(* Title: HOL/Tools/function_package/mutual.ML
ID: $Id$
Author: Alexander Krauss, TU Muenchen
A package for general recursive function definitions.
Tools for mutual recursive definitions.
*)
signature FUNDEF_MUTUAL =
sig
val prepare_fundef_mutual : ((string * typ) * mixfix) list
-> term list
-> string (* default, unparsed term *)
-> local_theory
-> ((FundefCommon.mutual_info * string * FundefCommon.prep_result) * local_theory)
val mk_partial_rules_mutual : Proof.context -> FundefCommon.mutual_info -> FundefCommon.prep_result -> thm ->
FundefCommon.fundef_mresult
val sort_by_function : FundefCommon.mutual_info -> string list -> 'a list -> 'a list list
end
structure FundefMutual: FUNDEF_MUTUAL =
struct
open FundefLib
open FundefCommon
(* Theory dependencies *)
val sum_case_rules = thms "Datatype.sum.cases"
val split_apply = thm "Product_Type.split"
fun mutual_induct_Pnames n =
if n < 5 then fst (chop n ["P","Q","R","S"])
else map (fn i => "P" ^ string_of_int i) (1 upto n)
fun open_all_all (Const ("all", _) $ Abs (n, T, b)) = apfst (cons (n, T)) (open_all_all b)
| open_all_all t = ([], t)
(* Builds a curried clause description in abstracted form *)
fun split_def ctxt fnames geq arities =
let
fun input_error msg = error (cat_lines [msg, ProofContext.string_of_term ctxt geq])
val (qs, imp) = open_all_all geq
val gs = Logic.strip_imp_prems imp
val eq = Logic.strip_imp_concl imp
val (f_args, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop eq)
val (head, args) = strip_comb f_args
val invalid_head_msg = "Head symbol of left hand side must be " ^ plural "" "one out of " fnames ^ commas_quote fnames
val fname = fst (dest_Free head)
handle TERM _ => input_error invalid_head_msg
val _ = if fname mem fnames then ()
else input_error invalid_head_msg
fun add_bvs t is = add_loose_bnos (t, 0, is)
val rvs = (add_bvs rhs [] \\ fold add_bvs args [])
|> map (fst o nth (rev qs))
val _ = if null rvs then ()
else input_error ("Variable" ^ plural " " "s " rvs ^ commas_quote rvs
^ " occur" ^ plural "s" "" rvs ^ " on right hand side only:")
val _ = (fold o fold_aterms)
(fn Free (n, _) => if n mem fnames
then input_error "Recursive Calls not allowed in premises:"
else I
| _ => I) gs ()
val k = length args
val arities' = case Symtab.lookup arities fname of
NONE => Symtab.update (fname, k) arities
| SOME i => if (i <> k)
then input_error ("Function " ^ quote fname ^ " has different numbers of arguments in different equations")
else arities
in
((fname, qs, gs, args, rhs), arities')
end
fun get_part fname =
the o find_first (fn (MutualPart {fvar=(n,_), ...}) => n = fname)
(* FIXME *)
fun mk_prod_abs e (t1, t2) =
let
val bTs = rev (map snd e)
val T1 = fastype_of1 (bTs, t1)
val T2 = fastype_of1 (bTs, t2)
in
HOLogic.pair_const T1 T2 $ t1 $ t2
end;
fun analyze_eqs ctxt fs eqs =
let
val fnames = map fst fs
val (fqgars, arities) = fold_map (split_def ctxt fnames) eqs Symtab.empty
fun curried_types (fname, fT) =
let
val k = the_default 1 (Symtab.lookup arities fname)
val (caTs, uaTs) = chop k (binder_types fT)
in
(caTs, uaTs ---> body_type fT)
end
val (caTss, resultTs) = split_list (map curried_types fs)
val argTs = map (foldr1 HOLogic.mk_prodT) caTss
val (RST,streeR, pthsR) = SumTools.mk_tree_distinct resultTs
val (ST, streeA, pthsA) = SumTools.mk_tree argTs
val def_name = foldr1 (fn (a,b) => a ^ "_" ^ b) (map Sign.base_name fnames)
val fsum_type = ST --> RST
val ([fsum_var_name], _) = Variable.add_fixes [ def_name ^ "_sum" ] ctxt
val fsum_var = (fsum_var_name, fsum_type)
fun define (fvar as (n, T)) caTs pthA pthR =
let
