41561
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(* Title: HOL/SPARK/Tools/spark_vcs.ML
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Author: Stefan Berghofer
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Copyright: secunet Security Networks AG
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Store for verification conditions generated by SPARK/Ada.
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*)
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signature SPARK_VCS =
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sig
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val set_vcs: Fdl_Parser.decls -> Fdl_Parser.rules -> Fdl_Parser.vcs ->
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Path.T -> theory -> theory
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val add_proof_fun: (typ option -> 'a -> term) ->
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string * ((string list * string) option * 'a) ->
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theory -> theory
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val lookup_vc: theory -> string -> (Element.context_i list *
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(string * bool * Element.context_i * Element.statement_i)) option
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val get_vcs: theory ->
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Element.context_i list * (binding * thm) list *
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(string * (string * bool * Element.context_i * Element.statement_i)) list
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val mark_proved: string -> theory -> theory
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val close: theory -> theory
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val is_closed: theory -> bool
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end;
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structure SPARK_VCs: SPARK_VCS =
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struct
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open Fdl_Parser;
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(** utilities **)
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fun mk_unop s t =
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let val T = fastype_of t
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in Const (s, T --> T) $ t end;
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fun mk_times (t, u) =
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let
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val setT = fastype_of t;
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val T = HOLogic.dest_setT setT;
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val U = HOLogic.dest_setT (fastype_of u)
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in
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Const (@{const_name Sigma}, setT --> (T --> HOLogic.mk_setT U) -->
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HOLogic.mk_setT (HOLogic.mk_prodT (T, U))) $ t $ Abs ("", T, u)
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end;
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fun mk_type _ "integer" = HOLogic.intT
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| mk_type _ "boolean" = HOLogic.boolT
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| mk_type thy ty = Syntax.check_typ (ProofContext.init_global thy)
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(Type (Sign.full_name thy (Binding.name ty), []));
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val booleanN = "boolean";
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val integerN = "integer";
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fun mk_qual_name thy s s' =
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Sign.full_name thy (Binding.qualify true s (Binding.name s'));
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fun define_overloaded (def_name, eq) lthy =
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let
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val ((c, _), rhs) = eq |> Syntax.check_term lthy |>
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Logic.dest_equals |>> dest_Free;
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val ((_, (_, thm)), lthy') = Local_Theory.define
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((Binding.name c, NoSyn), ((Binding.name def_name, []), rhs)) lthy
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val ctxt_thy = ProofContext.init_global (ProofContext.theory_of lthy');
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val thm' = singleton (ProofContext.export lthy' ctxt_thy) thm
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in (thm', lthy') end;
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fun strip_underscores s =
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strip_underscores (unsuffix "_" s) handle Fail _ => s;
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fun strip_tilde s =
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unsuffix "~" s ^ "_init" handle Fail _ => s;
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val mangle_name = strip_underscores #> strip_tilde;
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fun mk_variables thy xs ty (tab, ctxt) =
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let
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val T = mk_type thy ty;
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val (ys, ctxt') = Name.variants (map mangle_name xs) ctxt;
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val zs = map (Free o rpair T) ys;
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in (zs, (fold (Symtab.update o apsnd (rpair ty)) (xs ~~ zs) tab, ctxt')) end;
