include "basic_1/getl/props.ma".
-theorem getl_dec:
+lemma getl_dec:
\forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda
(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d:
C).((getl i c d) \to (\forall (P: Prop).P)))))
\def
- \lambda (c: C).(let TMP_6 \def (\lambda (c0: C).(\forall (i: nat).(let TMP_3
-\def (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(let TMP_1 \def (Bind b)
-in (let TMP_2 \def (CHead e TMP_1 v) in (getl i c0 TMP_2)))))) in (let TMP_4
-\def (ex_3 C B T TMP_3) in (let TMP_5 \def (\forall (d: C).((getl i c0 d) \to
-(\forall (P: Prop).P))) in (or TMP_4 TMP_5)))))) in (let TMP_14 \def (\lambda
-(n: nat).(\lambda (i: nat).(let TMP_10 \def (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(let TMP_7 \def (CSort n) in (let TMP_8 \def (Bind b) in
-(let TMP_9 \def (CHead e TMP_8 v) in (getl i TMP_7 TMP_9))))))) in (let
-TMP_11 \def (ex_3 C B T TMP_10) in (let TMP_12 \def (\forall (d: C).((getl i
-(CSort n) d) \to (\forall (P: Prop).P))) in (let TMP_13 \def (\lambda (d:
-C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P: Prop).(getl_gen_sort n i d
-H P)))) in (or_intror TMP_11 TMP_12 TMP_13))))))) in (let TMP_159 \def
-(\lambda (c0: C).(\lambda (H: ((\forall (i: nat).(or (ex_3 C B T (\lambda (e:
-C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) v))))))
-(\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (i: nat).(let TMP_21 \def (\lambda (n: nat).(let
-TMP_18 \def (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(let TMP_15 \def
-(CHead c0 k t) in (let TMP_16 \def (Bind b) in (let TMP_17 \def (CHead e
-TMP_16 v) in (getl n TMP_15 TMP_17))))))) in (let TMP_19 \def (ex_3 C B T
-TMP_18) in (let TMP_20 \def (\forall (d: C).((getl n (CHead c0 k t) d) \to
-(\forall (P: Prop).P))) in (or TMP_19 TMP_20))))) in (let TMP_28 \def
-(\lambda (k0: K).(let TMP_25 \def (\lambda (e: C).(\lambda (b: B).(\lambda
-(v: T).(let TMP_22 \def (CHead c0 k0 t) in (let TMP_23 \def (Bind b) in (let
-TMP_24 \def (CHead e TMP_23 v) in (getl O TMP_22 TMP_24))))))) in (let TMP_26
-\def (ex_3 C B T TMP_25) in (let TMP_27 \def (\forall (d: C).((getl O (CHead
-c0 k0 t) d) \to (\forall (P: Prop).P))) in (or TMP_26 TMP_27))))) in (let
-TMP_43 \def (\lambda (b: B).(let TMP_33 \def (\lambda (e: C).(\lambda (b0:
-B).(\lambda (v: T).(let TMP_29 \def (Bind b) in (let TMP_30 \def (CHead c0
-TMP_29 t) in (let TMP_31 \def (Bind b0) in (let TMP_32 \def (CHead e TMP_31
-v) in (getl O TMP_30 TMP_32)))))))) in (let TMP_34 \def (ex_3 C B T TMP_33)
-in (let TMP_35 \def (\forall (d: C).((getl O (CHead c0 (Bind b) t) d) \to
-(\forall (P: Prop).P))) in (let TMP_40 \def (\lambda (e: C).(\lambda (b0:
-B).(\lambda (v: T).(let TMP_36 \def (Bind b) in (let TMP_37 \def (CHead c0
-TMP_36 t) in (let TMP_38 \def (Bind b0) in (let TMP_39 \def (CHead e TMP_38
-v) in (getl O TMP_37 TMP_39)))))))) in (let TMP_41 \def (getl_refl b c0 t) in
-(let TMP_42 \def (ex_3_intro C B T TMP_40 c0 b t TMP_41) in (or_introl TMP_34
-TMP_35 TMP_42)))))))) in (let TMP_101 \def (\lambda (f: F).(let H_x \def (H
-O) in (let H0 \def H_x in (let TMP_46 \def (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(let TMP_44 \def (Bind b) in (let TMP_45 \def (CHead e
-TMP_44 v) in (getl O c0 TMP_45)))))) in (let TMP_47 \def (ex_3 C B T TMP_46)
-in (let TMP_48 \def (\forall (d: C).((getl O c0 d) \to (\forall (P:
-Prop).P))) in (let TMP_53 \def (\lambda (e: C).(\lambda (b: B).(\lambda (v:
-T).(let TMP_49 \def (Flat f) in (let TMP_50 \def (CHead c0 TMP_49 t) in (let
-TMP_51 \def (Bind b) in (let TMP_52 \def (CHead e TMP_51 v) in (getl O TMP_50
