author  haftmann 
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(* Title: HOL/Tools/inductive_set.ML 
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Author: Stefan Berghofer, TU Muenchen 
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Wrapper for defining inductive sets using package for inductive predicates, 
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including infrastructure for converting between predicates and sets. 
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*) 
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signature INDUCTIVE_SET = 
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sig 
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val to_set_att: thm list > attribute 
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val to_pred_att: thm list > attribute 
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val to_pred : thm list > Context.generic > thm > thm 
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val pred_set_conv_att: attribute 
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val add_inductive_i: 
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Inductive.inductive_flags > 
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((binding * typ) * mixfix) list > 
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type Attrib.binding abbreviates Name.binding without attributes;
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(string * typ) list > 
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type Attrib.binding abbreviates Name.binding without attributes;
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(Attrib.binding * term) list > thm list > 
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local_theory > Inductive.inductive_result * local_theory 
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type Attrib.binding abbreviates Name.binding without attributes;
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val add_inductive: bool > bool > 
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(binding * string option * mixfix) list > 
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(binding * string option * mixfix) list > 

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(Attrib.binding * string) list > (Facts.ref * Token.src list) list > 
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local_theory > Inductive.inductive_result * local_theory 
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val mono_add: attribute 
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val mono_del: attribute 
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end; 
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structure Inductive_Set: INDUCTIVE_SET = 
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struct 
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(***********************************************************************************) 
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(* simplifies (%x y. (x, y) : S & P x y) to (%x y. (x, y) : S Int {(x, y). P x y}) *) 
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(* and (%x y. (x, y) : S  P x y) to (%x y. (x, y) : S Un {(x, y). P x y}) *) 
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(* used for converting "strong" (co)induction rules *) 
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(***********************************************************************************) 
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val anyt = Free ("t", TFree ("'t", [])); 
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fun strong_ind_simproc tab = 
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Simplifier.make_simproc @{context} "strong_ind" 
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{lhss = [@{term "x::'a::{}"}], 

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proc = fn _ => fn ctxt => fn ct => 

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let 

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fun close p t f = 

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let val vs = Term.add_vars t [] 

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in Thm.instantiate' [] (rev (map (SOME o Thm.cterm_of ctxt o Var) vs)) 

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(p (fold (Logic.all o Var) vs t) f) 

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end; 

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fun mkop @{const_name HOL.conj} T x = 

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SOME (Const (@{const_name Lattices.inf}, T > T > T), x) 

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 mkop @{const_name HOL.disj} T x = 

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SOME (Const (@{const_name Lattices.sup}, T > T > T), x) 

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 mkop _ _ _ = NONE; 

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fun mk_collect p T t = 

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let val U = HOLogic.dest_setT T 

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in HOLogic.Collect_const U $ 

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HOLogic.mk_psplits (HOLogic.flat_tuple_paths p) U HOLogic.boolT t 

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end; 

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fun decomp (Const (s, _) $ ((m as Const (@{const_name Set.member}, 

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Type (_, [_, Type (_, [T, _])]))) $ p $ S) $ u) = 

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mkop s T (m, p, S, mk_collect p T (head_of u)) 

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 decomp (Const (s, _) $ u $ ((m as Const (@{const_name Set.member}, 

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Type (_, [_, Type (_, [T, _])]))) $ p $ S)) = 

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mkop s T (m, p, mk_collect p T (head_of u), S) 

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 decomp _ = NONE; 

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val simp = 

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full_simp_tac 

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(put_simpset HOL_basic_ss ctxt addsimps [mem_Collect_eq, @{thm split_conv}]) 1; 

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fun mk_rew t = (case strip_abs_vars t of 

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[] => NONE 

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 xs => (case decomp (strip_abs_body t) of 

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NONE => NONE 

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 SOME (bop, (m, p, S, S')) => 

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SOME (close (Goal.prove ctxt [] []) 

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(Logic.mk_equals (t, fold_rev Term.abs xs (m $ p $ (bop $ S $ S')))) 

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(K (EVERY 

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[resolve_tac ctxt [eq_reflection] 1, 