val vars = map_index (fn (i,T) => Free ("x" ^ string_of_int i, T)) caTs (* FIXME: Bind xs properly *)
val f_exp = SumTools.mk_proj streeR pthR (Free fsum_var $ SumTools.mk_inj streeA pthA (foldr1 HOLogic.mk_prod vars))
val def = Term.abstract_over (Free fsum_var, fold_rev lambda vars f_exp)
val rew = (n, fold_rev lambda vars f_exp)
in
(MutualPart {fvar=fvar,cargTs=caTs,pthA=pthA,pthR=pthR,f_def=def,f=NONE,f_defthm=NONE}, rew)
end
val (parts, rews) = split_list (map4 define fs caTss pthsA pthsR)
fun convert_eqs (f, qs, gs, args, rhs) =
let
val MutualPart {pthA, pthR, ...} = get_part f parts
in
(qs, gs, SumTools.mk_inj streeA pthA (foldr1 (mk_prod_abs qs) args),
SumTools.mk_inj streeR pthR (replace_frees rews rhs)
|> Envir.norm_term (Envir.empty 0))
end
val qglrs = map convert_eqs fqgars
in
Mutual {defname=def_name,fsum_var=fsum_var, ST=ST, RST=RST, streeA=streeA, streeR=streeR,
parts=parts, fqgars=fqgars, qglrs=qglrs, fsum=NONE}
end
fun define_projections fixes mutual fsum lthy =
let
fun def ((MutualPart {fvar=(fname, fT), cargTs, pthA, pthR, f_def, ...}), (_, mixfix)) lthy =
let
val ((f, (_, f_defthm)), lthy') = LocalTheory.def ((fname, mixfix),
((fname ^ "_def", []), Term.subst_bound (fsum, f_def)))
lthy
in
(MutualPart {fvar=(fname, fT), cargTs=cargTs, pthA=pthA, pthR=pthR, f_def=f_def,
f=SOME f, f_defthm=SOME f_defthm },
lthy')
end
val Mutual { defname, fsum_var, ST, RST, streeA, streeR, parts, fqgars, qglrs, ... } = mutual
val (parts', lthy') = fold_map def (parts ~~ fixes) lthy
in
(Mutual { defname=defname, fsum_var=fsum_var, ST=ST, RST=RST, streeA=streeA, streeR=streeR, parts=parts',
fqgars=fqgars, qglrs=qglrs, fsum=SOME fsum },
lthy')
end
fun prepare_fundef_mutual fixes eqss default lthy =
let
val mutual = analyze_eqs lthy (map fst fixes) eqss
val Mutual {defname, fsum_var=(n, T), qglrs, ...} = mutual
val (prep_result, fsum, lthy') =
FundefPrep.prepare_fundef defname (n, T, NoSyn) qglrs default lthy
val (mutual', lthy'') = define_projections fixes mutual fsum lthy'
in
((mutual', defname, prep_result), lthy'')
end
(* Beta-reduce both sides of a meta-equality *)
fun beta_norm_eq thm =
let
val (lhs, rhs) = dest_equals (cprop_of thm)
val lhs_conv = beta_conversion false lhs
val rhs_conv = beta_conversion false rhs
in
transitive (symmetric lhs_conv) (transitive thm rhs_conv)
end
fun beta_reduce thm = Thm.equal_elim (Thm.beta_conversion true (cprop_of thm)) thm
fun in_context ctxt (f, pre_qs, pre_gs, pre_args, pre_rhs) F =
let
val thy = ProofContext.theory_of ctxt
val oqnames = map fst pre_qs
val (qs, ctxt') = Variable.variant_fixes oqnames ctxt
|>> map2 (fn (_, T) => fn n => Free (n, T)) pre_qs
fun inst t = subst_bounds (rev qs, t)
val gs = map inst pre_gs
val args = map inst pre_args
val rhs = inst pre_rhs
val cqs = map (cterm_of thy) qs
val ags = map (assume o cterm_of thy) gs
val import = fold forall_elim cqs
#> fold implies_elim_swp ags
val export = fold_rev (implies_intr o cprop_of) ags
#> fold_rev forall_intr_rename (oqnames ~~ cqs)
in
F (f, qs, gs, args, rhs) import export
end
fun recover_mutual_psimp thy RST streeR all_f_defs parts (f, _, _, args, _) import (export : thm -> thm) sum_psimp_eq =
let
val (MutualPart {f_defthm=SOME f_def, pthR, ...}) = get_part f parts
val psimp = import sum_psimp_eq
val (simp, restore_cond) = case cprems_of psimp of
[] => (psimp, I)
| [cond] => (implies_elim psimp (assume cond), implies_intr cond)
| _ => sys_error "Too many conditions"
val x = Free ("x", RST)
val f_def_inst = fold (fn arg => fn thm => combination thm (reflexive (cterm_of thy arg))) args (Thm.freezeT f_def) (* FIXME *)
|> beta_reduce
in