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(** generate properties of enumeration types **)
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fun add_enum_type tyname els (tab, ctxt) thy =
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let
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val tyb = Binding.name tyname;
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val tyname' = Sign.full_name thy tyb;
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val T = Type (tyname', []);
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val case_name = mk_qual_name thy tyname (tyname ^ "_case");
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val cs = map (fn s => Const (mk_qual_name thy tyname s, T)) els;
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val k = length els;
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val p = Const (@{const_name pos}, T --> HOLogic.intT);
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val v = Const (@{const_name val}, HOLogic.intT --> T);
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val card = Const (@{const_name card},
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HOLogic.mk_setT T --> HOLogic.natT) $ HOLogic.mk_UNIV T;
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fun mk_binrel_def s f = Logic.mk_equals
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(Const (s, T --> T --> HOLogic.boolT),
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Abs ("x", T, Abs ("y", T,
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Const (s, HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $
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(f $ Bound 1) $ (f $ Bound 0))));
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val (((def1, def2), def3), lthy) = thy |>
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Datatype.add_datatype {strict = true, quiet = true} [tyname]
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[([], tyb, NoSyn,
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map (fn s => (Binding.name s, [], NoSyn)) els)] |> snd |>
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Class.instantiation ([tyname'], [], @{sort enum}) |>
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define_overloaded ("pos_" ^ tyname ^ "_def", Logic.mk_equals
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(p,
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list_comb (Const (case_name, replicate k HOLogic.intT @
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[T] ---> HOLogic.intT),
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map (HOLogic.mk_number HOLogic.intT) (0 upto k - 1)))) ||>>
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define_overloaded ("less_eq_" ^ tyname ^ "_def",
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mk_binrel_def @{const_name less_eq} p) ||>>
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define_overloaded ("less_" ^ tyname ^ "_def",
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mk_binrel_def @{const_name less} p);
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val UNIV_eq = Goal.prove lthy [] []
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(HOLogic.mk_Trueprop (HOLogic.mk_eq
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(HOLogic.mk_UNIV T, HOLogic.mk_set T cs)))
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(fn _ =>
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rtac @{thm subset_antisym} 1 THEN
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rtac @{thm subsetI} 1 THEN
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Datatype_Aux.exh_tac (K (#exhaust (Datatype_Data.the_info
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(ProofContext.theory_of lthy) tyname'))) 1 THEN
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ALLGOALS (asm_full_simp_tac (simpset_of lthy)));
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val finite_UNIV = Goal.prove lthy [] []
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(HOLogic.mk_Trueprop (Const (@{const_name finite},
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HOLogic.mk_setT T --> HOLogic.boolT) $ HOLogic.mk_UNIV T))
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(fn _ => simp_tac (simpset_of lthy addsimps [UNIV_eq]) 1);
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val card_UNIV = Goal.prove lthy [] []
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(HOLogic.mk_Trueprop (HOLogic.mk_eq
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(card, HOLogic.mk_number HOLogic.natT k)))
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(fn _ => simp_tac (simpset_of lthy addsimps [UNIV_eq]) 1);
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val range_pos = Goal.prove lthy [] []
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(HOLogic.mk_Trueprop (HOLogic.mk_eq
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(Const (@{const_name image}, (T --> HOLogic.intT) -->
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HOLogic.mk_setT T --> HOLogic.mk_setT HOLogic.intT) $
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p $ HOLogic.mk_UNIV T,
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Const (@{const_name atLeastLessThan}, HOLogic.intT -->
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HOLogic.intT --> HOLogic.mk_setT HOLogic.intT) $
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HOLogic.mk_number HOLogic.intT 0 $
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(@{term int} $ card))))
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(fn _ =>
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simp_tac (simpset_of lthy addsimps [card_UNIV]) 1 THEN
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simp_tac (simpset_of lthy addsimps [UNIV_eq, def1]) 1 THEN
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rtac @{thm subset_antisym} 1 THEN