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REPEAT (resolve_tac ctxt @{thms ext} 1), 

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resolve_tac ctxt [iffI] 1, 

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EVERY [eresolve_tac ctxt [conjE] 1, resolve_tac ctxt [IntI] 1, simp, simp, 

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eresolve_tac ctxt [IntE] 1, resolve_tac ctxt [conjI] 1, simp, simp] ORELSE 

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EVERY [eresolve_tac ctxt [disjE] 1, resolve_tac ctxt [UnI1] 1, simp, 

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resolve_tac ctxt [UnI2] 1, simp, 

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eresolve_tac ctxt [UnE] 1, resolve_tac ctxt [disjI1] 1, simp, 

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resolve_tac ctxt [disjI2] 1, simp]]))) 

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handle ERROR _ => NONE)) 

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in 

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(case strip_comb (Thm.term_of ct) of 

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(h as Const (name, _), ts) => 

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if Symtab.defined tab name then 

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let val rews = map mk_rew ts 

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in 

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if forall is_none rews then NONE 

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else SOME (fold (fn th1 => fn th2 => Thm.combination th2 th1) 

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(map2 (fn SOME r => K r  NONE => Thm.reflexive o Thm.cterm_of ctxt) 

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rews ts) (Thm.reflexive (Thm.cterm_of ctxt h))) 

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end 

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else NONE 

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 _ => NONE) 

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end, 

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identifier = []}; 

