include "basic_1/pr1/defs.ma".
-let rec pr1_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t t))) (f0:
-(\forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pr1 t2
-t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr1 t t0) on p: P
-t t0 \def match p with [(pr1_refl t1) \Rightarrow (f t1) | (pr1_sing t2 t1 p0
-t3 p1) \Rightarrow (let TMP_1 \def ((pr1_ind P f f0) t2 t3 p1) in (f0 t2 t1
-p0 t3 p1 TMP_1))].
+implied rec lemma pr1_ind (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t
+t))) (f0: (\forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3:
+T).((pr1 t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr1 t
+t0) on p: P t t0 \def match p with [(pr1_refl t1) \Rightarrow (f t1) |
+(pr1_sing t2 t1 p0 t3 p1) \Rightarrow (f0 t2 t1 p0 t3 p1 ((pr1_ind P f f0) t2
+t3 p1))].