Replaced blast by fast in proof of parts_singleton, since blast looped because of the new encoding of sets.

Adapted to encoding of sets as predicates

Replaced forward proofs of existential statements by backward proofs to avoid problems with HO unification

Adapted functions mk_setT and dest_setT to encoding of sets as predicates.

- Explicitely passed pred_subset_eq and pred_equals_eq as an argument to the to_set and to_pred attributes, because it is no longer applied automatically - Manually applied predicate1I in proof of accp_subset, because it is no longer part of the claset - Replaced psubset_def by less_le

Deleted instantiation "set :: (type) itself".

- Function dec in Trancl_Tac must eta-contract relation before calling decr, since it is now a function and could therefore be in eta-expanded form - The trancl prover now does more eta-contraction itself, so eta-contraction is no longer necessary in Tranclp_tac.

- Now uses Orderings as parent theory - "'a set" is now just a type abbreviation for "'a => bool" - The instantiation "set :: (type) ord" and the definition of (p)subset is no longer needed, since it is subsumed by the order on functions and booleans. The derived theorems (p)subset_eq can be used as a replacement. - mem_Collect_eq and Collect_mem_eq can now be derived from the definitions of mem and Collect. - Replaced the instantiation "set :: (type) minus" by the two instantiations "fun :: (type, minus) minus" and "bool :: minus". The theorem set_diff_eq can be used as a replacement for the definition set_diff_def - Replaced the instantiation "set :: (type) uminus" by the two instantiations "fun :: (type, uminus) uminus" and "bool :: uminus". The theorem Compl_eq can be used as a replacement for the definition Compl_def. - Variable P in rule split_if must be instantiated manually in proof of split_if_mem2 due to problems with HO unification - Moved definition of dense linear orders and proofs about LEAST from Orderings to Set - Deleted code setup for sets

Deleted instance "set :: (type) power" and moved instance "fun :: (type, type) power" to the beginning of the theory

split_beta is now declared as monotonicity rule, to allow bounded quantifiers in introduction rules of inductive predicates.

- Added mem_def and predicate1I in some of the proofs - pred_equals_eq and pred_subset_eq are no longer used in the conversion between sets and predicates, because sets and predicates can no longer be distinguished

- Now imports Code_Setup, rather than Set and Fun, since the theorems about orderings are already needed in Set - Moved "Dense orders" section to Set, since it requires set notation. - The "Order on sets" section is no longer necessary, since it is subsumed by the order on functions and booleans. - Moved proofs of Least_mono and Least_equality to Set, since they require set notation. - In proof of "instance fun :: (type, order) order", use ext instead of expand_fun_eq, since the latter is not yet available. - predicate1I is no longer declared as introduction rule, since it interferes with subsetI