include "basic_1/C/fwd.ma".
-let rec clear_ind (P: (C \to (C \to Prop))) (f: (\forall (b: B).(\forall (e:
-C).(\forall (u: T).(P (CHead e (Bind b) u) (CHead e (Bind b) u)))))) (f0:
-(\forall (e: C).(\forall (c: C).((clear e c) \to ((P e c) \to (\forall (f0:
-F).(\forall (u: T).(P (CHead e (Flat f0) u) c)))))))) (c: C) (c0: C) (c1:
-clear c c0) on c1: P c c0 \def match c1 with [(clear_bind b e u) \Rightarrow
-(f b e u) | (clear_flat e c2 c3 f1 u) \Rightarrow (f0 e c2 c3 ((clear_ind P f
-f0) e c2 c3) f1 u)].
+implied let rec clear_ind (P: (C \to (C \to Prop))) (f: (\forall (b:
+B).(\forall (e: C).(\forall (u: T).(P (CHead e (Bind b) u) (CHead e (Bind b)
+u)))))) (f0: (\forall (e: C).(\forall (c: C).((clear e c) \to ((P e c) \to
+(\forall (f0: F).(\forall (u: T).(P (CHead e (Flat f0) u) c)))))))) (c: C)
+(c0: C) (c1: clear c c0) on c1: P c c0 \def match c1 with [(clear_bind b e u)
+\Rightarrow (f b e u) | (clear_flat e c2 c3 f1 u) \Rightarrow (f0 e c2 c3
+((clear_ind P f f0) e c2 c3) f1 u)].
-theorem clear_gen_sort:
+lemma clear_gen_sort:
\forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P:
Prop).P)))
\def
[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n)
H3) in (False_ind P H4))))))))) y x H0))) H)))).
-theorem clear_gen_bind:
+lemma clear_gen_bind:
\forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
\def
b) u) H3) in (False_ind (eq C c (CHead e0 (Flat f) u0)) H4))))))))) y x H0)))
H))))).
-theorem clear_gen_flat:
+lemma clear_gen_flat:
\forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
(CHead e (Flat f) u) x) \to (clear e x)))))
\def
(\lambda (c0: C).(clear c0 c)) H1 e H8) in H10))))) H5)) H4))))))))) y x
H0))) H))))).
-theorem clear_gen_flat_r:
+lemma clear_gen_flat_r:
\forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x
(CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
\def
(c0: C).(clear e0 c0)) H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C
(CHead e (Flat f) u)))))))))))) x y H0))) H)))))).
-theorem clear_gen_all:
+lemma clear_gen_all:
\forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b:
B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))))))
\def
c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f
c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))) k H0 H1))))))))) c).
-theorem clear_cle:
+lemma clear_cle:
\forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
\def
\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to