author  berghofe 
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permissions  rwrr 
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(* Title: HOL/Tools/Function/size.ML 
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Author: Stefan Berghofer, Florian Haftmann, TU Muenchen 
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Size functions for datatypes. 

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*) 

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signature SIZE = 

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sig 

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val size_thms: theory > string > thm list 

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val setup: theory > theory 

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end; 

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structure Size: SIZE = 

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struct 

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structure SizeData = Theory_Data 
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( 
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type T = (string * thm list) Symtab.table; 
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val empty = Symtab.empty; 
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val extend = I 

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fun merge data = Symtab.merge (K true) data; 
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); 
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val lookup_size = SizeData.get #> Symtab.lookup; 
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fun plus (t1, t2) = Const (@{const_name Groups.plus}, 
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HOLogic.natT > HOLogic.natT > HOLogic.natT) $ t1 $ t2; 
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fun size_of_type f g h (T as Type (s, Ts)) = 
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(case f s of 
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SOME t => SOME t 
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 NONE => (case g s of 
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SOME size_name => 
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SOME (list_comb (Const (size_name, 
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map (fn U => U > HOLogic.natT) Ts @ [T] > HOLogic.natT), 
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map (size_of_type' f g h) Ts)) 
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 NONE => NONE)) 
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 size_of_type f g h (TFree (s, _)) = h s 
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and size_of_type' f g h T = (case size_of_type f g h T of 
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NONE => Abs ("x", T, HOLogic.zero) 
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 SOME t => t); 
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fun is_poly thy (Datatype.DtType (name, dts)) = 
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(case lookup_size thy name of 
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NONE => false 
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 SOME _ => exists (is_poly thy) dts) 
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 is_poly _ _ = true; 
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fun constrs_of thy name = 
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let 
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val {descr, index, ...} = Datatype.the_info thy name 
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val SOME (_, _, constrs) = AList.lookup op = descr index 
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in constrs end; 
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val app = curry (list_comb o swap); 
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fun prove_size_thms (info : Datatype.info) new_type_names thy = 
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let 
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val {descr, rec_names, rec_rewrites, induct, ...} = info; 
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val l = length new_type_names; 
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val descr' = List.take (descr, l); 
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val (rec_names1, rec_names2) = chop l rec_names; 
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val recTs = Datatype_Aux.get_rec_types descr; 
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val (recTs1, recTs2) = chop l recTs; 
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val (_, (_, paramdts, _)) :: _ = descr; 
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val paramTs = map (Datatype_Aux.typ_of_dtyp descr) paramdts; 
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val ((param_size_fs, param_size_fTs), f_names) = paramTs > 
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map (fn T as TFree (s, _) => 
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let 
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val name = "f" ^ unprefix "'" s; 
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val U = T > HOLogic.natT 
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in 
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(((s, Free (name, U)), U), name) 
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end) > split_list >> split_list; 
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val param_size = AList.lookup op = param_size_fs; 
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val extra_rewrites = descr > map (#1 o snd) > distinct op = > 
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map_filter (Option.map snd o lookup_size thy) > flat; 
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val extra_size = Option.map fst o lookup_size thy; 
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val (((size_names, size_fns), def_names), def_names') = 
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recTs1 > map (fn T as Type (s, _) => 
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let 
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val s' = Long_Name.base_name s ^ "_size"; 
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val s'' = Sign.full_bname thy s'; 

