alias nat        /Coq/Init/Datatypes/nat.ind#1/1
alias eq         /Coq/Init/Logic/eq.ind#1/1
alias eq_ind     /Coq/Init/Logic/eq_ind.con
alias eqT        /Coq/Init/Logic_Type/eqT.ind#1/1
alias O          /Coq/Init/Datatypes/nat.ind#1/1/1
alias S          /Coq/Init/Datatypes/nat.ind#1/1/2
alias plus       /Coq/Init/Peano/plus.con
alias mult       /Coq/Init/Peano/mult.con
alias le         /Coq/Init/Peano/le.ind#1/1
alias lt         /Coq/Init/Peano/lt.con
alias not        /Coq/Init/Logic/not.con
alias and        /Coq/Init/Logic/and.ind#1/1
alias prod       /Coq/Init/Datatypes/prod.ind#1/1 
alias list       /Coq/Lists/PolyList/list.ind#1/1
alias AllS_assoc /Coq/Lists/TheoryList/AllS_assoc.ind#1/1
alias V          /Coq/Lists/PolyList/Lists/A.var
alias VA         /Coq/Lists/TheoryList/Lists/A.var
alias VB         /Coq/Lists/TheoryList/Lists/Assoc_sec/B.var

!A:Set.!B:Set.!P:!a:A.Prop.!l:list{V := (prod A B)}.
 !H:(AllS_assoc {VA := A ; VB := B} P l).
  (and
   (eq list{V := (prod A B)} l l)
   (eqT !n:A.Prop P P))

(* Intros; Elim H:

?1: (A,B:Set; P:(A->Prop); l:(list A*B))
     (AllS_assoc A B P l) -> (nil A*B)=(nil A*B)/\P==P
?2: (A,B:Set; P:(A->Prop); l:(list A*B))
     (AllS_assoc A B P l) ->
      (a:A; b:B; l0:(list A*B))
       (P a) -> (AllS_assoc A B P l0) -> l0=l0/\P==P ->
        (cons (a,b) l0)=(cons (a,b) l0)/\P==P
[A,B:Set; P:(A->Prop); l:(list A*B); H:(AllS_assoc A B P l)]
 (AllS_assoc_ind A B P [l0:(list A*B)]l0=l0/\P==P
  (?1 A B P l H) (?2 A B P l H) l H)

*)