include "basic_1/clear/fwd.ma".
-theorem clear_clear:
+lemma clear_clear:
\forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
\def
- \lambda (c1: C).(let TMP_1 \def (\lambda (c: C).(\forall (c2: C).((clear c
-c2) \to (clear c2 c2)))) in (let TMP_3 \def (\lambda (n: nat).(\lambda (c2:
-C).(\lambda (H: (clear (CSort n) c2)).(let TMP_2 \def (clear c2 c2) in
-(clear_gen_sort c2 n H TMP_2))))) in (let TMP_13 \def (\lambda (c:
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to
+(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear
+(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c:
C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2
c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear
-(CHead c k t) c2)).(let TMP_4 \def (\lambda (k0: K).((clear (CHead c k0 t)
-c2) \to (clear c2 c2))) in (let TMP_10 \def (\lambda (b: B).(\lambda (H1:
-(clear (CHead c (Bind b) t) c2)).(let TMP_5 \def (Bind b) in (let TMP_6 \def
-(CHead c TMP_5 t) in (let TMP_7 \def (\lambda (c0: C).(clear c0 c0)) in (let
-TMP_8 \def (clear_bind b c t) in (let TMP_9 \def (clear_gen_bind b c c2 t H1)
-in (eq_ind_r C TMP_6 TMP_7 TMP_8 c2 TMP_9)))))))) in (let TMP_12 \def
-(\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(let TMP_11
-\def (clear_gen_flat f c c2 t H1) in (H c2 TMP_11)))) in (K_ind TMP_4 TMP_10
-TMP_12 k H0)))))))))) in (C_ind TMP_1 TMP_3 TMP_13 c1)))).
+(CHead c k t) c2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 t) c2) \to
+(clear c2 c2))) (\lambda (b: B).(\lambda (H1: (clear (CHead c (Bind b) t)
+c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0))
+(clear_bind b c t) c2 (clear_gen_bind b c c2 t H1)))) (\lambda (f:
+F).(\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c
+c2 t H1)))) k H0))))))) c1).
theorem clear_trans:
\forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c
c2) \to (clear c1 c2)))))
\def
- \lambda (c1: C).(let TMP_1 \def (\lambda (c: C).(\forall (c0: C).((clear c
-c0) \to (\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) in (let TMP_4
-\def (\lambda (n: nat).(\lambda (c: C).(\lambda (H: (clear (CSort n)
-c)).(\lambda (c2: C).(\lambda (_: (clear c c2)).(let TMP_2 \def (CSort n) in
-(let TMP_3 \def (clear TMP_2 c2) in (clear_gen_sort c n H TMP_3)))))))) in
-(let TMP_22 \def (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0)
-\to (\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k:
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to
+(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2:
+C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n)
+c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to
+(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k:
K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t)
-c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(let TMP_6 \def (\lambda
-(k0: K).((clear (CHead c k0 t) c0) \to (let TMP_5 \def (CHead c k0 t) in
-(clear TMP_5 c2)))) in (let TMP_18 \def (\lambda (b: B).(\lambda (H2: (clear
-(CHead c (Bind b) t) c0)).(let TMP_7 \def (\lambda (c3: C).(clear c3 c2)) in
-(let TMP_8 \def (Bind b) in (let TMP_9 \def (CHead c TMP_8 t) in (let TMP_10
-\def (clear_gen_bind b c c0 t H2) in (let H3 \def (eq_ind C c0 TMP_7 H1 TMP_9
-TMP_10) in (let TMP_11 \def (Bind b) in (let TMP_12 \def (CHead c TMP_11 t)
-in (let TMP_15 \def (\lambda (c3: C).(let TMP_13 \def (Bind b) in (let TMP_14
-\def (CHead c TMP_13 t) in (clear TMP_14 c3)))) in (let TMP_16 \def
-(clear_bind b c t) in (let TMP_17 \def (clear_gen_bind b c c2 t H3) in
-(eq_ind_r C TMP_12 TMP_15 TMP_16 c2 TMP_17))))))))))))) in (let TMP_21 \def
-(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(let TMP_19
-\def (clear_gen_flat f c c0 t H2) in (let TMP_20 \def (H c0 TMP_19 c2 H1) in
-(clear_flat c c2 TMP_20 f t))))) in (K_ind TMP_6 TMP_18 TMP_21 k
-H0)))))))))))) in (C_ind TMP_1 TMP_4 TMP_22 c1)))).
+c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).(K_ind (\lambda (k0:
+K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) (\lambda (b:
+B).(\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0
+(\lambda (c3: C).(clear c3 c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c
+c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead
+c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3)))))
+(\lambda (f: F).(\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c
+c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))) k H0))))))))) c1).
