind b) in (let TMP_26 \def (CHead c TMP_25 v) in (let 
-H5 \def (eq_ind C c0 TMP_24 H1 TMP_26 H4) in (let TMP_27 \def (Bind Abbr) in 
-(let TMP_28 \def (CHead d TMP_27 u) in (let H_x \def (getl_gen_bind b c 
-TMP_28 v i H5) in (let H6 \def H_x in (let TMP_29 \def (eq nat i O) in (let 
-TMP_30 \def (Bind Abbr) in (let TMP_31 \def (CHead d TMP_30 u) in (let TMP_32 
-\def (Bind b) in (let TMP_33 \def (CHead c TMP_32 v) in (let TMP_34 \def (eq 
-C TMP_31 TMP_33) in (let TMP_35 \def (land TMP_29 TMP_34) in (let TMP_37 \def 
-(\lambda (j: nat).(let TMP_36 \def (S j) in (eq nat i TMP_36))) in (let 
-TMP_40 \def (\lambda (j: nat).(let TMP_38 \def (Bind Abbr) in (let TMP_39 
-\def (CHead d TMP_38 u) in (getl j c TMP_39)))) in (let TMP_41 \def (ex2 nat 
-TMP_37 TMP_40) in (let TMP_42 \def (Bind b) in (let TMP_43 \def (THead TMP_42 
-v t3) in (let TMP_44 \def (Bind b) in (let TMP_45 \def (THead TMP_44 v t) in 
-(let TMP_46 \def (pr2 c TMP_43 TMP_45) in (let TMP_90 \def (\lambda (H7: 
-(land (eq nat i O) (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(let 
-TMP_47 \def (eq nat i O) in (let TMP_48 \def (Bind Abbr) in (let TMP_49 \def 
-(CHead d TMP_48 u) in (let TMP_50 \def (Bind b) in (let TMP_51 \def (CHead c 
-TMP_50 v) in (let TMP_52 \def (eq C TMP_49 TMP_51) in (let TMP_53 \def (Bind 
-b) in (let TMP_54 \def (THead TMP_53 v t3) in (let TMP_55 \def (Bind b) in 
-(let TMP_56 \def (THead TMP_55 v t) in (let TMP_57 \def (pr2 c TMP_54 TMP_56) 
-in (let TMP_89 \def (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead d 
-(Bind Abbr) u) (CHead c (Bind b) v))).(let TMP_58 \def (\lambda (e: C).(match 
-e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) in (let 
-TMP_59 \def (Bind Abbr) in (let TMP_60 \def (CHead d TMP_59 u) in (let TMP_61 
-\def (Bind b) in (let TMP_62 \def (CHead c TMP_61 v) in (let H10 \def 
-(f_equal C C TMP_58 TMP_60 TMP_62 H9) in (let TMP_63 \def (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow 
-(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) in 
-(let TMP_64 \def (Bind Abbr) in (let TMP_65 \def (CHead d TMP_64 u) in (let 
-TMP_66 \def (Bind b) in (let TMP_67 \def (CHead c TMP_66 v) in (let H11 \def 
-(f_equal C B TMP_63 TMP_65 TMP_67 H9) in (let TMP_68 \def (\lambda (e: 
-C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) 
-in (let TMP_69 \def (Bind Abbr) in (let TMP_70 \def (CHead d TMP_69 u) in 
-(let TMP_71 \def (Bind b) in (let TMP_72 \def (CHead c TMP_71 v) in (let H12 
-\def (f_equal C T TMP_68 TMP_70 TMP_72 H9) in (let TMP_87 \def (\lambda (H13: 
-(eq B Abbr b)).(\lambda (_: (eq C d c)).(let TMP_73 \def (\lambda (n: 
-nat).(subst0 n u t4 t)) in (let H15 \def (eq_ind nat i TMP_73 H3 O H8) in 
-(let TMP_74 \def (\lambda (t0: T).(subst0 O t0 t4 t)) in (let H16 \def 
-(eq_ind T u TMP_74 H15 v H12) in (let TMP_79 \def (\lambda (b0: B).(let 
-TMP_75 \def (Bind b0) in (let TMP_76 \def (THead TMP_75 v t3) in (let TMP_77 
-\def (Bind b0) in (let TMP_78 \def (THead TMP_77 v t) in (pr2 c TMP_76 
-TMP_78)))))) in (let TMP_80 \def (Bind Abbr) in (let TMP_81 \def (THead 
-TMP_80 v t3) in (let TMP_82 \def (Bind Abbr) in (let TMP_83 \def (THead 
-TMP_82 v t) in (let TMP_84 \def (pr0_refl v) in (let TMP_85 \def (pr0_delta v 
-v TMP_84 t3 t4 H2 t H16) in (let TMP_86 \def (pr2_free c TMP_81 TMP_83 
-TMP_85) in (eq_ind B Abbr TMP_79 TMP_86 b H13))))))))))))))) in (let TMP_88 
-\def (TMP_87 H11) in (TMP_88 H10))))))))))))))))))))))) in (land_ind TMP_47 
-TMP_52 TMP_57 TMP_89 H7)))))))))))))) in (let TMP_117 \def (\lambda (H7: (ex2 
-nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d 
-(Bind Abbr) u))))).(let TMP_92 \def (\lambda (j: nat).(let TMP_91 \def (S j) 
-in (eq nat i TMP_91))) in (let TMP_95 \def (\lambda (j: nat).(let TMP_93 \def 
-(Bind Abbr) in (let TMP_94 \def (CHead d TMP_93 u) in (getl j c TMP_94)))) in 
-(let TMP_96 \def (Bind b) in (let TMP_97 \def (THead TMP_96 v t3) in (let 
-TMP_98 \def (Bind b) in (let TMP_99 \def (THead TMP_98 v t) in (let TMP_100 
-\def (pr2 c TMP_97 TMP_99) in (let TMP_116 \def (\lambda (x: nat).(\lambda 
-(H8: (eq nat i (S x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let 
-TMP_101 \def (\lambda (e: nat).e) in (let TMP_102 \def (S x) in (let H10 \def 
-(f_equal nat nat TMP_101 i TMP_102 H8) in (let TMP_103 \def (\lambda (n: 
-nat).(subst0 n u t4 t)) in (let TMP_104 \def (S x) in (let H11 \def (eq_ind 
-nat i TMP_103 H3 TMP_104 H10) in (let TMP_105 \def (Bind b) in (let TMP_106 
-\def (Bind b) in (let TMP_107 \def (CHead c TMP_106 v) in (let TMP_108 \def 
-(S x) in (let TMP_109 \def (Bind b) in (let TMP_110 \def (CHead c TMP_109 v) 
-in (let TMP_111 \def (clear_bind b c v) in (let TMP_112 \def (Bind Abbr) in 
-(let TMP_113 \def (CHead d TMP_112 u) in (let TMP_114 \def (getl_clear_bind b 
-TMP_110 c v TMP_111 TMP_113 x H9) in (let TMP_115 \def (pr2_delta TMP_107 d u 
-TMP_108 TMP_114 t3 t4 H2 t H11) in (pr2_head_2 c v t3 t TMP_105 
-TMP_115))))))))))))))))))))) in (ex2_ind nat TMP_92 TMP_95 TMP_100 TMP_116 
-H7)))))))))) in (or_ind TMP_35 TMP_41 TMP_46 TMP_90 TMP_117 
