Section pcp_hand_dec.
Variable (X : Type) (lc : list (list X * list X)).
Notation pcp_hand := (fun s t => lc ⊳ s∕t).
(* Any hand is either a card or of the for x++p/y++q where x/y is a non-void card and p/q is a hand *)

Lemma pcp_hand_inv p q :
lc ⊳ p∕q -> In (p,q) lc
\/ exists x y p' q', In (x,y) lc /\ lc ⊳ p'∕q'
/\ p = x++p' /\ q = y++q'
/\  (x <> nil /\ y = nil
\/ x = nil /\ y <> nil
\/ x <> nil /\ y <> nil ).