(* Title: Pure/Isar/local_defs.ML
Author: Makarius
Local definitions.
*)
signature LOCAL_DEFS =
sig
val cert_def: Proof.context -> term -> (string * typ) * term
val abs_def: term -> (string * typ) * term
val mk_def: Proof.context -> (string * term) list -> term list
val expand: cterm list -> thm -> thm
val def_export: Assumption.export
val add_defs: ((Binding.T * mixfix) * ((Binding.T * attribute list) * term)) list ->
Proof.context -> (term * (string * thm)) list * Proof.context
val add_def: (Binding.T * mixfix) * term -> Proof.context -> (term * thm) * Proof.context
val fixed_abbrev: (Binding.T * mixfix) * term -> Proof.context ->
(term * term) * Proof.context
val export: Proof.context -> Proof.context -> thm -> thm list * thm
val export_cterm: Proof.context -> Proof.context -> cterm -> cterm * thm
val trans_terms: Proof.context -> thm list -> thm
val trans_props: Proof.context -> thm list -> thm
val print_rules: Proof.context -> unit
val defn_add: attribute
val defn_del: attribute
val meta_rewrite_conv: Proof.context -> conv
val meta_rewrite_rule: Proof.context -> thm -> thm
val unfold: Proof.context -> thm list -> thm -> thm
val unfold_goals: Proof.context -> thm list -> thm -> thm
val unfold_tac: Proof.context -> thm list -> tactic
val fold: Proof.context -> thm list -> thm -> thm
val fold_tac: Proof.context -> thm list -> tactic
val derived_def: Proof.context -> bool -> term ->
((string * typ) * term) * (Proof.context -> thm -> thm)
end;
structure LocalDefs: LOCAL_DEFS =
struct
(** primitive definitions **)
(* prepare defs *)
fun cert_def ctxt eq =
let
fun err msg = cat_error msg ("The error(s) above occurred in definition: " ^
quote (Syntax.string_of_term ctxt eq));
val ((lhs, _), eq') = eq
|> Sign.no_vars (Syntax.pp ctxt)
|> PrimitiveDefs.dest_def ctxt Term.is_Free (Variable.is_fixed ctxt) (K true)
handle TERM (msg, _) => err msg | ERROR msg => err msg;
in (Term.dest_Free (Term.head_of lhs), eq') end;
val abs_def = PrimitiveDefs.abs_def #>> Term.dest_Free;
fun mk_def ctxt args =
let
val (xs, rhss) = split_list args;
val (bind, _) = ProofContext.bind_fixes xs ctxt;
val lhss = map (fn (x, rhs) => bind (Free (x, Term.fastype_of rhs))) args;
in map Logic.mk_equals (lhss ~~ rhss) end;
(* export defs *)
val head_of_def =
#1 o Term.dest_Free o Term.head_of o #1 o Logic.dest_equals o Term.strip_all_body;
(*
[x, x == a]
:
B x
-----------
B a
*)
fun expand defs =
Drule.implies_intr_list defs
#> Drule.generalize ([], map (head_of_def o Thm.term_of) defs)
#> funpow (length defs) (fn th => Drule.reflexive_thm RS th);
val expand_term = Envir.expand_term_frees o map (abs_def o Thm.term_of);
fun def_export _ defs = (expand defs, expand_term defs);
(* add defs *)
fun add_defs defs ctxt =
let
val ((bvars, mxs), specs) = defs |> split_list |>> split_list;
val ((bfacts, atts), rhss) = specs |> split_list |>> split_list;
val xs = map Binding.base_name bvars;
val names = map2 (Binding.map_base o Thm.def_name_optional) xs bfacts;
val eqs = mk_def ctxt (xs ~~ rhss);
val lhss = map (fst o Logic.dest_equals) eqs;
in
ctxt
|> ProofContext.add_fixes_i (map2 (fn x => fn mx => (x, NONE, mx)) bvars mxs) |> #2
|> fold Variable.declare_term eqs
|> ProofContext.add_assms_i def_export
(map2 (fn a => fn eq => (a, [(eq, [])])) (names ~~ atts) eqs)
|>> map2 (fn lhs => fn (name, [th]) => (lhs, (name, th))) lhss
end;
fun add_def (var, rhs) ctxt =
let val ([(lhs, (_, th))], ctxt') = add_defs [(var, ((Binding.empty, []), rhs))] ctxt
in ((lhs, th), ctxt') end;
(* fixed_abbrev *)
fun fixed_abbrev ((x, mx), rhs) ctxt =
let
val T = Term.fastype_of rhs;
val ([x'], ctxt') = ctxt