-TMP_52)))))))) in (let TMP_54 \def (ex_3 C B T TMP_53) in (let TMP_55 \def
-(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))
-in (let TMP_56 \def (or TMP_54 TMP_55) in (let TMP_85 \def (\lambda (H1:
-(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead
-e (Bind b) v))))))).(let TMP_59 \def (\lambda (e: C).(\lambda (b: B).(\lambda
-(v: T).(let TMP_57 \def (Bind b) in (let TMP_58 \def (CHead e TMP_57 v) in
-(getl O c0 TMP_58)))))) in (let TMP_64 \def (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(let TMP_60 \def (Flat f) in (let TMP_61 \def (CHead c0
-TMP_60 t) in (let TMP_62 \def (Bind b) in (let TMP_63 \def (CHead e TMP_62 v)
-in (getl O TMP_61 TMP_63)))))))) in (let TMP_65 \def (ex_3 C B T TMP_64) in
-(let TMP_66 \def (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to
-(\forall (P: Prop).P))) in (let TMP_67 \def (or TMP_65 TMP_66) in (let TMP_84
-\def (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O
-c0 (CHead x0 (Bind x1) x2))).(let TMP_72 \def (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(let TMP_68 \def (Flat f) in (let TMP_69 \def (CHead c0
-TMP_68 t) in (let TMP_70 \def (Bind b) in (let TMP_71 \def (CHead e TMP_70 v)
-in (getl O TMP_69 TMP_71)))))))) in (let TMP_73 \def (ex_3 C B T TMP_72) in
-(let TMP_74 \def (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to
-(\forall (P: Prop).P))) in (let TMP_79 \def (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(let TMP_75 \def (Flat f) in (let TMP_76 \def (CHead c0
-TMP_75 t) in (let TMP_77 \def (Bind b) in (let TMP_78 \def (CHead e TMP_77 v)
-in (getl O TMP_76 TMP_78)))))))) in (let TMP_80 \def (Bind x1) in (let TMP_81
-\def (CHead x0 TMP_80 x2) in (let TMP_82 \def (getl_flat c0 TMP_81 O H2 f t)
-in (let TMP_83 \def (ex_3_intro C B T TMP_79 x0 x1 x2 TMP_82) in (or_introl
-TMP_73 TMP_74 TMP_83))))))))))))) in (ex_3_ind C B T TMP_59 TMP_67 TMP_84
-H1)))))))) in (let TMP_100 \def (\lambda (H1: ((\forall (d: C).((getl O c0 d)
-\to (\forall (P: Prop).P))))).(let TMP_90 \def (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(let TMP_86 \def (Flat f) in (let TMP_87 \def (CHead c0
-TMP_86 t) in (let TMP_88 \def (Bind b) in (let TMP_89 \def (CHead e TMP_88 v)
-in (getl O TMP_87 TMP_89)))))))) in (let TMP_91 \def (ex_3 C B T TMP_90) in
-(let TMP_92 \def (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to
-(\forall (P: Prop).P))) in (let TMP_99 \def (\lambda (d: C).(\lambda (H2:
-(getl O (CHead c0 (Flat f) t) d)).(\lambda (P: Prop).(let TMP_93 \def
-(drop_refl c0) in (let TMP_94 \def (Flat f) in (let TMP_95 \def (CHead c0
-TMP_94 t) in (let TMP_96 \def (getl_gen_O TMP_95 d H2) in (let TMP_97 \def
-(clear_gen_flat f c0 d t TMP_96) in (let TMP_98 \def (getl_intro O c0 d c0
-TMP_93 TMP_97) in (H1 d TMP_98 P)))))))))) in (or_intror TMP_91 TMP_92
-TMP_99)))))) in (or_ind TMP_47 TMP_48 TMP_56 TMP_85 TMP_100 H0)))))))))))))
-in (let TMP_102 \def (K_ind TMP_28 TMP_43 TMP_101 k) in (let TMP_158 \def
-(\lambda (n: nat).(\lambda (_: (or (ex_3 C B T (\lambda (e: C).(\lambda (b:
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b)
+v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P))))))
+(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e:
+C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b)
+v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P)))
+(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P:
+Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall
+(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
+T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to