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simp_tac (simpset_of lthy) 1 THEN
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rtac @{thm subsetI} 1 THEN
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asm_full_simp_tac (simpset_of lthy addsimps @{thms interval_expand}
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delsimps @{thms atLeastLessThan_iff}) 1);
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val lthy' =
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Class.prove_instantiation_instance (fn _ =>
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Class.intro_classes_tac [] THEN
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rtac finite_UNIV 1 THEN
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rtac range_pos 1 THEN
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simp_tac (HOL_basic_ss addsimps [def3]) 1 THEN
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simp_tac (HOL_basic_ss addsimps [def2]) 1) lthy;
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val (pos_eqs, val_eqs) = split_list (map_index (fn (i, c) =>
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let
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val n = HOLogic.mk_number HOLogic.intT i;
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val th = Goal.prove lthy' [] []
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(HOLogic.mk_Trueprop (HOLogic.mk_eq (p $ c, n)))
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(fn _ => simp_tac (simpset_of lthy' addsimps [def1]) 1);
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val th' = Goal.prove lthy' [] []
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(HOLogic.mk_Trueprop (HOLogic.mk_eq (v $ n, c)))
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(fn _ =>
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rtac (@{thm inj_pos} RS @{thm injD}) 1 THEN
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simp_tac (simpset_of lthy' addsimps
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[@{thm pos_val}, range_pos, card_UNIV, th]) 1)
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in (th, th') end) cs);
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val first_el = Goal.prove lthy' [] []
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(HOLogic.mk_Trueprop (HOLogic.mk_eq
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(Const (@{const_name first_el}, T), hd cs)))
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(fn _ => simp_tac (simpset_of lthy' addsimps
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[@{thm first_el_def}, hd val_eqs]) 1);
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val last_el = Goal.prove lthy' [] []
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(HOLogic.mk_Trueprop (HOLogic.mk_eq
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(Const (@{const_name last_el}, T), List.last cs)))
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(fn _ => simp_tac (simpset_of lthy' addsimps
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[@{thm last_el_def}, List.last val_eqs, card_UNIV]) 1);
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val simp_att = [Attrib.internal (K Simplifier.simp_add)]
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in
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((fold (Symtab.update_new o apsnd (rpair tyname)) (els ~~ cs) tab,
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fold Name.declare els ctxt),
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lthy' |>
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Local_Theory.note
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((Binding.name (tyname ^ "_card_UNIV"), simp_att), [card_UNIV]) ||>>
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Local_Theory.note
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((Binding.name (tyname ^ "_pos"), simp_att), pos_eqs) ||>>
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Local_Theory.note
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((Binding.name (tyname ^ "_val"), simp_att), val_eqs) ||>>
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Local_Theory.note
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((Binding.name (tyname ^ "_first_el"), simp_att), [first_el]) ||>>
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Local_Theory.note
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((Binding.name (tyname ^ "_last_el"), simp_att), [last_el]) |> snd |>
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Local_Theory.exit_global)
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end;
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fun add_type_def (s, Basic_Type ty) (ids, thy) =
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(ids,
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Typedecl.abbrev_global (Binding.name s, [], NoSyn)
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(mk_type thy ty) thy |> snd)
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| add_type_def (s, Enum_Type els) (ids, thy) = add_enum_type s els ids thy
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| add_type_def (s, Array_Type (argtys, resty)) (ids, thy) =
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(ids,
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Typedecl.abbrev_global (Binding.name s, [], NoSyn)
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(foldr1 HOLogic.mk_prodT (map (mk_type thy) argtys) -->
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mk_type thy resty) thy |> snd)
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| add_type_def (s, Record_Type fldtys) (ids, thy) =
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(ids,
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Record.add_record true ([], Binding.name s) NONE