-theorem clear_ctail:
+lemma clear_ctail:
\forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1
(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k
u1 c1) (CHead (CTail k u1 c2) (Bind b) u2))))))))
\def
- \lambda (b: B).(\lambda (c1: C).(let TMP_5 \def (\lambda (c: C).(\forall
-(c2: C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k:
-K).(\forall (u1: T).(let TMP_1 \def (CTail k u1 c) in (let TMP_2 \def (CTail
-k u1 c2) in (let TMP_3 \def (Bind b) in (let TMP_4 \def (CHead TMP_2 TMP_3
-u2) in (clear TMP_1 TMP_4))))))))))) in (let TMP_34 \def (\lambda (n:
-nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: (clear (CSort n) (CHead
-c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: T).(let TMP_11 \def (\lambda
-(k0: K).(let TMP_6 \def (CSort n) in (let TMP_7 \def (CHead TMP_6 k0 u1) in
-(let TMP_8 \def (CTail k0 u1 c2) in (let TMP_9 \def (Bind b) in (let TMP_10
-\def (CHead TMP_8 TMP_9 u2) in (clear TMP_7 TMP_10))))))) in (let TMP_22 \def
-(\lambda (b0: B).(let TMP_12 \def (Bind b) in (let TMP_13 \def (CHead c2
-TMP_12 u2) in (let TMP_14 \def (CSort n) in (let TMP_15 \def (Bind b0) in
-(let TMP_16 \def (CHead TMP_14 TMP_15 u1) in (let TMP_17 \def (Bind b0) in
-(let TMP_18 \def (CTail TMP_17 u1 c2) in (let TMP_19 \def (Bind b) in (let
-TMP_20 \def (CHead TMP_18 TMP_19 u2) in (let TMP_21 \def (clear TMP_16
-TMP_20) in (clear_gen_sort TMP_13 n H TMP_21)))))))))))) in (let TMP_33 \def
-(\lambda (f: F).(let TMP_23 \def (Bind b) in (let TMP_24 \def (CHead c2
-TMP_23 u2) in (let TMP_25 \def (CSort n) in (let TMP_26 \def (Flat f) in (let
-TMP_27 \def (CHead TMP_25 TMP_26 u1) in (let TMP_28 \def (Flat f) in (let
-TMP_29 \def (CTail TMP_28 u1 c2) in (let TMP_30 \def (Bind b) in (let TMP_31
-\def (CHead TMP_29 TMP_30 u2) in (let TMP_32 \def (clear TMP_27 TMP_31) in
-(clear_gen_sort TMP_24 n H TMP_32)))))))))))) in (K_ind TMP_11 TMP_22 TMP_33
-k)))))))))) in (let TMP_102 \def (\lambda (c: C).(\lambda (H: ((\forall (c2:
+ \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k:
K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b)
-u2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2:
-T).(\lambda (H0: (clear (CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0:
-K).(\lambda (u1: T).(let TMP_40 \def (\lambda (k1: K).((clear (CHead c k1 t)
-(CHead c2 (Bind b) u2)) \to (let TMP_35 \def (CTail k0 u1 c) in (let TMP_36
-\def (CHead TMP_35 k1 t) in (let TMP_37 \def (CTail k0 u1 c2) in (let TMP_38
-\def (Bind b) in (let TMP_39 \def (CHead TMP_37 TMP_38 u2) in (clear TMP_36
-TMP_39)))))))) in (let TMP_92 \def (\lambda (b0: B).(\lambda (H1: (clear
-(CHead c (Bind b0) t) (CHead c2 (Bind b) u2))).(let TMP_41 \def (\lambda (e:
-C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0]))
-in (let TMP_42 \def (Bind b) in (let TMP_43 \def (CHead c2 TMP_42 u2) in (let
-TMP_44 \def (Bind b0) in (let TMP_45 \def (CHead c TMP_44 t) in (let TMP_46
-\def (Bind b) in (let TMP_47 \def (CHead c2 TMP_46 u2) in (let TMP_48 \def
-(clear_gen_bind b0 c TMP_47 t H1) in (let H2 \def (f_equal C C TMP_41 TMP_43
-TMP_45 TMP_48) in (let TMP_49 \def (\lambda (e: C).(match e with [(CSort _)
+u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H:
+(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1:
+T).(K_ind (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1
+c2) (Bind b) u2))) (\lambda (b0: B).(clear_gen_sort (CHead c2 (Bind b) u2) n
+H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind
+b) u2)))) (\lambda (f: F).(clear_gen_sort (CHead c2 (Bind b) u2) n H (clear
+(CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2))))
+k))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2:
+T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1:
+T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear
+(CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1:
+T).(K_ind (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to
+(clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2))))
