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What are the ways in which one can add a decidable equivalence relation in a type system with undecidable type checking/extensional equality?

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    $\begingroup$ Some motivation and background for the question would be helpful. Please see the tips about how to write a better question section of our FAQ. $\endgroup$ – Kaveh Apr 14 '13 at 17:48
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Your question is very brief, so my answer will be as well:

Structure DecidableEquivalenceRelation :=
  {
    carrier :> Type ;
    rel :> carrier -> carrier -> Prop ;
    reflexive : forall x, rel x x ;
    symmetric : forall x y, rel x y -> rel y x ;
    transitive : forall x y z, rel x y -> rel y z -> rel x y ;
    decidable : forall x y, (rel x y) + ~ (rel x y)
  }.
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