include "basic_1/drop1/defs.ma".
-let rec drop1_ind (P: (PList \to (C \to (C \to Prop)))) (f: (\forall (c:
-C).(P PNil c c))) (f0: (\forall (c1: C).(\forall (c2: C).(\forall (h:
-nat).(\forall (d: nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds:
-PList).((drop1 hds c2 c3) \to ((P hds c2 c3) \to (P (PCons h d hds) c1
-c3))))))))))) (p: PList) (c: C) (c0: C) (d: drop1 p c c0) on d: P p c c0 \def
-match d with [(drop1_nil c1) \Rightarrow (f c1) | (drop1_cons c1 c2 h d0 d1
-c3 hds d2) \Rightarrow (f0 c1 c2 h d0 d1 c3 hds d2 ((drop1_ind P f f0) hds c2
-c3 d2))].
+implied rec lemma drop1_ind (P: (PList \to (C \to (C \to Prop)))) (f:
+(\forall (c: C).(P PNil c c))) (f0: (\forall (c1: C).(\forall (c2:
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c2) \to (\forall (c3:
+C).(\forall (hds: PList).((drop1 hds c2 c3) \to ((P hds c2 c3) \to (P (PCons
+h d hds) c1 c3))))))))))) (p: PList) (c: C) (c0: C) (d: drop1 p c c0) on d: P
+p c c0 \def match d with [(drop1_nil c1) \Rightarrow (f c1) | (drop1_cons c1
+c2 h d0 d1 c3 hds d2) \Rightarrow (f0 c1 c2 h d0 d1 c3 hds d2 ((drop1_ind P f
+f0) hds c2 c3 d2))].
-theorem drop1_gen_pnil:
+lemma drop1_gen_pnil:
\forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2)))
\def
\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq
\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H4) in
(False_ind (eq C c3 c5) H5)))))))))))) y c1 c2 H0))) H))).
-theorem drop1_gen_pcons:
+lemma drop1_gen_pcons:
\forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h:
nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda
(c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3))))))))