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(maps (fn (flds, ty) =>
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let val T = mk_type thy ty
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in map (fn fld => (Binding.name fld, T, NoSyn)) flds
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end) fldtys) thy)
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| add_type_def (s, Pending_Type) (ids, thy) =
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(ids, Typedecl.typedecl_global (Binding.name s, [], NoSyn) thy |> snd);
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fun term_of_expr thy types funs pfuns =
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let
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fun tm_of vs (Funct ("->", [e, e'])) =
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(HOLogic.mk_imp (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
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| tm_of vs (Funct ("<->", [e, e'])) =
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(HOLogic.mk_eq (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
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| tm_of vs (Funct ("or", [e, e'])) =
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(HOLogic.mk_disj (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
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| tm_of vs (Funct ("and", [e, e'])) =
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(HOLogic.mk_conj (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
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| tm_of vs (Funct ("not", [e])) =
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(HOLogic.mk_not (fst (tm_of vs e)), booleanN)
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| tm_of vs (Funct ("=", [e, e'])) =
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(HOLogic.mk_eq (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
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| tm_of vs (Funct ("<>", [e, e'])) = (HOLogic.mk_not
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(HOLogic.mk_eq (fst (tm_of vs e), fst (tm_of vs e'))), booleanN)
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| tm_of vs (Funct ("<", [e, e'])) = (HOLogic.mk_binrel @{const_name less}
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(fst (tm_of vs e), fst (tm_of vs e')), booleanN)
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| tm_of vs (Funct (">", [e, e'])) = (HOLogic.mk_binrel @{const_name less}
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(fst (tm_of vs e'), fst (tm_of vs e)), booleanN)
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| tm_of vs (Funct ("<=", [e, e'])) = (HOLogic.mk_binrel @{const_name less_eq}
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(fst (tm_of vs e), fst (tm_of vs e')), booleanN)
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| tm_of vs (Funct (">=", [e, e'])) = (HOLogic.mk_binrel @{const_name less_eq}
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(fst (tm_of vs e'), fst (tm_of vs e)), booleanN)
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| tm_of vs (Funct ("+", [e, e'])) = (HOLogic.mk_binop @{const_name plus}
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(fst (tm_of vs e), fst (tm_of vs e')), integerN)
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| tm_of vs (Funct ("-", [e, e'])) = (HOLogic.mk_binop @{const_name minus}
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(fst (tm_of vs e), fst (tm_of vs e')), integerN)
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| tm_of vs (Funct ("*", [e, e'])) = (HOLogic.mk_binop @{const_name times}
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(fst (tm_of vs e), fst (tm_of vs e')), integerN)
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| tm_of vs (Funct ("/", [e, e'])) = (HOLogic.mk_binop @{const_name divide}
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(fst (tm_of vs e), fst (tm_of vs e')), integerN)
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| tm_of vs (Funct ("div", [e, e'])) = (HOLogic.mk_binop @{const_name sdiv}
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(fst (tm_of vs e), fst (tm_of vs e')), integerN)
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| tm_of vs (Funct ("mod", [e, e'])) = (HOLogic.mk_binop @{const_name smod}
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(fst (tm_of vs e), fst (tm_of vs e')), integerN)
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| tm_of vs (Funct ("-", [e])) =
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(mk_unop @{const_name uminus} (fst (tm_of vs e)), integerN)
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| tm_of vs (Funct ("**", [e, e'])) =
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(Const (@{const_name power}, HOLogic.intT --> HOLogic.natT -->
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HOLogic.intT) $ fst (tm_of vs e) $
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(@{const nat} $ fst (tm_of vs e')), integerN)
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| tm_of (tab, _) (Ident s) =
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(case Symtab.lookup tab s of
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SOME t_ty => t_ty
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| NONE => error ("Undeclared identifier " ^ s))
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| tm_of _ (Number i) = (HOLogic.mk_number HOLogic.intT i, integerN)
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| tm_of vs (Quantifier (s, xs, ty, e)) =
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let
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val (ys, vs') = mk_variables thy xs ty vs;
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val q = (case s of
