(* Title: HOL/Tools/datatype_prop.ML ID: $Id: datatype_prop.ML,v 1.27 2005/09/20 14:17:34 haftmann Exp $ Author: Stefan Berghofer, TU Muenchen Characteristic properties of datatypes. *) signature DATATYPE_PROP = sig val dtK : int ref val indexify_names: string list -> string list val make_tnames: typ list -> string list val make_injs : DatatypeAux.descr list -> (string * sort) list -> term list list val make_ind : DatatypeAux.descr list -> (string * sort) list -> term val make_casedists : DatatypeAux.descr list -> (string * sort) list -> term list val make_primrec_Ts : DatatypeAux.descr list -> (string * sort) list -> string list -> typ list * typ list val make_primrecs : string list -> DatatypeAux.descr list -> (string * sort) list -> theory -> term list val make_cases : string list -> DatatypeAux.descr list -> (string * sort) list -> theory -> term list list val make_distincts : string list -> DatatypeAux.descr list -> (string * sort) list -> theory -> term list list val make_splits : string list -> DatatypeAux.descr list -> (string * sort) list -> theory -> (term * term) list val make_size : DatatypeAux.descr list -> (string * sort) list -> theory -> term list val make_weak_case_congs : string list -> DatatypeAux.descr list -> (string * sort) list -> theory -> term list val make_case_congs : string list -> DatatypeAux.descr list -> (string * sort) list -> theory -> term list val make_nchotomys : DatatypeAux.descr list -> (string * sort) list -> term list end; structure DatatypeProp : DATATYPE_PROP = struct open DatatypeAux; (*the kind of distinctiveness axioms depends on number of constructors*) val dtK = ref 7; fun indexify_names names = let fun index (x :: xs) tab = (case AList.lookup (op =) tab x of NONE => if x mem xs then (x ^ "1") :: index xs ((x, 2) :: tab) else x :: index xs tab | SOME i => (x ^ Library.string_of_int i) :: index xs ((x, i + 1) :: tab)) | index [] _ = []; in index names [] end; fun make_tnames Ts = let fun type_name (TFree (name, _)) = implode (tl (explode name)) | type_name (Type (name, _)) = let val name' = Sign.base_name name in if Syntax.is_identifier name' then name' else "x" end; in indexify_names (map type_name Ts) end; (************************* injectivity of constructors ************************) fun make_injs descr sorts = let val descr' = List.concat descr; fun make_inj T ((cname, cargs), injs) = if null cargs then injs else let val Ts = map (typ_of_dtyp descr' sorts) cargs; val constr_t = Const (cname, Ts ---> T); val tnames = make_tnames Ts; val frees = map Free (tnames ~~ Ts); val frees' = map Free ((map ((op ^) o (rpair "'")) tnames) ~~ Ts); in (HOLogic.mk_Trueprop (HOLogic.mk_eq (HOLogic.mk_eq (list_comb (constr_t, frees), list_comb (constr_t, frees')), foldr1 (HOLogic.mk_binop "op &") (map HOLogic.mk_eq (frees ~~ frees')))))::injs end; in map (fn (d, T) => foldr (make_inj T) [] (#3 (snd d))) ((hd descr) ~~ Library.take (length (hd descr), get_rec_types descr' sorts)) end; (********************************* induction **********************************) fun make_ind descr sorts = let val descr' = List.concat descr; val recTs = get_rec_types descr' sorts; val pnames = if length descr' = 1 then ["P"] else map (fn i => "P" ^ string_of_int i) (1 upto length descr'); fun make_pred i T = let val T' = T --> HOLogic.boolT in Free (List.nth (pnames, i), T') end; fun make_ind_prem k T (cname, cargs) = let fun mk_prem ((dt, s), T) = let val (Us, U) = strip_type T in list_all (map (pair "x") Us, HOLogic.mk_Trueprop (make_pred (body_index dt) U $ app_bnds (Free (s, T)) (length Us))) end; val recs = List.filter is_rec_type cargs; val Ts = map (typ_of_dtyp descr' sorts) cargs; val recTs' = map (typ_of_dtyp descr' sorts) recs; val tnames = variantlist (make_tnames Ts, pnames); val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs)); val frees = tnames ~~ Ts; val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs'); in list_all_free (frees, Logic.list_implies (prems, HOLogic.mk_Trueprop (make_pred k T $ list_comb (Const (cname, Ts ---> T), map Free frees)))) end; val prems = List.concat (map (fn ((i, (_, _, constrs)), T) => map (make_ind_prem i T) constrs) (descr' ~~ recTs)); val tnames = make_tnames recTs; val concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &") (map (fn (((i, _), T), tname) => make_pred i T $ Free (tname, T)) (descr' ~~ recTs ~~ tnames))) in Logic.list_implies (prems, concl) end; (******************************* case distinction *****************************) fun make_casedists descr sorts = let val descr' = List.concat descr; fun make_casedist_prem T (cname, cargs) = let val Ts = map (typ_of_dtyp descr' sorts) cargs; val frees = variantlist (make_tnames Ts, ["P", "y"]) ~~ Ts; val free_ts = map Free frees in list_all_free (frees, Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (Free ("y", T), list_comb (Const (cname, Ts ---> T), free_ts))), HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT)))) end; fun make_casedist ((_, (_, _, constrs)), T) = let val prems = map (make_casedist_prem T) constrs in Logic.list_implies (prems, HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT))) end in map make_casedist ((hd descr) ~~ Library.take (length (hd descr), get_rec_types descr' sorts)) end; (*************** characteristic equations for primrec combinator **************) fun make_primrec_Ts descr sorts used = let val descr' = List.concat descr; val rec_result_Ts = map TFree (variantlist (replicate (length descr') "'t", used) ~~ replicate (length descr') HOLogic.typeS); val reccomb_fn_Ts = List.concat (map (fn (i, (_, _, constrs)) => map (fn (_, cargs) => let val Ts = map (typ_of_dtyp descr' sorts) cargs; val recs = List.filter (is_rec_type o fst) (cargs ~~ Ts); fun mk_argT (dt, T) = binder_types T ---> List.nth (rec_result_Ts, body_index dt); val argTs = Ts @ map mk_argT recs in argTs ---> List.nth (rec_result_Ts, i) end) constrs) descr'); in (rec_result_Ts, reccomb_fn_Ts) end; fun make_primrecs new_type_names descr sorts thy = let val sign = Theory.sign_of thy; val descr' = List.concat descr; val recTs = get_rec_types descr' sorts; val used = foldr add_typ_tfree_names [] recTs; val (rec_result_Ts, reccomb_fn_Ts) = make_primrec_Ts descr sorts used; val rec_fns = map (uncurry (mk_Free "f")) (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts))); val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec"; val reccomb_names = map (Sign.intern_const sign) (if length descr' = 1 then [big_reccomb_name] else (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int) (1 upto (length descr')))); val reccombs = map (fn ((name, T), T') => list_comb (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns)) (reccomb_names ~~ recTs ~~ rec_result_Ts); fun make_primrec T comb_t ((ts, f::fs), (cname, cargs)) = let val recs = List.filter is_rec_type cargs; val Ts = map (typ_of_dtyp descr' sorts) cargs; val recTs' = map (typ_of_dtyp descr' sorts) recs; val tnames = make_tnames Ts; val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs)); val frees = map Free (tnames ~~ Ts); val frees' = map Free (rec_tnames ~~ recTs'); fun mk_reccomb ((dt, T), t) = let val (Us, U) = strip_type T in list_abs (map (pair "x") Us, List.nth (reccombs, body_index dt) $ app_bnds t (length Us)) end; val reccombs' = map mk_reccomb (recs ~~ recTs' ~~ frees') in (ts @ [HOLogic.mk_Trueprop (HOLogic.mk_eq (comb_t $ list_comb (Const (cname, Ts ---> T), frees), list_comb (f, frees @ reccombs')))], fs) end in fst (Library.foldl (fn (x, ((dt, T), comb_t)) => Library.foldl (make_primrec T comb_t) (x, #3 (snd dt))) (([], rec_fns), descr' ~~ recTs ~~ reccombs)) end; (****************** make terms of form t_case f1 ... fn *********************) fun make_case_combs new_type_names descr sorts thy fname = let val descr' = List.concat descr; val recTs = get_rec_types descr' sorts; val used = foldr add_typ_tfree_names [] recTs; val newTs = Library.take (length (hd descr), recTs); val T' = TFree (variant used "'t", HOLogic.typeS); val case_fn_Ts = map (fn (i, (_, _, constrs)) => map (fn (_, cargs) => let val Ts = map (typ_of_dtyp descr' sorts) cargs in Ts ---> T' end) constrs) (hd descr); val case_names = map (fn s => Sign.intern_const (Theory.sign_of thy) (s ^ "_case")) new_type_names in map (fn ((name, Ts), T) => list_comb (Const (name, Ts @ [T] ---> T'), map (uncurry (mk_Free fname)) (Ts ~~ (1 upto length Ts)))) (case_names ~~ case_fn_Ts ~~ newTs) end; (**************** characteristic equations for case combinator ****************) fun make_cases new_type_names descr sorts thy = let val descr' = List.concat descr; val recTs = get_rec_types descr' sorts; val newTs = Library.take (length (hd descr), recTs); fun make_case T comb_t ((cname, cargs), f) = let val Ts = map (typ_of_dtyp descr' sorts) cargs; val frees = map Free ((make_tnames Ts) ~~ Ts) in HOLogic.mk_Trueprop (HOLogic.mk_eq (comb_t $ list_comb (Const (cname, Ts ---> T), frees), list_comb (f, frees))) end in map (fn (((_, (_, _, constrs)), T), comb_t) => map (make_case T comb_t) (constrs ~~ (snd (strip_comb comb_t)))) ((hd descr) ~~ newTs ~~ (make_case_combs new_type_names descr sorts thy "f")) end; (************************* distinctness of constructors ***********************) fun make_distincts new_type_names descr sorts thy = let val descr' = List.concat descr; val recTs = get_rec_types descr' sorts; val newTs = Library.take (length (hd descr), recTs); (**** number of constructors < dtK : C_i ... ~= C_j ... ****) fun make_distincts_1 _ [] = [] | make_distincts_1 T ((cname, cargs)::constrs) = let val Ts = map (typ_of_dtyp descr' sorts) cargs; val frees = map Free ((make_tnames Ts) ~~ Ts); val t = list_comb (Const (cname, Ts ---> T), frees); fun make_distincts' [] = [] | make_distincts' ((cname', cargs')::constrs') = let val Ts' = map (typ_of_dtyp descr' sorts) cargs'; val frees' = map Free ((map ((op ^) o (rpair "'")) (make_tnames Ts')) ~~ Ts'); val t' = list_comb (Const (cname', Ts' ---> T), frees') in (HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.mk_eq (t, t'))):: (HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.mk_eq (t', t))):: (make_distincts' constrs') end in (make_distincts' constrs) @ (make_distincts_1 T constrs) end; in map (fn (((_, (_, _, constrs)), T), tname) => if length constrs < !dtK then make_distincts_1 T constrs else []) ((hd descr) ~~ newTs ~~ new_type_names) end; (*************************** the "split" - equations **************************) fun make_splits new_type_names descr sorts thy = let val descr' = List.concat descr; val recTs = get_rec_types descr' sorts; val used' = foldr add_typ_tfree_names [] recTs; val newTs = Library.take (length (hd descr), recTs); val T' = TFree (variant used' "'t", HOLogic.typeS); val P = Free ("P", T' --> HOLogic.boolT); fun make_split (((_, (_, _, constrs)), T), comb_t) = let val (_, fs) = strip_comb comb_t; val used = ["P", "x"] @ (map (fst o dest_Free) fs); fun process_constr (((cname, cargs), f), (t1s, t2s)) = let val Ts = map (typ_of_dtyp descr' sorts) cargs; val frees = map Free (variantlist (make_tnames Ts, used) ~~ Ts); val eqn = HOLogic.mk_eq (Free ("x", T), list_comb (Const (cname, Ts ---> T), frees)); val P' = P $ list_comb (f, frees) in ((foldr (fn (Free (s, T), t) => HOLogic.mk_all (s, T, t)) (HOLogic.imp $ eqn $ P') frees)::t1s, (foldr (fn (Free (s, T), t) => HOLogic.mk_exists (s, T, t)) (HOLogic.conj $ eqn $ (HOLogic.Not $ P')) frees)::t2s) end; val (t1s, t2s) = foldr process_constr ([], []) (constrs ~~ fs); val lhs = P $ (comb_t $ Free ("x", T)) in (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, mk_conj t1s)), HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, HOLogic.Not $ mk_disj t2s))) end in map make_split ((hd descr) ~~ newTs ~~ (make_case_combs new_type_names descr sorts thy "f")) end; (******************************* size functions *******************************) fun make_size descr sorts thy = let val descr' = List.concat descr; val recTs = get_rec_types descr' sorts; val size_name = "Nat.size"; val size_names = replicate (length (hd descr)) size_name @ map (Sign.intern_const (Theory.sign_of thy)) (indexify_names (map (fn T => name_of_typ T ^ "_size") (Library.drop (length (hd descr), recTs)))); val size_consts = map (fn (s, T) => Const (s, T --> HOLogic.natT)) (size_names ~~ recTs); fun plus (t1, t2) = Const ("op +", [HOLogic.natT, HOLogic.natT] ---> HOLogic.natT) $ t1 $ t2; fun make_size_eqn size_const T (cname, cargs) = let val recs = List.filter is_rec_type cargs; val Ts = map (typ_of_dtyp descr' sorts) cargs; val recTs = map (typ_of_dtyp descr' sorts) recs; val tnames = make_tnames Ts; val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs)); val ts = map (fn ((r, s), T) => List.nth (size_consts, dest_DtRec r) $ Free (s, T)) (recs ~~ rec_tnames ~~ recTs); val t = if ts = [] then HOLogic.zero else foldl1 plus (ts @ [HOLogic.mk_nat 1]) in HOLogic.mk_Trueprop (HOLogic.mk_eq (size_const $ list_comb (Const (cname, Ts ---> T), map Free (tnames ~~ Ts)), t)) end in List.concat (map (fn (((_, (_, _, constrs)), size_const), T) => map (make_size_eqn size_const T) constrs) (descr' ~~ size_consts ~~ recTs)) end; (************************* additional rules for TFL ***************************) fun make_weak_case_congs new_type_names descr sorts thy = let val case_combs = make_case_combs new_type_names descr sorts thy "f"; fun mk_case_cong comb = let val Type ("fun", [T, _]) = fastype_of comb; val M = Free ("M", T); val M' = Free ("M'", T); in Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (M, M')), HOLogic.mk_Trueprop (HOLogic.mk_eq (comb $ M, comb $ M'))) end in map mk_case_cong case_combs end; (*--------------------------------------------------------------------------- * Structure of case congruence theorem looks like this: * * (M = M') * ==> (!!x1,...,xk. (M' = C1 x1..xk) ==> (f1 x1..xk = g1 x1..xk)) * ==> ... * ==> (!!x1,...,xj. (M' = Cn x1..xj) ==> (fn x1..xj = gn x1..xj)) * ==> * (ty_case f1..fn M = ty_case g1..gn M') *---------------------------------------------------------------------------*) fun make_case_congs new_type_names descr sorts thy = let val case_combs = make_case_combs new_type_names descr sorts thy "f"; val case_combs' = make_case_combs new_type_names descr sorts thy "g"; fun mk_case_cong ((comb, comb'), (_, (_, _, constrs))) = let val Type ("fun", [T, _]) = fastype_of comb; val (_, fs) = strip_comb comb; val (_, gs) = strip_comb comb'; val used = ["M", "M'"] @ map (fst o dest_Free) (fs @ gs); val M = Free ("M", T); val M' = Free ("M'", T); fun mk_clause ((f, g), (cname, _)) = let val (Ts, _) = strip_type (fastype_of f); val tnames = variantlist (make_tnames Ts, used); val frees = map Free (tnames ~~ Ts) in list_all_free (tnames ~~ Ts, Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (M', list_comb (Const (cname, Ts ---> T), frees))), HOLogic.mk_Trueprop (HOLogic.mk_eq (list_comb (f, frees), list_comb (g, frees))))) end in Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (M, M')) :: map mk_clause (fs ~~ gs ~~ constrs), HOLogic.mk_Trueprop (HOLogic.mk_eq (comb $ M, comb' $ M'))) end in map mk_case_cong (case_combs ~~ case_combs' ~~ hd descr) end; (*--------------------------------------------------------------------------- * Structure of exhaustion theorem looks like this: * * !v. (? y1..yi. v = C1 y1..yi) | ... | (? y1..yj. v = Cn y1..yj) *---------------------------------------------------------------------------*) fun make_nchotomys descr sorts = let val descr' = List.concat descr; val recTs = get_rec_types descr' sorts; val newTs = Library.take (length (hd descr), recTs); fun mk_eqn T (cname, cargs) = let val Ts = map (typ_of_dtyp descr' sorts) cargs; val tnames = variantlist (make_tnames Ts, ["v"]); val frees = tnames ~~ Ts in foldr (fn ((s, T'), t) => HOLogic.mk_exists (s, T', t)) (HOLogic.mk_eq (Free ("v", T), list_comb (Const (cname, Ts ---> T), map Free frees))) frees end in map (fn ((_, (_, _, constrs)), T) => HOLogic.mk_Trueprop (HOLogic.mk_all ("v", T, mk_disj (map (mk_eqn T) constrs)))) (hd descr ~~ newTs) end; end;