-H6))))))))))))))))))))))))))))))))))))) in (pr2_ind TMP_13 TMP_21 TMP_118 y 
-t1 t2 H0)))))) in (insert_eq C TMP_2 TMP_3 TMP_8 TMP_119 H))))))))))).
+(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c 
+(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead 
+(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y 
+t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 
+(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v 
+t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead 
+(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1 
+(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: 
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def 
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead 
+c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) 
+v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d 
+(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S 
+j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead 
+(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3) 
+(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead 
+d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda 
+(e: C).(match e with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow 
+c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H9) in ((let H11 \def 
+(f_equal C B (\lambda (e: C).(match e with [(CSort _) \Rightarrow Abbr | 
+(CHead _ k _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) 
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H9) in 
+((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) v) H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C 
+d c)).(let H15 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O 
+H8) in (let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v 
+H12) in (eq_ind B Abbr (\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead 
+(Bind b0) v t))) (pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) 
+(pr0_delta v v (pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) 
+(\lambda (H7: (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: 
+nat).(getl j c (CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq 
+nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c 
+(THead (Bind b) v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: 
+(eq nat i (S x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 
+\def (f_equal nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def 
+(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in 
+(pr2_head_2 c v t3 t (Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) 
+(getl_clear_bind b (CHead c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind 
+Abbr) u) x H9) t3 t4 H2 t H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) 
+H)))))).
 
-theorem pr2_gen_cflat:
+lemma pr2_gen_cflat:
  \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall 
 (t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
 \def
  \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let TMP_1 \def (Flat 
-f) in (let TMP_2 \def (CHead c TMP_1 v) in (let TMP_3 \def (\lambda (c0: 
-C).(pr2 c0 t1 t2)) in (let TMP_4 \def (\lambda (_: C).(pr2 c t1 t2)) in (let 
-TMP_15 \def (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(let TMP_5 \def 
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) 
-v)) \to (pr2 c t t0))))) in (let TMP_6 \def (\lambda (c0: C).(\lambda (t3: 
-T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c 
-(Flat f) v))).(pr2_free c t3 t4 H1)))))) in (let TMP_14 \def (\lambda (c0: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 
-(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: 
-(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u t4 t)).(\lambda (H4: 
-(eq C c0 (CHead c (Flat f) v))).(let TMP_9 \def (\lambda (c1: C).(let TMP_7 
-\def (Bind Abbr) in (let TMP_8 \def (CHead d TMP_7 u) in (getl i c1 TMP_8)))) 
-in (let TMP_10 \def (Flat f) in (let TMP_11 \def (CHead c TMP_10 v) in (let 
-H5 \def (eq_ind C c0 TMP_9 H1 TMP_11 H4) in (let TMP_12 \def (Bind Abbr) in 
-(let TMP_13 \def (CHead d TMP_12 u) in (let H_y \def (getl_gen_flat f c 
-TMP_13 v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3))))))))))))))))))) in 
-(pr2_ind TMP_5 TMP_6 TMP_14 y t1 t2 H0)))))) in (insert_eq C TMP_2 TMP_3 
-TMP_4 TMP_15 H))))))))))).
+(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c 
+(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2)) 
+(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: 
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2 
+c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: 
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4 
+H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: 
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: 
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: 
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def 
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead 
+c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) 
+v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0))) 
+H)))))).