include "basic_1/wcpr0/defs.ma".
-let rec wcpr0_ind (P: (C \to (C \to Prop))) (f: (\forall (c: C).(P c c)))
-(f0: (\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to ((P c1 c2) \to
-(\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(P (CHead
-c1 k u1) (CHead c2 k u2))))))))))) (c: C) (c0: C) (w: wcpr0 c c0) on w: P c
-c0 \def match w with [(wcpr0_refl c1) \Rightarrow (f c1) | (wcpr0_comp c1 c2
-w0 u1 u2 p k) \Rightarrow (f0 c1 c2 w0 ((wcpr0_ind P f f0) c1 c2 w0) u1 u2 p
-k)].
+implied rec lemma wcpr0_ind (P: (C \to (C \to Prop))) (f: (\forall (c: C).(P
+c c))) (f0: (\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to ((P c1 c2)
+\to (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(P
+(CHead c1 k u1) (CHead c2 k u2))))))))))) (c: C) (c0: C) (w: wcpr0 c c0) on
+w: P c c0 \def match w with [(wcpr0_refl c1) \Rightarrow (f c1) | (wcpr0_comp
+c1 c2 w0 u1 u2 p k) \Rightarrow (f0 c1 c2 w0 ((wcpr0_ind P f f0) c1 c2 w0) u1
+u2 p k)].
-theorem wcpr0_gen_sort:
+lemma wcpr0_gen_sort:
\forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort
n))))
\def
| (CHead _ _ _) \Rightarrow True])) I (CSort n) H4) in (False_ind (eq C
(CHead c2 k u2) (CHead c1 k u1)) H5))))))))))) y x H0))) H))).
-theorem wcpr0_gen_head:
+lemma wcpr0_gen_head:
\forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0
(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2:
C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: