Section Case2_.
Hypothesis Even_u : even u.
Remark Alt_v : alt b v.
End Case2_.
(* We recall from the prelude the definition of the conditional : Definition IF := [P,Q,R:Prop](P /\ Q) \/ ((~P) /\ R) Syntax IF "IF _ then _ else _" *)
End Context.
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(* Gilbreath's trick *)
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Theorem Gilbreath :
forall x : word,
even x ->
alternate x ->
forall u v : word,
conc u v x ->
forall w : word,
shuffle u v w -> IF opposite u v then paired w else paired (rotate w).