+(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i:
+nat).(nat_ind (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b:
B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall
-(d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))).(let TMP_103
-\def (r k n) in (let H_x \def (H TMP_103) in (let H1 \def H_x in (let TMP_107
-\def (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(let TMP_104 \def (r k
-n) in (let TMP_105 \def (Bind b) in (let TMP_106 \def (CHead e TMP_105 v) in
-(getl TMP_104 c0 TMP_106))))))) in (let TMP_108 \def (ex_3 C B T TMP_107) in
-(let TMP_109 \def (\forall (d: C).((getl (r k n) c0 d) \to (\forall (P:
-Prop).P))) in (let TMP_114 \def (\lambda (e: C).(\lambda (b: B).(\lambda (v:
-T).(let TMP_110 \def (S n) in (let TMP_111 \def (CHead c0 k t) in (let
-TMP_112 \def (Bind b) in (let TMP_113 \def (CHead e TMP_112 v) in (getl
-TMP_110 TMP_111 TMP_113)))))))) in (let TMP_115 \def (ex_3 C B T TMP_114) in
-(let TMP_116 \def (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall
-(P: Prop).P))) in (let TMP_117 \def (or TMP_115 TMP_116) in (let TMP_147 \def
-(\lambda (H2: (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
-T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(let TMP_121 \def (\lambda (e:
-C).(\lambda (b: B).(\lambda (v: T).(let TMP_118 \def (r k n) in (let TMP_119
-\def (Bind b) in (let TMP_120 \def (CHead e TMP_119 v) in (getl TMP_118 c0
-TMP_120))))))) in (let TMP_126 \def (\lambda (e: C).(\lambda (b: B).(\lambda
-(v: T).(let TMP_122 \def (S n) in (let TMP_123 \def (CHead c0 k t) in (let
-TMP_124 \def (Bind b) in (let TMP_125 \def (CHead e TMP_124 v) in (getl
-TMP_122 TMP_123 TMP_125)))))))) in (let TMP_127 \def (ex_3 C B T TMP_126) in
-(let TMP_128 \def (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall
-(P: Prop).P))) in (let TMP_129 \def (or TMP_127 TMP_128) in (let TMP_146 \def
-(\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H3: (getl (r k
-n) c0 (CHead x0 (Bind x1) x2))).(let TMP_134 \def (\lambda (e: C).(\lambda
-(b: B).(\lambda (v: T).(let TMP_130 \def (S n) in (let TMP_131 \def (CHead c0
-k t) in (let TMP_132 \def (Bind b) in (let TMP_133 \def (CHead e TMP_132 v)
-in (getl TMP_130 TMP_131 TMP_133)))))))) in (let TMP_135 \def (ex_3 C B T
-TMP_134) in (let TMP_136 \def (\forall (d: C).((getl (S n) (CHead c0 k t) d)
-\to (\forall (P: Prop).P))) in (let TMP_141 \def (\lambda (e: C).(\lambda (b:
-B).(\lambda (v: T).(let TMP_137 \def (S n) in (let TMP_138 \def (CHead c0 k
-t) in (let TMP_139 \def (Bind b) in (let TMP_140 \def (CHead e TMP_139 v) in
-(getl TMP_137 TMP_138 TMP_140)))))))) in (let TMP_142 \def (Bind x1) in (let
-TMP_143 \def (CHead x0 TMP_142 x2) in (let TMP_144 \def (getl_head k n c0
-TMP_143 H3 t) in (let TMP_145 \def (ex_3_intro C B T TMP_141 x0 x1 x2
-TMP_144) in (or_introl TMP_135 TMP_136 TMP_145))))))))))))) in (ex_3_ind C B
-T TMP_121 TMP_129 TMP_146 H2)))))))) in (let TMP_157 \def (\lambda (H2:
-((\forall (d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))))).(let
-TMP_152 \def (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(let TMP_148
-\def (S n) in (let TMP_149 \def (CHead c0 k t) in (let TMP_150 \def (Bind b)
-in (let TMP_151 \def (CHead e TMP_150 v) in (getl TMP_148 TMP_149
-TMP_151)))))))) in (let TMP_153 \def (ex_3 C B T TMP_152) in (let TMP_154
-\def (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P:
-Prop).P))) in (let TMP_156 \def (\lambda (d: C).(\lambda (H3: (getl (S n)
-(CHead c0 k t) d)).(\lambda (P: Prop).(let TMP_155 \def (getl_gen_S k c0 d t