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in 
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(s'', 
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(list_comb (Const (s'', param_size_fTs @ [T] > HOLogic.natT), 
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map snd param_size_fs), 
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(s' ^ "_def", s' ^ "_overloaded_def"))) 
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end) > split_list >> split_list >> split_list; 
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val overloaded_size_fns = map HOLogic.size_const recTs1; 
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(* instantiation for primrec combinator *) 
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fun size_of_constr b size_ofp ((_, cargs), (_, cargs')) = 
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let 
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val Ts = map (Datatype_Aux.typ_of_dtyp descr) cargs; 
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val k = length (filter Datatype_Aux.is_rec_type cargs); 
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val (ts, _, _) = fold_rev (fn ((dt, dt'), T) => fn (us, i, j) => 
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if Datatype_Aux.is_rec_type dt then (Bound i :: us, i + 1, j + 1) 
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else 
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(if b andalso is_poly thy dt' then 
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case size_of_type (K NONE) extra_size size_ofp T of 
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NONE => us  SOME sz => sz $ Bound j :: us 
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else us, i, j + 1)) 
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(cargs ~~ cargs' ~~ Ts) ([], 0, k); 
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val t = 
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if null ts andalso (not b orelse not (exists (is_poly thy) cargs')) 
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then HOLogic.zero 
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else foldl1 plus (ts @ [HOLogic.Suc_zero]) 
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in 
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fold_rev (fn T => fn t' => Abs ("x", T, t')) (Ts @ replicate k HOLogic.natT) t 
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end; 
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val fs = maps (fn (_, (name, _, constrs)) => 
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map (size_of_constr true param_size) (constrs ~~ constrs_of thy name)) descr; 
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val fs' = maps (fn (n, (name, _, constrs)) => 
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map (size_of_constr (l <= n) (K NONE)) (constrs ~~ constrs_of thy name)) descr; 
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val fTs = map fastype_of fs; 
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val (rec_combs1, rec_combs2) = chop l (map (fn (T, rec_name) => 
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Const (rec_name, fTs @ [T] > HOLogic.natT)) 
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(recTs ~~ rec_names)); 
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fun define_overloaded (def_name, eq) lthy = 
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let 

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val (Free (c, _), rhs) = (Logic.dest_equals o Syntax.check_term lthy) eq; 

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val (thm, lthy') = lthy 
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> Local_Theory.define ((Binding.name c, NoSyn), ((Binding.name def_name, []), rhs)) 
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> (fn (t, (_, thm)) => Spec_Rules.add Spec_Rules.Equational ([t], [thm]) #> pair thm); 
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val ctxt_thy = Proof_Context.init_global (Proof_Context.theory_of lthy'); 
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val thm' = singleton (Proof_Context.export lthy' ctxt_thy) thm; 

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in (thm', lthy') end; 
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val ((size_def_thms, size_def_thms'), thy') = 
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> Sign.add_consts_i (map (fn (s, T) => 
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(Binding.name (Long_Name.base_name s), param_size_fTs @ [T] > HOLogic.natT, NoSyn)) 
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(size_names ~~ recTs1)) 
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> Global_Theory.add_defs false 
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(map (Thm.no_attributes o apsnd (Logic.mk_equals o apsnd (app fs))) 
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(map Binding.name def_names ~~ (size_fns ~~ rec_combs1))) 
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> Class.instantiation 
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(map (#1 o snd) descr', map dest_TFree paramTs, [HOLogic.class_size]) 
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>> fold_map define_overloaded 
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(def_names' ~~ map Logic.mk_equals (overloaded_size_fns ~~ map (app fs') rec_combs1)) 

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> Class.prove_instantiation_instance (K (Class.intro_classes_tac [])) 

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> Local_Theory.exit_global; 
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val ctxt = Proof_Context.init_global thy'; 
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val simpset1 = HOL_basic_ss addsimps @{thm Nat.add_0} :: @{thm Nat.add_0_right} :: 
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size_def_thms @ size_def_thms' @ rec_rewrites @ extra_rewrites; 
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val xs = map (fn i => "x" ^ string_of_int i) (1 upto length recTs2); 
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fun mk_unfolded_size_eq tab size_ofp fs (p as (x, T), r) = 
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HOLogic.mk_eq (app fs r $ Free p, 
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the (size_of_type tab extra_size size_ofp T) $ Free p); 
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fun prove_unfolded_size_eqs size_ofp fs = 
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if null recTs2 then [] 
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else Datatype_Aux.split_conj_thm (Skip_Proof.prove ctxt xs [] 
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(HOLogic.mk_Trueprop (Datatype_Aux.mk_conj (replicate l @{term True} @ 
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map (mk_unfolded_size_eq (AList.lookup op = 
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(new_type_names ~~ map (app fs) rec_combs1)) size_ofp fs) 
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(xs ~~ recTs2 ~~ rec_combs2)))) 
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(fn _ => (Datatype_Aux.ind_tac induct xs THEN_ALL_NEW asm_simp_tac simpset1) 1)); 
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val unfolded_size_eqs1 = prove_unfolded_size_eqs param_size fs; 
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val unfolded_size_eqs2 = prove_unfolded_size_eqs (K NONE) fs'; 
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(* characteristic equations for size functions *) 
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fun gen_mk_size_eq p size_of size_ofp size_const T (cname, cargs) = 
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let 
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val Ts = map (Datatype_Aux.typ_of_dtyp descr) cargs; 
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val tnames = Name.variant_list f_names (Datatype_Prop.make_tnames Ts); 
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val ts = map_filter (fn (sT as (s, T), dt) => 
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Option.map (fn sz => sz $ Free sT) 
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(if p dt then size_of_type size_of extra_size size_ofp T 
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else NONE)) (tnames ~~ Ts ~~ cargs) 
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in 
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HOLogic.mk_Trueprop (HOLogic.mk_eq 
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(size_const $ list_comb (Const (cname, Ts > T), 
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map2 (curry Free) tnames Ts), 
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if null ts then HOLogic.zero 
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else foldl1 plus (ts @ [HOLogic.Suc_zero]))) 
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end; 
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val simpset2 = HOL_basic_ss addsimps 
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rec_rewrites @ size_def_thms @ unfolded_size_eqs1; 
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val simpset3 = HOL_basic_ss addsimps 
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rec_rewrites @ size_def_thms' @ unfolded_size_eqs2; 
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fun prove_size_eqs p size_fns size_ofp simpset = 
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maps (fn (((_, (_, _, constrs)), size_const), T) => 
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map (fn constr => Drule.export_without_context (Skip_Proof.prove ctxt [] [] 
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(gen_mk_size_eq p (AList.lookup op = (new_type_names ~~ size_fns)) 
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size_ofp size_const T constr) 
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(fn _ => simp_tac simpset 1))) constrs) 
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(descr' ~~ size_fns ~~ recTs1); 
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val size_eqns = prove_size_eqs (is_poly thy') size_fns param_size simpset2 @ 
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prove_size_eqs Datatype_Aux.is_rec_type overloaded_size_fns (K NONE) simpset3; 
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val ([(_, size_thms)], thy'') = thy' 
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> Global_Theory.note_thmss "" 
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[((Binding.name "size", 
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[Simplifier.simp_add, Nitpick_Simps.add, 
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Thm.declaration_attribute (fn thm => Context.mapping (Code.add_default_eqn thm) I)]), 
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[(size_eqns, [])])]; 
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in 

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SizeData.map (fold (Symtab.update_new o apsnd (rpair size_thms)) 
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(new_type_names ~~ size_names)) thy'' 
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end; 
216 

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fun add_size_thms config (new_type_names as name :: _) thy = 
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let 
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val info as {descr, ...} = Datatype.the_info thy name; 
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val prefix = 

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Long_Name.map_base_name (K (space_implode "_" (map Long_Name.base_name new_type_names))) name; 

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val no_size = exists (fn (_, (_, _, constrs)) => exists (fn (_, cargs) => exists (fn dt => 
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Datatype_Aux.is_rec_type dt andalso 
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not (null (fst (Datatype_Aux.strip_dtyp dt)))) cargs) constrs) descr 

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in 
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if no_size then thy 

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else 
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thy 

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> Sign.root_path 
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> Sign.add_path prefix 
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> Theory.checkpoint 
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> prove_size_thms info new_type_names 
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> Sign.restore_naming thy 
24710  234 
end; 
235 

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val size_thms = snd oo (the oo lookup_size); 
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val setup = Datatype.interpretation add_size_thms; 
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end; 