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"for_all" => HOLogic.mk_all
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| "for_some" => HOLogic.mk_exists)
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in
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(fold_rev (fn Free (x, T) => fn t => q (x, T, t))
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ys (fst (tm_of vs' e)),
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booleanN)
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end
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| tm_of vs (Funct (s, es)) =
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(* record field selection *)
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(case try (unprefix "fld_") s of
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SOME fname => (case es of
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[e] =>
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let val (t, rcdty) = tm_of vs e
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in case lookup types rcdty of
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SOME (Record_Type fldtys) =>
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(case get_first (fn (flds, fldty) =>
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|
330 |
if member (op =) flds fname then SOME fldty
|
|
331 |
else NONE) fldtys of
|
|
332 |
SOME fldty =>
|
|
333 |
(Const (mk_qual_name thy rcdty fname,
|
|
334 |
mk_type thy rcdty --> mk_type thy fldty) $ t,
|
|
335 |
fldty)
|
|
336 |
| NONE => error ("Record " ^ rcdty ^
|
|
337 |
" has no field named " ^ fname))
|
|
338 |
| _ => error (rcdty ^ " is not a record type")
|
|
339 |
end
|
|
340 |
| _ => error ("Function " ^ s ^ " expects one argument"))
|
|
341 |
| NONE =>
|
|
342 |
|
|
343 |
(* record field update *)
|
|
344 |
(case try (unprefix "upf_") s of
|
|
345 |
SOME fname => (case es of
|
|
346 |
[e, e'] =>
|
|
347 |
let
|
|
348 |
val (t, rcdty) = tm_of vs e;
|
|
349 |
val rT = mk_type thy rcdty;
|
|
350 |
val (u, fldty) = tm_of vs e';
|
|
351 |
val fT = mk_type thy fldty
|
|
352 |
in case lookup types rcdty of
|
|
353 |
SOME (Record_Type fldtys) =>
|
|
354 |
(case get_first (fn (flds, fldty) =>
|
|
355 |
if member (op =) flds fname then SOME fldty
|
|
356 |
else NONE) fldtys of
|
|
357 |
SOME fldty' =>
|
|
358 |
if fldty = fldty' then
|
|
359 |
(Const (mk_qual_name thy rcdty (fname ^ "_update"),
|
|
360 |
(fT --> fT) --> rT --> rT) $
|
|
361 |
Abs ("x", fT, u) $ t,
|
|
362 |
rcdty)
|
|
363 |
else error ("Type " ^ fldty ^
|
|
364 |
" does not match type " ^ fldty' ^ " of field " ^
|
|
365 |
fname)
|
|
366 |
| NONE => error ("Record " ^ rcdty ^
|
|
367 |
" has no field named " ^ fname))
|
|
368 |
| _ => error (rcdty ^ " is not a record type")
|
|
369 |
end
|
|
370 |
| _ => error ("Function " ^ s ^ " expects two arguments"))
|
|
371 |
| NONE =>
|
|
372 |
|
|
373 |
(* enumeration type to integer *)
|
|
374 |
(case try (unsuffix "__pos") s of
|
|
375 |
SOME tyname => (case es of
|
|
376 |
[e] => (Const (@{const_name pos},
|
|
377 |
mk_type thy tyname --> HOLogic.intT) $ fst (tm_of vs e), integerN)
|
|
378 |
| _ => error ("Function " ^ s ^ " expects one argument"))
|
|
379 |
| NONE =>
|
|
380 |
|
|
381 |
(* integer to enumeration type *)
|
|
382 |
(case try (unsuffix "__val") s of
|
|
383 |
SOME tyname => (case es of
|
|
384 |
[e] => (Const (@{const_name val},
|
|
385 |
HOLogic.intT --> mk_type thy tyname) $ fst (tm_of vs e), tyname)
|
|
386 |
| _ => error ("Function " ^ s ^ " expects one argument"))
|
|
387 |
| NONE =>
|
|
388 |
|
|
389 |
(* successor / predecessor of enumeration type element *)
|
|
390 |
if s = "succ" orelse s = "pred" then (case es of
|
|
391 |
[e] =>
|
|
392 |
let
|
|
393 |
val (t, tyname) = tm_of vs e;
|
|
394 |
val T = mk_type thy tyname
|
|
395 |
in (Const
|
|
396 |
(if s = "succ" then @{const_name succ}
|
|
397 |
else @{const_name pred}, T --> T) $ t, tyname)
|
|
398 |
end
|
|
399 |
| _ => error ("Function " ^ s ^ " expects one argument"))
|
|
400 |
|
|
401 |
(* user-defined proof function *)
|
|
402 |
else
|
|
403 |
(case Symtab.lookup pfuns s of
|
|
404 |
SOME (SOME (_, resty), t) =>
|
|
405 |
(list_comb (t, map (fst o tm_of vs) es), resty)
|
|
406 |
| _ => error ("Undeclared proof function " ^ s))))))
|
|
407 |
|
|
408 |
| tm_of vs (Element (e, es)) =
|
|
409 |
let val (t, ty) = tm_of vs e
|
|
410 |
in case lookup types ty of
|
|
411 |
SOME (Array_Type (_, elty)) =>
|
|
412 |
(t $ foldr1 HOLogic.mk_prod (map (fst o tm_of vs) es), elty)
|
|
413 |
| _ => error (ty ^ " is not an array type")
|
|
414 |
end
|
|
415 |
|
|
416 |
| tm_of vs (Update (e, es, e')) =
|
|
417 |
let val (t, ty) = tm_of vs e
|
|
418 |
in case lookup types ty of
|
|
419 |
SOME (Array_Type (idxtys, elty)) =>
|
|
420 |
let
|
|
421 |
val T = foldr1 HOLogic.mk_prodT (map (mk_type thy) idxtys);
|
|
422 |
val U = mk_type thy elty;
|
|
423 |
val fT = T --> U
|
|
424 |
in
|
|
425 |
(Const (@{const_name fun_upd}, fT --> T --> U --> fT) $
|
|
426 |
t $ foldr1 HOLogic.mk_prod (map (fst o tm_of vs) es) $
|
|
427 |
fst (tm_of vs e'),
|
|
428 |
ty)
|
|
429 |
end
|
|
430 |
| _ => error (ty ^ " is not an array type")
|
|
431 |
end
|
|
432 |
|
|
433 |
| tm_of vs (Record (s, flds)) =
|
|
434 |
(case lookup types s of
|
|
435 |
SOME (Record_Type fldtys) =>
|
|
436 |
let
|
|
437 |
val flds' = map (apsnd (tm_of vs)) flds;
|
|
438 |
val fnames = maps fst fldtys;
|
|
439 |
val fnames' = map fst flds;
|
|
440 |
val (fvals, ftys) = split_list (map (fn s' =>
|
|
441 |
case AList.lookup (op =) flds' s' of
|
|
442 |
SOME fval_ty => fval_ty
|
|
443 |
| NONE => error ("Field " ^ s' ^ " missing in record " ^ s))
|
|
444 |
fnames);
|
|
445 |
val _ = (case subtract (op =) fnames fnames' of
|
|
446 |
[] => ()
|
|
447 |
| xs => error ("Extra field(s) " ^ commas xs ^
|
|
448 |
" in record " ^ s));
