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Parameter ff: (n,m:nat)~n=m -> ~(S n)=(S m).
Parameter discr_r : (n:nat) ~(O=(S n)).
Parameter discr_l : (n:nat) ~((S n)=O).
Fixpoint eqdec [n:nat] : (m:nat) n=m \/ ~n=m :=
[m:nat]
<[n,m:nat] n=m \/ ~n=m>Cases n m of
O O => (or_introl ? ~O=O (refl_equal ? O))
| O (S x) => (or_intror O=(S x) ? (discr_r x))
| (S x) O => (or_intror ? ~(S x)=O (discr_l x))
| ((S x) as N) ((S y) as M) =>
<N=M\/~N=M>Cases (eqdec x y) of
(or_introl h) => (or_introl ? ~N=M (f_equal nat nat S x y h))
| (or_intror h) => (or_intror N=M ? (ff x y h))
end
end.
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