Definition loc := nat.Definition val := option nat.Definition var := Ident.t.Definition env := var -> val.Definition heap := loc -> val.Definition loc := loc.Instance Join_loc : Join loc := @Join_equiv _.Instance Perm_loc : Perm_alg loc := _.Instance Sep_loc : Sep_alg loc := _.Instance Canc_loc : Canc_alg loc := _.Instance Join_nat : Join nat := @Join_discrete _.Instance Pos_nat : @Pos_alg nat (@Join_discrete _) := @psa_discrete _.Instance Canc_nat : @Canc_alg nat Join_nat.Definition val := val.Instance Join_val : Join val := Join_lower Join_nat.Instance Perm_val : Perm_alg val := _.Instance Sep_val : Sep_alg val := _.Instance Canc_val : Canc_alg val := @Canc_lower _ _ _ Canc_nat.

Lemma identity_None: @identity _ Join_val None.