|
|
449 |
val _ = (case duplicates (op =) fnames' of
|
|
450 |
[] => ()
|
|
451 |
| xs => error ("Duplicate field(s) " ^ commas xs ^
|
|
452 |
" in record " ^ s))
|
|
453 |
in
|
|
454 |
(list_comb
|
|
455 |
(Const (mk_qual_name thy s (s ^ "_ext"),
|
|
456 |
map (mk_type thy) ftys @ [HOLogic.unitT] --->
|
|
457 |
mk_type thy s),
|
|
458 |
fvals @ [HOLogic.unit]),
|
|
459 |
s)
|
|
460 |
end
|
|
461 |
| _ => error (s ^ " is not a record type"))
|
|
462 |
|
|
463 |
| tm_of vs (Array (s, default, assocs)) =
|
|
464 |
(case lookup types s of
|
|
465 |
SOME (Array_Type (idxtys, elty)) =>
|
|
466 |
let
|
|
467 |
val Ts = map (mk_type thy) idxtys;
|
|
468 |
val T = foldr1 HOLogic.mk_prodT Ts;
|
|
469 |
val U = mk_type thy elty;
|
|
470 |
fun mk_idx' T (e, NONE) = HOLogic.mk_set T [fst (tm_of vs e)]
|
|
471 |
| mk_idx' T (e, SOME e') = Const (@{const_name atLeastAtMost},
|
|
472 |
T --> T --> HOLogic.mk_setT T) $
|
|
473 |
fst (tm_of vs e) $ fst (tm_of vs e');
|
|
474 |
fun mk_idx idx =
|
|
475 |
if length Ts <> length idx then
|
|
476 |
error ("Arity mismatch in construction of array " ^ s)
|
|
477 |
else foldr1 mk_times (map2 mk_idx' Ts idx);
|
|
478 |
fun mk_upd (idxs, e) t =
|
|
479 |
if length idxs = 1 andalso forall (is_none o snd) (hd idxs)
|
|
480 |
then
|
|
481 |
Const (@{const_name fun_upd}, (T --> U) -->
|
|
482 |
T --> U --> T --> U) $ t $
|
|
483 |
foldl1 HOLogic.mk_prod
|
|
484 |
(map (fst o tm_of vs o fst) (hd idxs)) $
|
|
485 |
fst (tm_of vs e)
|
|
486 |
else
|
|
487 |
Const (@{const_name fun_upds}, (T --> U) -->
|
|
488 |
HOLogic.mk_setT T --> U --> T --> U) $ t $
|
|
489 |
foldl1 (HOLogic.mk_binop @{const_name sup})
|
|
490 |
(map mk_idx idxs) $
|
|
491 |
fst (tm_of vs e)
|
|
492 |
in
|
|
493 |
(fold mk_upd assocs
|
|
494 |
(case default of
|
|
495 |
SOME e => Abs ("x", T, fst (tm_of vs e))
|
|
496 |
| NONE => Const (@{const_name undefined}, T --> U)),
|
|
497 |
s)
|
|
498 |
end
|
|
499 |
| _ => error (s ^ " is not an array type"))
|
|
500 |
|
|
501 |
in tm_of end;
|
|
502 |
|
|
503 |
|
|
504 |
fun term_of_rule thy types funs pfuns ids rule =
|
|
505 |
let val tm_of = fst o term_of_expr thy types funs pfuns ids
|
|
506 |
in case rule of
|
|
507 |
Inference_Rule (es, e) => Logic.list_implies
|
|
508 |
(map (HOLogic.mk_Trueprop o tm_of) es, HOLogic.mk_Trueprop (tm_of e))
|
|
509 |
| Substitution_Rule (es, e, e') => Logic.list_implies
|
|
510 |
(map (HOLogic.mk_Trueprop o tm_of) es,
|
|
511 |
HOLogic.mk_Trueprop (HOLogic.mk_eq (tm_of e, tm_of e')))
|
|
512 |
end;
|
|
513 |
|
|
514 |
|
|
515 |
val builtin = Symtab.make (map (rpair ())
|
|
516 |
["->", "<->", "or", "and", "not", "=", "<>", "<", ">", "<=", ">=",
|
|
517 |
"+", "-", "*", "/", "div", "mod", "**"]);
|
|
518 |
|
|
519 |
fun complex_expr (Number _) = false
|
|
520 |
| complex_expr (Ident _) = false
|
|
521 |
| complex_expr (Funct (s, es)) =
|
|
522 |
not (Symtab.defined builtin s) orelse exists complex_expr es
|
|
523 |
| complex_expr (Quantifier (_, _, _, e)) = complex_expr e
|
|
524 |
| complex_expr _ = true;
|
|
525 |
|
|
526 |
fun complex_rule (Inference_Rule (es, e)) =
|
|
527 |
complex_expr e orelse exists complex_expr es
|
|
528 |
| complex_rule (Substitution_Rule (es, e, e')) =
|
|
529 |
complex_expr e orelse complex_expr e' orelse
|
|
530 |
exists complex_expr es;
|
|
531 |
|
|
532 |
val is_pfun =
|
|
533 |
Symtab.defined builtin orf
|
|
534 |
can (unprefix "fld_") orf can (unprefix "upf_") orf
|
|
535 |
can (unsuffix "__pos") orf can (unsuffix "__val") orf
|
|
536 |
equal "succ" orf equal "pred";
|
|
537 |
|
|
538 |
fun fold_opt f = the_default I o Option.map f;
|
|
539 |
fun fold_pair f g (x, y) = f x #> g y;
|
|
540 |
|
|
541 |
fun fold_expr f g (Funct (s, es)) = f s #> fold (fold_expr f g) es
|
|
542 |
| fold_expr f g (Ident s) = g s
|
|
543 |
| fold_expr f g (Number _) = I
|
|
544 |
| fold_expr f g (Quantifier (_, _, _, e)) = fold_expr f g e
|
|
545 |
| fold_expr f g (Element (e, es)) =
|
|
546 |
fold_expr f g e #> fold (fold_expr f g) es
|
|
547 |
| fold_expr f g (Update (e, es, e')) =
|
|
548 |
fold_expr f g e #> fold (fold_expr f g) es #> fold_expr f g e'
|
|
549 |
| fold_expr f g (Record (_, flds)) = fold (fold_expr f g o snd) flds
|
|
550 |
| fold_expr f g (Array (_, default, assocs)) =
|
|
551 |
fold_opt (fold_expr f g) default #>
|
|
552 |
fold (fold_pair
|
|
553 |
(fold (fold (fold_pair
|
|
554 |
(fold_expr f g) (fold_opt (fold_expr f g)))))
|
|
555 |
(fold_expr f g)) assocs;
|
|
556 |
|
|
557 |
val add_expr_pfuns = fold_expr
|
|
558 |
(fn s => if is_pfun s then I else insert (op =) s) (K I);
|
|
559 |
|
|
560 |
val add_expr_idents = fold_expr (K I) (insert (op =));
|
|
561 |
|
|
562 |
fun pfun_type thy (argtys, resty) =
|
|
563 |
map (mk_type thy) argtys ---> mk_type thy resty;
|
|
564 |
|
|
565 |
fun check_pfun_type thy s t optty1 optty2 =
|
|
566 |
let
|
|
567 |
val T = fastype_of t;
|
|
568 |
fun check ty =
|
|
569 |
let val U = pfun_type thy ty
|
|
570 |
in
|
|
571 |
T = U orelse
|
|
572 |
error ("Type\n" ^
|
|
573 |
Syntax.string_of_typ_global thy T ^
|
|
574 |
"\nof function " ^
|
|
575 |
Syntax.string_of_term_global thy t ^
|
|
576 |
" associated with proof function " ^ s ^
|
|
577 |
"\ndoes not match declared type\n" ^
|
|
578 |
Syntax.string_of_typ_global thy U)
|
|
579 |
end
|
|
580 |
in (Option.map check optty1; Option.map check optty2; ()) end;
|
|
581 |
|
|
582 |
fun upd_option x y = if is_some x then x else y;
|
|
583 |
|
|
584 |
fun check_pfuns_types thy funs =
|
|
585 |
Symtab.map (fn s => fn (optty, t) =>
|
|
586 |
let val optty' = lookup funs s
|
|
587 |
in
|
|
588 |
(check_pfun_type thy s t optty optty';
|
|
589 |
(NONE |> upd_option optty |> upd_option optty', t))
|
|
590 |
end);
|
|
591 |
|
|
592 |
|
|
593 |
(** the VC store **)
|
|
594 |
|
|
595 |
fun err_unfinished () = error "An unfinished SPARK environment is still open."