reflexive (cterm_of thy (lambda x (SumTools.mk_proj streeR pthR x))) (* PR(x) == PR(x) *)
|> (fn it => combination it (simp RS eq_reflection))
|> beta_norm_eq (* PR(S(I(as))) == PR(IR(...)) *)
|> transitive f_def_inst (* f ... == PR(IR(...)) *)
|> simplify (HOL_basic_ss addsimps [SumTools.projl_inl, SumTools.projr_inr]) (* f ... == ... *)
|> simplify (HOL_basic_ss addsimps all_f_defs) (* f ... == ... *)
|> (fn it => it RS meta_eq_to_obj_eq)
|> restore_cond
|> export
end
(* FIXME HACK *)
fun mk_applied_form ctxt caTs thm =
let
val thy = ProofContext.theory_of ctxt
val xs = map_index (fn (i,T) => cterm_of thy (Free ("x" ^ string_of_int i, T))) caTs (* FIXME: Bind xs properly *)
in
fold (fn x => fn thm => combination thm (reflexive x)) xs thm
|> beta_reduce
|> fold_rev forall_intr xs
|> forall_elim_vars 0
end
fun mutual_induct_rules thy induct all_f_defs (Mutual {RST, parts, streeA, ...}) =
let
val newPs = map2 (fn Pname => fn MutualPart {cargTs, ...} =>
Free (Pname, cargTs ---> HOLogic.boolT))
(mutual_induct_Pnames (length parts))
parts
fun mk_P (MutualPart {cargTs, ...}) P =
let
val avars = map_index (fn (i,T) => Var (("a", i), T)) cargTs
val atup = foldr1 HOLogic.mk_prod avars
in
tupled_lambda atup (list_comb (P, avars))
end
val Ps = map2 mk_P parts newPs
val case_exp = SumTools.mk_sumcases streeA HOLogic.boolT Ps
val induct_inst =
forall_elim (cterm_of thy case_exp) induct
|> full_simplify (HOL_basic_ss addsimps (split_apply :: sum_case_rules))
|> full_simplify (HOL_basic_ss addsimps all_f_defs)
fun mk_proj rule (MutualPart {cargTs, pthA, ...}) =
let
val afs = map_index (fn (i,T) => Free ("a" ^ string_of_int i, T)) cargTs
val inj = SumTools.mk_inj streeA pthA (foldr1 HOLogic.mk_prod afs)
in
rule
|> forall_elim (cterm_of thy inj)
|> full_simplify (HOL_basic_ss addsimps (split_apply :: sum_case_rules))
|> fold_rev (forall_intr o cterm_of thy) (afs @ newPs)
end
in
map (mk_proj induct_inst) parts
end
fun mk_partial_rules_mutual lthy (m as Mutual {RST, parts, streeR, fqgars, ...}) data prep_result =
let
val thy = ProofContext.theory_of lthy
(* FIXME !? *)
val expand = Assumption.export false lthy (LocalTheory.target_of lthy)
val expand_term = Drule.term_rule thy expand
val result = FundefProof.mk_partial_rules thy data prep_result
val FundefResult {f, G, R, completeness, psimps, subset_pinduct,simple_pinduct,total_intro,dom_intros} = result
val all_f_defs = map (fn MutualPart {f_defthm = SOME f_def, cargTs, ...} =>
mk_applied_form lthy cargTs (symmetric (Thm.freezeT f_def)))
parts
fun mk_mpsimp fqgar sum_psimp =
in_context lthy fqgar (recover_mutual_psimp thy RST streeR all_f_defs parts) sum_psimp
val mpsimps = map2 mk_mpsimp fqgars psimps
val minducts = mutual_induct_rules thy simple_pinduct all_f_defs m
val termination = full_simplify (HOL_basic_ss addsimps all_f_defs) total_intro
in
FundefMResult { f=expand_term f, G=expand_term G, R=expand_term R,
psimps=map expand mpsimps, subset_pinducts=[expand subset_pinduct], simple_pinducts=map expand minducts,
cases=expand completeness, termination=expand termination,
domintros=map expand dom_intros }
end
(* puts an object in the "right bucket" *)
fun store_grouped P x [] = []
| store_grouped P x ((l, xs)::bs) =
if P (x, l) then ((l, x::xs)::bs) else ((l, xs)::store_grouped P x bs)
fun sort_by_function (Mutual {fqgars, ...}) names xs =
fold_rev (store_grouped (eq_str o apfst fst)) (* fill *)
(map name_of_fqgar fqgars ~~ xs) (* the name-thm pairs *)
(map (rpair []) names) (* in the empty buckets labeled with names *)
|> map (snd #> map snd) (* and remove the labels afterwards *)
end