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(* only eta contract terms occurring as arguments of functions satisfying p *) 
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fun eta_contract p = 
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let 
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fun eta b (Abs (a, T, body)) = 
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(case eta b body of 
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body' as (f $ Bound 0) => 
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if Term.is_dependent f orelse not b then Abs (a, T, body') 
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else incr_boundvars ~1 f 
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 body' => Abs (a, T, body')) 
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 eta b (t $ u) = eta b t $ eta (p (head_of t)) u 
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 eta b t = t 
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in eta false end; 
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fun eta_contract_thm ctxt p = 
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Conv.fconv_rule (Conv.then_conv (Thm.beta_conversion true, fn ct => 
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Thm.transitive (Thm.eta_conversion ct) 
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(Thm.symmetric (Thm.eta_conversion (Thm.cterm_of ctxt (eta_contract p (Thm.term_of ct))))))); 
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(***********************************************************) 
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(* rules for converting between predicate and set notation *) 
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(* *) 
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(* rules for converting predicates to sets have the form *) 
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(* P (%x y. (x, y) : s) = (%x y. (x, y) : S s) *) 
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(* *) 
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(* rules for converting sets to predicates have the form *) 
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(* S {(x, y). p x y} = {(x, y). P p x y} *) 
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(* *) 
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(* where s and p are parameters *) 
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(***********************************************************) 
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structure Data = Generic_Data 
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( 
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type T = 
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{(* rules for converting predicates to sets *) 
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to_set_simps: thm list, 
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(* rules for converting sets to predicates *) 
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to_pred_simps: thm list, 
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(* arities of functions of type t set => ... => u set *) 
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set_arities: (typ * (int list list option list * int list list option)) list Symtab.table, 
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(* arities of functions of type (t => ... => bool) => u => ... => bool *) 
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pred_arities: (typ * (int list list option list * int list list option)) list Symtab.table}; 
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val empty = {to_set_simps = [], to_pred_simps = [], 
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set_arities = Symtab.empty, pred_arities = Symtab.empty}; 
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val extend = I; 
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fun merge 
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({to_set_simps = to_set_simps1, to_pred_simps = to_pred_simps1, 
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set_arities = set_arities1, pred_arities = pred_arities1}, 
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{to_set_simps = to_set_simps2, to_pred_simps = to_pred_simps2, 
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set_arities = set_arities2, pred_arities = pred_arities2}) : T = 
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{to_set_simps = Thm.merge_thms (to_set_simps1, to_set_simps2), 
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to_pred_simps = Thm.merge_thms (to_pred_simps1, to_pred_simps2), 
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set_arities = Symtab.merge_list (op =) (set_arities1, set_arities2), 
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pred_arities = Symtab.merge_list (op =) (pred_arities1, pred_arities2)}; 
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); 
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fun name_type_of (Free p) = SOME p 
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 name_type_of (Const p) = SOME p 
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 name_type_of _ = NONE; 
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fun map_type f (Free (s, T)) = Free (s, f T) 
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 map_type f (Var (ixn, T)) = Var (ixn, f T) 
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 map_type f _ = error "map_type"; 
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fun find_most_specific is_inst f eq xs T = 
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find_first (fn U => is_inst (T, f U) 
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andalso forall (fn U' => eq (f U, f U') orelse not 
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(is_inst (T, f U') andalso is_inst (f U', f U))) 
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xs) xs; 
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fun lookup_arity thy arities (s, T) = case Symtab.lookup arities s of 
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NONE => NONE 
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 SOME xs => find_most_specific (Sign.typ_instance thy) fst (op =) xs T; 
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fun lookup_rule thy f rules = find_most_specific 
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(swap #> Pattern.matches thy) (f #> fst) (op aconv) rules; 
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fun infer_arities thy arities (optf, t) fs = case strip_comb t of 
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(Abs (_, _, u), []) => infer_arities thy arities (NONE, u) fs 
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 (Abs _, _) => infer_arities thy arities (NONE, Envir.beta_norm t) fs 
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 (u, ts) => (case Option.map (lookup_arity thy arities) (name_type_of u) of 
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SOME (SOME (_, (arity, _))) => 
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(fold (infer_arities thy arities) (arity ~~ List.take (ts, length arity)) fs 
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handle General.Subscript => error "infer_arities: bad term") 
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 _ => fold (infer_arities thy arities) (map (pair NONE) ts) 
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(case optf of 
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NONE => fs 
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 SOME f => AList.update op = (u, the_default f 
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(Option.map (fn g => inter (op =) g f) (AList.lookup op = fs u))) fs)); 
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(**************************************************************) 
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(* derive the to_pred equation from the to_set equation *) 
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(* *) 
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(* 1. instantiate each set parameter with {(x, y). p x y} *) 
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(* 2. apply %P. {(x, y). P x y} to both sides of the equation *) 
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(* 3. simplify *) 
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(**************************************************************) 
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fun mk_to_pred_inst ctxt fs = 
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map (fn (x, ps) => 
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let 
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val (Ts, T) = strip_type (fastype_of x); 
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val U = HOLogic.dest_setT T; 
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val x' = map_type 
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(K (Ts @ HOLogic.strip_ptupleT ps U > HOLogic.boolT)) x; 
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in 
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(dest_Var x, 
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Thm.cterm_of ctxt (fold_rev (Term.abs o pair "x") Ts 
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(HOLogic.Collect_const U $ 
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HOLogic.mk_psplits ps U HOLogic.boolT 
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(list_comb (x', map Bound (length Ts  1 downto 0)))))) 
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end) fs; 
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fun mk_to_pred_eq ctxt p fs optfs' T thm = 
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let 
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val insts = mk_to_pred_inst ctxt fs; 
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val thm' = Thm.instantiate ([], insts) thm; 
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val thm'' = 
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(case optfs' of 
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NONE => thm' RS sym 
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 SOME fs' => 
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let 
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val U = HOLogic.dest_setT (body_type T); 
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val Ts = HOLogic.strip_ptupleT fs' U; 
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val arg_cong' = Thm.incr_indexes (Thm.maxidx_of thm + 1) arg_cong; 
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val (Var (arg_cong_f, _), _) = arg_cong' > Thm.concl_of > 
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dest_comb > snd > strip_comb > snd > hd > dest_comb; 