|
|
596 |
|
|
597 |
fun err_vcs names = error (Pretty.string_of
|
|
598 |
(Pretty.big_list "The following verification conditions have not been proved:"
|
|
599 |
(map Pretty.str names)))
|
|
600 |
|
|
601 |
val strip_number = pairself implode o take_suffix Fdl_Lexer.is_digit o raw_explode;
|
|
602 |
|
|
603 |
val name_ord = prod_ord string_ord (option_ord int_ord) o
|
|
604 |
pairself (strip_number ##> Int.fromString);
|
|
605 |
|
|
606 |
structure VCtab = Table(type key = string val ord = name_ord);
|
|
607 |
|
|
608 |
structure VCs = Theory_Data
|
|
609 |
(
|
|
610 |
type T =
|
|
611 |
{pfuns: ((string list * string) option * term) Symtab.table,
|
|
612 |
env:
|
|
613 |
{ctxt: Element.context_i list,
|
|
614 |
defs: (binding * thm) list,
|
|
615 |
types: fdl_type tab,
|
|
616 |
funs: (string list * string) tab,
|
|
617 |
ids: (term * string) Symtab.table * Name.context,
|
|
618 |
proving: bool,
|
|
619 |
vcs: (string * bool *
|
|
620 |
(string * expr) list * (string * expr) list) VCtab.table,
|
|
621 |
path: Path.T} option}
|
|
622 |
val empty : T = {pfuns = Symtab.empty, env = NONE}
|
|
623 |
val extend = I
|
|
624 |
fun merge ({pfuns = pfuns1, env = NONE}, {pfuns = pfuns2, env = NONE}) =
|
|
625 |
{pfuns = Symtab.merge (eq_pair (op =) (op aconv)) (pfuns1, pfuns2),
|
|
626 |
env = NONE}
|
|
627 |
| merge _ = err_unfinished ()
|
|
628 |
)
|
|
629 |
|
|
630 |
fun set_env (env as {funs, ...}) thy = VCs.map (fn
|
|
631 |
{pfuns, env = NONE} =>
|
|
632 |
{pfuns = check_pfuns_types thy funs pfuns, env = SOME env}
|
|
633 |
| _ => err_unfinished ()) thy;
|
|
634 |
|
|
635 |
fun mk_pat s = (case Int.fromString s of
|
|
636 |
SOME i => [HOLogic.mk_Trueprop (Var (("C", i), HOLogic.boolT))]
|
|
637 |
| NONE => error ("Bad conclusion identifier: C" ^ s));
|
|
638 |
|
|
639 |
fun mk_vc thy types funs pfuns ids (tr, proved, ps, cs) =
|
|
640 |
let val prop_of =
|
|
641 |
HOLogic.mk_Trueprop o fst o term_of_expr thy types funs pfuns ids
|
|
642 |
in
|
|
643 |
(tr, proved,
|
|
644 |
Element.Assumes (map (fn (s', e) =>
|
|
645 |
((Binding.name ("H" ^ s'), []), [(prop_of e, [])])) ps),
|
|
646 |
Element.Shows (map (fn (s', e) =>
|
|
647 |
(Attrib.empty_binding, [(prop_of e, mk_pat s')])) cs))
|
|
648 |
end;
|
|
649 |
|
|
650 |
fun fold_vcs f vcs =
|
|
651 |
VCtab.fold (fn (_, (_, _, ps, cs)) => fold f ps #> fold f cs) vcs;
|
|
652 |
|
|
653 |
fun pfuns_of_vcs pfuns vcs =
|
|
654 |
fold_vcs (add_expr_pfuns o snd) vcs [] |>
|
|
655 |
filter_out (Symtab.defined pfuns);
|
|
656 |
|
|
657 |
fun declare_missing_pfuns thy funs pfuns vcs (tab, ctxt) =
|
|
658 |
let
|
|
659 |
val (fs, (tys, Ts)) =
|
|
660 |
pfuns_of_vcs pfuns vcs |>
|
|
661 |
map_filter (fn s => lookup funs s |>
|
|
662 |
Option.map (fn ty => (s, (SOME ty, pfun_type thy ty)))) |>
|
|
663 |
split_list ||> split_list;
|
|
664 |
val (fs', ctxt') = Name.variants fs ctxt
|
|
665 |
in
|
|
666 |
(fold Symtab.update_new (fs ~~ (tys ~~ map Free (fs' ~~ Ts))) pfuns,
|
|
667 |
Element.Fixes (map2 (fn s => fn T =>
|
|
668 |
(Binding.name s, SOME T, NoSyn)) fs' Ts),
|
|
669 |
(tab, ctxt'))
|
|
670 |
end;
|
|
671 |
|
|
672 |
fun add_proof_fun prep (s, (optty, raw_t)) thy =
|
|
673 |
VCs.map (fn
|
|
674 |
{env = SOME {proving = true, ...}, ...} => err_unfinished ()
|
|
675 |
| {pfuns, env} =>
|
|
676 |
let
|
|
677 |
val optty' = (case env of
|
|
678 |
SOME {funs, ...} => lookup funs s
|
|
679 |
| NONE => NONE);
|
|
680 |
val optty'' = NONE |> upd_option optty |> upd_option optty';
|
|
681 |
val t = prep (Option.map (pfun_type thy) optty'') raw_t
|
|
682 |
in
|
|
683 |
(check_pfun_type thy s t optty optty';
|
|
684 |