|> Variable.declare_term rhs
|> ProofContext.add_fixes_i [(x, SOME T, mx)];
val lhs = Free (x', T);
val _ = cert_def ctxt' (Logic.mk_equals (lhs, rhs));
fun abbrev_export _ _ = (I, Envir.expand_term_frees [((x', T), rhs)]);
val (_, ctxt'') = Assumption.add_assms abbrev_export [] ctxt';
in ((lhs, rhs), ctxt'') end;
(* specific export -- result based on educated guessing *)
(*
[xs, xs == as]
:
B xs
--------------
B as
*)
fun export inner outer = (*beware of closure sizes*)
let
val exp = Assumption.export false inner outer;
val prems = Assumption.prems_of inner;
in fn th =>
let
val th' = exp th;
val still_fixed = map #1 (Thm.fold_terms Term.add_frees th' []);
val defs = prems |> filter_out (fn prem =>
(case try (head_of_def o Thm.prop_of) prem of
SOME x => member (op =) still_fixed x
| NONE => true));
in (map Drule.abs_def defs, th') end
end;
(*
[xs, xs == as]
:
TERM b xs
-------------- and --------------
TERM b as b xs == b as
*)
fun export_cterm inner outer ct =
let val (defs, ct') = export inner outer (Drule.mk_term ct) ||> Drule.dest_term
in (ct', MetaSimplifier.rewrite true defs ct) end;
(* basic transitivity reasoning -- modulo beta-eta *)
local
val is_trivial = Pattern.aeconv o Logic.dest_equals o Thm.prop_of;
fun trans_list _ _ [] = raise Empty
| trans_list trans ctxt (th :: raw_eqs) =
(case filter_out is_trivial raw_eqs of
[] => th
| eqs =>
let val ((_, th' :: eqs'), ctxt') = Variable.import_thms true (th :: eqs) ctxt
in singleton (Variable.export ctxt' ctxt) (fold trans eqs' th') end);
in
val trans_terms = trans_list (fn eq2 => fn eq1 => eq2 COMP (eq1 COMP Drule.transitive_thm));
val trans_props = trans_list (fn eq => fn th => th COMP (eq COMP Drule.equal_elim_rule1));
end;
(** defived definitions **)
(* transformation rules *)
structure Rules = GenericDataFun
(
type T = thm list;
val empty = []
val extend = I;
fun merge _ = Thm.merge_thms;
);
fun print_rules ctxt =
Pretty.writeln (Pretty.big_list "definitional transformations:"
(map (ProofContext.pretty_thm ctxt) (Rules.get (Context.Proof ctxt))));
val defn_add = Thm.declaration_attribute (Rules.map o Thm.add_thm);
val defn_del = Thm.declaration_attribute (Rules.map o Thm.del_thm);
(* meta rewrite rules *)
fun meta_rewrite_conv ctxt =
MetaSimplifier.rewrite_cterm (false, false, false) (K (K NONE))
(MetaSimplifier.context ctxt MetaSimplifier.empty_ss
addeqcongs [Drule.equals_cong] (*protect meta-level equality*)
addsimps (Rules.get (Context.Proof ctxt)));
val meta_rewrite_rule = Conv.fconv_rule o meta_rewrite_conv;
(* rewriting with object-level rules *)
fun meta f ctxt = f o map (meta_rewrite_rule ctxt);
val unfold = meta MetaSimplifier.rewrite_rule;
val unfold_goals = meta MetaSimplifier.rewrite_goals_rule;
val unfold_tac = meta MetaSimplifier.rewrite_goals_tac;
val fold = meta MetaSimplifier.fold_rule;
val fold_tac = meta MetaSimplifier.fold_goals_tac;
(* derived defs -- potentially within the object-logic *)
fun derived_def ctxt conditional prop =
let
val ((c, T), rhs) = prop
|> Thm.cterm_of (ProofContext.theory_of ctxt)
|> meta_rewrite_conv ctxt
|> (snd o Logic.dest_equals o Thm.prop_of)
|> conditional ? Logic.strip_imp_concl
|> (abs_def o #2 o cert_def ctxt);
fun prove ctxt' def =
let
val frees = Term.fold_aterms (fn Free (x, _) =>
if Variable.is_fixed ctxt' x then I else insert (op =) x | _ => I) prop [];
in
Goal.prove ctxt' frees [] prop (K (ALLGOALS
(CONVERSION (meta_rewrite_conv ctxt') THEN'
MetaSimplifier.rewrite_goal_tac [def] THEN'
resolve_tac [Drule.reflexive_thm])))
handle ERROR msg => cat_error msg "Failed to prove definitional specification."
end;
in (((c, T), rhs), prove) end;
end;