-n H3) in (H2 d TMP_155 P))))) in (or_intror TMP_153 TMP_154 TMP_156)))))) in
-(or_ind TMP_108 TMP_109 TMP_117 TMP_147 TMP_157 H1))))))))))))))) in (nat_ind
-TMP_21 TMP_102 TMP_158 i)))))))))))) in (C_ind TMP_6 TMP_14 TMP_159 c)))).
+(d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))) (K_ind
+(\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
+T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O
+(CHead c0 k0 t) d) \to (\forall (P: Prop).P))))) (\lambda (b: B).(or_introl
+(ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead
+c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0
+(Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e:
+C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e
+(Bind b0) v))))) c0 b t (getl_refl b c0 t)))) (\lambda (f: F).(let H_x \def
+(H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b:
+B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e:
+C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e
+(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to
+(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda
+(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b)
+v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl
+O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O
+(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0:
+C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind
+x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
+T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro
+C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat
+f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2)
+O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to
+(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b:
+B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v))))))
+(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))
+(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P:
+Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t
+(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))) k) (\lambda (n:
+nat).(\lambda (_: (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda
+(v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))).(let H_x \def (H
+(r k n)) in (let H1 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda
+(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall
+(d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t)
+(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to
+(\forall (P: Prop).P)))) (\lambda (H2: (ex_3 C B T (\lambda (e: C).(\lambda
+(b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind
+C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead
+e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda
+(v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0:
+C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H3: (getl (r k n) c0 (CHead x0
+(Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b:
+B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v))))))
+(\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))
+(ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n)
+(CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0
+(Bind x1) x2) H3 t))))))) H2)) (\lambda (H2: ((\forall (d: C).((getl (r k n)
+c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e:
+C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind
+b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P:
+Prop).P))) (\lambda (d: C).(\lambda (H3: (getl (S n) (CHead c0 k t)
+d)).(\lambda (P: Prop).(H2 d (getl_gen_S k c0 d t n H3) P)))))) H1)))))
+i)))))) c).
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