+(\lambda (b0: B).(\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind
+b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _)
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind b) u2)
+(CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in
+((let H3 \def (f_equal C B (\lambda (e: C).(match e with [(CSort _)
\Rightarrow b | (CHead _ k1 _) \Rightarrow (match k1 with [(Bind b1)
-\Rightarrow b1 | (Flat _) \Rightarrow b])])) in (let TMP_50 \def (Bind b) in
-(let TMP_51 \def (CHead c2 TMP_50 u2) in (let TMP_52 \def (Bind b0) in (let
-TMP_53 \def (CHead c TMP_52 t) in (let TMP_54 \def (Bind b) in (let TMP_55
-\def (CHead c2 TMP_54 u2) in (let TMP_56 \def (clear_gen_bind b0 c TMP_55 t
-H1) in (let H3 \def (f_equal C B TMP_49 TMP_51 TMP_53 TMP_56) in (let TMP_57
-\def (\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 | (CHead _ _ t0)
-\Rightarrow t0])) in (let TMP_58 \def (Bind b) in (let TMP_59 \def (CHead c2
-TMP_58 u2) in (let TMP_60 \def (Bind b0) in (let TMP_61 \def (CHead c TMP_60
-t) in (let TMP_62 \def (Bind b) in (let TMP_63 \def (CHead c2 TMP_62 u2) in
-(let TMP_64 \def (clear_gen_bind b0 c TMP_63 t H1) in (let H4 \def (f_equal C
-T TMP_57 TMP_59 TMP_61 TMP_64) in (let TMP_90 \def (\lambda (H5: (eq B b
-b0)).(\lambda (H6: (eq C c2 c)).(let TMP_71 \def (\lambda (t0: T).(let TMP_65
-\def (CTail k0 u1 c) in (let TMP_66 \def (Bind b0) in (let TMP_67 \def (CHead
-TMP_65 TMP_66 t) in (let TMP_68 \def (CTail k0 u1 c2) in (let TMP_69 \def
-(Bind b) in (let TMP_70 \def (CHead TMP_68 TMP_69 t0) in (clear TMP_67
-TMP_70)))))))) in (let TMP_78 \def (\lambda (c0: C).(let TMP_72 \def (CTail
-k0 u1 c) in (let TMP_73 \def (Bind b0) in (let TMP_74 \def (CHead TMP_72
-TMP_73 t) in (let TMP_75 \def (CTail k0 u1 c0) in (let TMP_76 \def (Bind b)
-in (let TMP_77 \def (CHead TMP_75 TMP_76 t) in (clear TMP_74 TMP_77))))))))
-in (let TMP_85 \def (\lambda (b1: B).(let TMP_79 \def (CTail k0 u1 c) in (let
-TMP_80 \def (Bind b1) in (let TMP_81 \def (CHead TMP_79 TMP_80 t) in (let
-TMP_82 \def (CTail k0 u1 c) in (let TMP_83 \def (Bind b) in (let TMP_84 \def
-(CHead TMP_82 TMP_83 t) in (clear TMP_81 TMP_84)))))))) in (let TMP_86 \def
-(CTail k0 u1 c) in (let TMP_87 \def (clear_bind b TMP_86 t) in (let TMP_88
-\def (eq_ind B b TMP_85 TMP_87 b0 H5) in (let TMP_89 \def (eq_ind_r C c
-TMP_78 TMP_88 c2 H6) in (eq_ind_r T t TMP_71 TMP_89 u2 H4)))))))))) in (let
-TMP_91 \def (TMP_90 H3) in (TMP_91 H2)))))))))))))))))))))))))))))))) in (let
-TMP_101 \def (\lambda (f: F).(\lambda (H1: (clear (CHead c (Flat f) t) (CHead
-c2 (Bind b) u2))).(let TMP_93 \def (CTail k0 u1 c) in (let TMP_94 \def (CTail
-k0 u1 c2) in (let TMP_95 \def (Bind b) in (let TMP_96 \def (CHead TMP_94
-TMP_95 u2) in (let TMP_97 \def (Bind b) in (let TMP_98 \def (CHead c2 TMP_97
-u2) in (let TMP_99 \def (clear_gen_flat f c TMP_98 t H1) in (let TMP_100 \def
-(H c2 u2 TMP_99 k0 u1) in (clear_flat TMP_93 TMP_96 TMP_100 f t))))))))))) in
-(K_ind TMP_40 TMP_92 TMP_101 k H0))))))))))))) in (C_ind TMP_5 TMP_34 TMP_102
-c1))))).
+\Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead c
+(Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H4
+\def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u2 |
+(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t)
+(clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b
+b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead
+(CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r
+C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail
+k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0
+u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail
+k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1:
+(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1
+c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead
+c2 (Bind b) u2) t H1) k0 u1) f t))) k H0)))))))))) c1)).
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