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in 
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thm' RS (infer_instantiate ctxt [(arg_cong_f, 
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Thm.cterm_of ctxt (Abs ("P", Ts > HOLogic.boolT, 
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HOLogic.Collect_const U $ HOLogic.mk_psplits fs' U 
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HOLogic.boolT (Bound 0))))] arg_cong' RS sym) 
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end) 
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in 
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Simplifier.simplify (put_simpset HOL_basic_ss ctxt addsimps [mem_Collect_eq, @{thm split_conv}] 
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addsimprocs [@{simproc Collect_mem}]) thm'' > 
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zero_var_indexes > eta_contract_thm ctxt (equal p) 
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end; 
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(**** declare rules for converting predicates to sets ****) 
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exception Malformed of string; 
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fun add context thm (tab as {to_set_simps, to_pred_simps, set_arities, pred_arities}) = 
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(case Thm.prop_of thm of 
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Const (@{const_name Trueprop}, _) $ (Const (@{const_name HOL.eq}, Type (_, [T, _])) $ lhs $ rhs) => 
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(case body_type T of 
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@{typ bool} => 
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let 
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val thy = Context.theory_of context; 
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val ctxt = Context.proof_of context; 
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fun factors_of t fs = case strip_abs_body t of 
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Const (@{const_name Set.member}, _) $ u $ S => 
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if is_Free S orelse is_Var S then 
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let val ps = HOLogic.flat_tuple_paths u 
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in (SOME ps, (S, ps) :: fs) end 
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else (NONE, fs) 
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 _ => (NONE, fs); 
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val (h, ts) = strip_comb lhs 
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val (pfs, fs) = fold_map factors_of ts []; 
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val ((h', ts'), fs') = (case rhs of 
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Abs _ => (case strip_abs_body rhs of 
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Const (@{const_name Set.member}, _) $ u $ S => 
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(strip_comb S, SOME (HOLogic.flat_tuple_paths u)) 
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 _ => raise Malformed "member symbol on righthand side expected") 
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 _ => (strip_comb rhs, NONE)) 
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in 
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case (name_type_of h, name_type_of h') of 
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(SOME (s, T), SOME (s', T')) => 
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if exists (fn (U, _) => 
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Sign.typ_instance thy (T', U) andalso 
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Sign.typ_instance thy (U, T')) 
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(Symtab.lookup_list set_arities s') 
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then 
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(if Context_Position.is_really_visible ctxt then 
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warning ("Ignoring conversion rule for operator " ^ s') 
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else (); tab) 
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else 
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{to_set_simps = thm :: to_set_simps, 
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to_pred_simps = 
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mk_to_pred_eq ctxt h fs fs' T' thm :: to_pred_simps, 
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set_arities = Symtab.insert_list op = (s', 
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(T', (map (AList.lookup op = fs) ts', fs'))) set_arities, 
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pred_arities = Symtab.insert_list op = (s, 
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(T, (pfs, fs'))) pred_arities} 
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 _ => raise Malformed "set / predicate constant expected" 
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end 
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 _ => raise Malformed "equation between predicates expected") 
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 _ => raise Malformed "equation expected") 
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handle Malformed msg => 
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let 
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val ctxt = Context.proof_of context 
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val _ = 
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if Context_Position.is_really_visible ctxt then 
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warning ("Ignoring malformed set / predicate conversion rule: " ^ msg ^ 
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"\n" ^ Display.string_of_thm ctxt thm) 
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else (); 
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in tab end; 
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val pred_set_conv_att = Thm.declaration_attribute 
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(fn thm => fn ctxt => Data.map (add ctxt thm) ctxt); 
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(**** convert theorem in set notation to predicate notation ****) 
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fun is_pred tab t = 
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case Option.map (Symtab.lookup tab o fst) (name_type_of t) of 
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SOME (SOME _) => true  _ => false; 
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fun to_pred_simproc rules = 
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let val rules' = map mk_meta_eq rules 
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in 
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Simplifier.make_simproc @{context} "to_pred" 
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{lhss = [anyt], 

320 
proc = fn _ => fn ctxt => fn ct => 

321 
lookup_rule (Proof_Context.theory_of ctxt) 

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(Thm.prop_of #> Logic.dest_equals) rules' (Thm.term_of ct), 

323 
identifier = []} 

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end; 
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fun to_pred_proc thy rules t = 
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case lookup_rule thy I rules t of 

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NONE => NONE 
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 SOME (lhs, rhs) => 
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SOME (Envir.subst_term 
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(Pattern.match thy (lhs, t) (Vartab.empty, Vartab.empty)) rhs); 
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fun to_pred thms context thm = 
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let 
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val thy = Context.theory_of context; 
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val ctxt = Context.proof_of context; 
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val {to_pred_simps, set_arities, pred_arities, ...} = 
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fold (add context) thms (Data.get context); 
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val fs = filter (is_Var o fst) 
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(infer_arities thy set_arities (NONE, Thm.prop_of thm) []); 
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(* instantiate each set parameter with {(x, y). p x y} *) 
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val insts = mk_to_pred_inst ctxt fs 
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in 
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thm > 
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Thm.instantiate ([], insts) > 
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Simplifier.full_simplify (put_simpset HOL_basic_ss ctxt addsimprocs 
37136  347 
[to_pred_simproc (mem_Collect_eq :: @{thm split_conv} :: to_pred_simps)]) > 
60328  348 
eta_contract_thm ctxt (is_pred pred_arities) > 
33368  349 
Rule_Cases.save thm 
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end; 
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val to_pred_att = Thm.rule_attribute o to_pred; 
45979  353 

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(**** convert theorem in predicate notation to set notation ****) 
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fun to_set thms context thm = 
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let 
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val thy = Context.theory_of context; 
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val ctxt = Context.proof_of context; 
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val {to_set_simps, pred_arities, ...} = 
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fold (add context) thms (Data.get context); 
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val fs = filter (is_Var o fst) 
59582  364 
(infer_arities thy pred_arities (NONE, Thm.prop_of thm) []); 
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(* instantiate each predicate parameter with %x y. (x, y) : s *) 
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val insts = map (fn (x, ps) => 
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let 
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val Ts = binder_types (fastype_of x); 
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val l = length Ts; 
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val k = length ps; 
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val (Rs, Us) = chop (l  k  1) Ts; 
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val T = HOLogic.mk_ptupleT ps Us; 
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val x' = map_type (K (Rs > HOLogic.mk_setT T)) x 
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in 
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(dest_Var x, 
59642  376 
Thm.cterm_of ctxt (fold_rev (Term.abs o pair "x") Ts 
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(HOLogic.mk_mem (HOLogic.mk_ptuple ps T (map Bound (k downto 0)), 
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list_comb (x', map Bound (l  1 downto k + 1)))))) 
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end) fs; 
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in 
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thm > 
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Thm.instantiate ([], insts) > 
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Simplifier.full_simplify (put_simpset HOL_basic_ss ctxt addsimps to_set_simps 
56512  384 
addsimprocs [strong_ind_simproc pred_arities, @{simproc Collect_mem}]) > 
33368  385 
Rule_Cases.save thm 
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end; 
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val to_set_att = Thm.rule_attribute o to_set; 
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(**** definition of inductive sets ****) 
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29389  393 
fun add_ind_set_def 
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{quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono} 
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cs intros monos params cnames_syn lthy = 
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let 
42361  397 
val thy = Proof_Context.theory_of lthy; 
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val {set_arities, pred_arities, to_pred_simps, ...} = 
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Data.get (Context.Proof lthy); 
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fun infer (Abs (_, _, t)) = infer t 
37677  401 
 infer (Const (@{const_name Set.member}, _) $ t $ u) = 
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infer_arities thy set_arities (SOME (HOLogic.flat_tuple_paths t), u) 
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 infer (t $ u) = infer t #> infer u 
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 infer _ = I; 
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val new_arities = filter_out 
45979  406 
(fn (x as Free (_, T), _) => member (op =) params x andalso length (binder_types T) > 0 
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 _ => false) (fold (snd #> infer) intros []); 
33278  408 
val params' = map (fn x => 
409 
(case AList.lookup op = new_arities x of 

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SOME fs => 
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let 
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val T = HOLogic.dest_setT (fastype_of x); 
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val Ts = HOLogic.strip_ptupleT fs T; 
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val x' = map_type (K (Ts > HOLogic.boolT)) x 
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in 
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416 
(x, (x', 
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(HOLogic.Collect_const T $ 
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HOLogic.mk_psplits fs T HOLogic.boolT x', 
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fold_rev (Term.abs o pair "x") Ts 
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(HOLogic.mk_mem 
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(HOLogic.mk_ptuple fs T (map Bound (length fs downto 0)), x))))) 
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422 
end 
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 NONE => (x, (x, (x, x))))) params; 
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val (params1, (params2, params3)) = 
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params' > map snd > split_list > split_list; 
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426 
val paramTs = map fastype_of params; 
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427 

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428 
(* equations for converting sets to predicates *) 
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val ((cs', cs_info), eqns) = cs > map (fn c as Free (s, T) => 
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430 
let 
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val fs = the_default [] (AList.lookup op = new_arities c); 
45979  432 
val (Us, U) = strip_type T > apsnd HOLogic.dest_setT; 
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val _ = Us = paramTs orelse error (Pretty.string_of (Pretty.chunks 
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[Pretty.str "Argument types", 
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Pretty.block (Pretty.commas (map (Syntax.pretty_typ lthy) Us)), 
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436 
Pretty.str ("of " ^ s ^ " do not agree with types"), 
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437 
Pretty.block (Pretty.commas (map (Syntax.pretty_typ lthy) paramTs)), 
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438 
Pretty.str "of declared parameters"])); 
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439 
val Ts = HOLogic.strip_ptupleT fs U; 
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440 
val c' = Free (s ^ "p", 
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441 
map fastype_of params1 @ Ts > HOLogic.boolT) 
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442 
in 
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443 
((c', (fs, U, Ts)), 
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444 
(list_comb (c, params2), 
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445 
HOLogic.Collect_const U $ HOLogic.mk_psplits fs U HOLogic.boolT 
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446 
(list_comb (c', params1)))) 
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447 
end) > split_list >> split_list; 
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448 
val eqns' = eqns @ 
59582  449 
map (Thm.prop_of #> HOLogic.dest_Trueprop #> HOLogic.dest_eq) 
37136  450 
(mem_Collect_eq :: @{thm split_conv} :: to_pred_simps); 
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451 

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452 
(* predicate version of the introduction rules *) 
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453 
val intros' = 
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454 
map (fn (name_atts, t) => (name_atts, 
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455 
t > 
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456 
map_aterms (fn u => 
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457 
(case AList.lookup op = params' u of 
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458 
SOME (_, (u', _)) => u' 
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459 
 NONE => u)) > 
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460 
Pattern.rewrite_term thy [] [to_pred_proc thy eqns'] > 
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461 
eta_contract (member op = cs' orf is_pred pred_arities))) intros; 
30345  462 
val cnames_syn' = map (fn (b, _) => (Binding.suffix_name "p" b, NoSyn)) cnames_syn; 
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463 
val monos' = map (to_pred [] (Context.Proof lthy)) monos; 
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464 
val ({preds, intrs, elims, raw_induct, eqs, ...}, lthy1) = 
31723
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465 
Inductive.add_ind_def 
33669  466 
{quiet_mode = quiet_mode, verbose = verbose, alt_name = Binding.empty, 
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467 
coind = coind, no_elim = no_elim, no_ind = no_ind, skip_mono = skip_mono} 
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468 
cs' intros' monos' params1 cnames_syn' lthy; 
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469 

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470 
(* define inductive sets using previously defined predicates *) 
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471 
val (defs, lthy2) = lthy1 
33766
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472 
> fold_map Local_Theory.define 
46909  473 
(map (fn (((c, syn), (fs, U, _)), p) => ((c, syn), ((Thm.def_binding c, []), 
33278  474 
fold_rev lambda params (HOLogic.Collect_const U $ 
475 
HOLogic.mk_psplits fs U HOLogic.boolT (list_comb (p, params3)))))) 

476 
(cnames_syn ~~ cs_info ~~ preds)) 

59880
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477 
> Proof_Context.restore_naming lthy1; 
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478 

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479 
(* prove theorems for converting predicate to set notation *) 
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480 
val lthy3 = fold 
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481 
(fn (((p, c as Free (s, _)), (fs, U, Ts)), (_, (_, def))) => fn lthy => 
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482 
let val conv_thm = 
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483 
Goal.prove lthy (map (fst o dest_Free) params) [] 
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484 
(HOLogic.mk_Trueprop (HOLogic.mk_eq 
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485 
(list_comb (p, params3), 
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486 
fold_rev (Term.abs o pair "x") Ts 
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487 
(HOLogic.mk_mem (HOLogic.mk_ptuple fs U (map Bound (length fs downto 0)), 
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488 
list_comb (c, params)))))) 
59498
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489 
(K (REPEAT (resolve_tac lthy @{thms ext} 1) THEN 
58839  490 
simp_tac (put_simpset HOL_basic_ss lthy addsimps 
491 
[def, mem_Collect_eq, @{thm split_conv}]) 1)) 

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492 
in 
33671  493 
lthy > Local_Theory.note ((Binding.name (s ^ "p_" ^ s ^ "_eq"), 
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[Attrib.internal (K pred_set_conv_att)]), 
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[conv_thm]) > snd 
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end) (preds ~~ cs ~~ cs_info ~~ defs) lthy2; 
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497 

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498 
(* convert theorems to set notation *) 
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val rec_name = 
28965  500 
if Binding.is_empty alt_name then 
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Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn)) 
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else alt_name; 
33671  503 
val cnames = map (Local_Theory.full_name lthy3 o #1) cnames_syn; (* FIXME *) 
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val (intr_names, intr_atts) = split_list (map fst intros); 
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val raw_induct' = to_set [] (Context.Proof lthy3) raw_induct; 
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val (intrs', elims', eqs', induct, inducts, lthy4) = 
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507 
Inductive.declare_rules rec_name coind no_ind cnames (map fst defs) 
33459  508 
(map (to_set [] (Context.Proof lthy3)) intrs) intr_names intr_atts 
509 
(map (fn th => (to_set [] (Context.Proof lthy3) th, 

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map (fst o fst) (fst (Rule_Cases.get th)), 
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Rule_Cases.get_constraints th)) elims) 
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(map (to_set [] (Context.Proof lthy3)) eqs) raw_induct' lthy3; 
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513 
in 
35646  514 
({intrs = intrs', elims = elims', induct = induct, inducts = inducts, 
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raw_induct = raw_induct', preds = map fst defs, eqs = eqs'}, 
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516 
lthy4) 
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end; 
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518 

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val add_inductive_i = Inductive.gen_add_inductive_i add_ind_set_def; 
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val add_inductive = Inductive.gen_add_inductive add_ind_set_def; 
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521 

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fun mono_att att = 
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523 
Thm.declaration_attribute (fn thm => fn context => 
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Thm.attribute_declaration att (to_pred [] context thm) context); 
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525 

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val mono_add = mono_att Inductive.mono_add; 
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val mono_del = mono_att Inductive.mono_del; 
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528 

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529 

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530 
(** package setup **) 
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531 

56512  532 
(* attributes *) 
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533 

56512  534 
val _ = 
535 
Theory.setup 

536 
(Attrib.setup @{binding pred_set_conv} (Scan.succeed pred_set_conv_att) 

537 
"declare rules for converting between predicate and set notation" #> 

538 
Attrib.setup @{binding to_set} (Attrib.thms >> to_set_att) 

539 
"convert rule to set notation" #> 

540 
Attrib.setup @{binding to_pred} (Attrib.thms >> to_pred_att) 

541 
"convert rule to predicate notation" #> 

542 
Attrib.setup @{binding mono_set} (Attrib.add_del mono_add mono_del) 

543 
"declare of monotonicity rule for set operators"); 

30528  544 

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545 

56512  546 
(* commands *) 
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547 

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val ind_set_decl = Inductive.gen_ind_decl add_ind_set_def; 
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549 

24867  550 
val _ = 
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551 
Outer_Syntax.local_theory @{command_keyword inductive_set} "define inductive sets" 
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552 
(ind_set_decl false); 
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553 

24867  554 
val _ = 
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Outer_Syntax.local_theory @{command_keyword coinductive_set} "define coinductive sets" 
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556 
(ind_set_decl true); 
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557 

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558 
end; 