if is_some optty'' orelse is_none env then
|
|
685 |
{pfuns = Symtab.update_new (s, (optty'', t)) pfuns,
|
|
686 |
env = env}
|
|
687 |
handle Symtab.DUP _ => error ("Proof function " ^ s ^
|
|
688 |
" already associated with function")
|
|
689 |
else error ("Undeclared proof function " ^ s))
|
|
690 |
end) thy;
|
|
691 |
|
|
692 |
val is_closed = is_none o #env o VCs.get;
|
|
693 |
|
|
694 |
fun lookup_vc thy name =
|
|
695 |
(case VCs.get thy of
|
|
696 |
{env = SOME {vcs, types, funs, ids, ctxt, ...}, pfuns} =>
|
|
697 |
(case VCtab.lookup vcs name of
|
|
698 |
SOME vc =>
|
|
699 |
let val (pfuns', ctxt', ids') =
|
|
700 |
declare_missing_pfuns thy funs pfuns vcs ids
|
|
701 |
in SOME (ctxt @ [ctxt'], mk_vc thy types funs pfuns' ids' vc) end
|
|
702 |
| NONE => NONE)
|
|
703 |
| _ => NONE);
|
|
704 |
|
|
705 |
fun get_vcs thy = (case VCs.get thy of
|
|
706 |
{env = SOME {vcs, types, funs, ids, ctxt, defs, ...}, pfuns} =>
|
|
707 |
let val (pfuns', ctxt', ids') =
|
|
708 |
declare_missing_pfuns thy funs pfuns vcs ids
|
|
709 |
in
|
|
710 |
(ctxt @ [ctxt'], defs,
|
|
711 |
VCtab.dest vcs |>
|
|
712 |
map (apsnd (mk_vc thy types funs pfuns' ids')))
|
|
713 |
end
|
|
714 |
| _ => ([], [], []));
|
|
715 |
|
|
716 |
fun mark_proved name = VCs.map (fn
|
|
717 |
{pfuns, env = SOME {ctxt, defs, types, funs, ids, vcs, path, ...}} =>
|
|
718 |
{pfuns = pfuns,
|
|
719 |
env = SOME {ctxt = ctxt, defs = defs,
|
|
720 |
types = types, funs = funs, ids = ids,
|
|
721 |
proving = true,
|
|
722 |
vcs = VCtab.map_entry name (fn (trace, _, ps, cs) =>
|
|
723 |
(trace, true, ps, cs)) vcs,
|
|
724 |
path = path}}
|
|
725 |
| x => x);
|
|
726 |
|
|
727 |
fun close thy = VCs.map (fn
|
|
728 |
{pfuns, env = SOME {vcs, path, ...}} =>
|
|
729 |
(case VCtab.fold_rev (fn (s, (_, p, _, _)) =>
|
|
730 |
(if p then apfst else apsnd) (cons s)) vcs ([], []) of
|
|
731 |
(proved, []) =>
|
|
732 |
(File.write (Path.ext "prv" path)
|
|
733 |
(concat (map (fn s => snd (strip_number s) ^
|
|
734 |
" -- proved by " ^ Distribution.version ^ "\n") proved));
|
|
735 |
{pfuns = pfuns, env = NONE})
|
|
736 |
| (_, unproved) => err_vcs unproved)
|
|
737 |
| x => x) thy;
|
|
738 |
|
|
739 |
|
|
740 |
(** set up verification conditions **)
|
|
741 |
|
|
742 |
fun partition_opt f =
|
|
743 |
let
|
|
744 |
fun part ys zs [] = (rev ys, rev zs)
|
|
745 |
| part ys zs (x :: xs) = (case f x of
|
|
746 |
SOME y => part (y :: ys) zs xs
|
|
747 |
| NONE => part ys (x :: zs) xs)
|
|
748 |
in part [] [] end;
|
|
749 |
|
|
750 |
fun dest_def (id, (Substitution_Rule ([], Ident s, rhs))) = SOME (id, (s, rhs))
|
|
751 |
| dest_def _ = NONE;
|
|
752 |
|
|
753 |
fun mk_rulename (s, i) = Binding.name (s ^ string_of_int i);
|
|
754 |
|
|
755 |
fun add_const (s, ty) ((tab, ctxt), thy) =
|
|
756 |
let
|
|
757 |
val T = mk_type thy ty;
|
|
758 |
val b = Binding.name s;
|
|
759 |
val c = Const (Sign.full_name thy b, T)
|
|
760 |
in
|
|
761 |
(c,
|
|
762 |
((Symtab.update (s, (c, ty)) tab, Name.declare s ctxt),
|
|
763 |
Sign.add_consts_i [(b, T, NoSyn)] thy))
|
|
764 |
end;
|
|
765 |
|
|
766 |
fun add_def types funs pfuns consts (id, (s, e)) (ids as (tab, ctxt), thy) =
|
|
767 |
(case lookup consts s of
|
|
768 |
SOME ty =>
|
|
769 |
let
|
|
770 |
val (t, ty') = term_of_expr thy types funs pfuns ids e;
|
|
771 |
val _ = ty = ty' orelse
|
|
772 |
error ("Declared type " ^ ty ^ " of " ^ s ^
|
|
773 |
"\ndoes not match type " ^ ty' ^ " in definition");
|
|
774 |
val id' = mk_rulename id;
|
|
775 |
val lthy = Named_Target.theory_init thy;
|
|
776 |
val ((t', (_, th)), lthy') = Specification.definition
|
|
777 |
(NONE, ((id', []), HOLogic.mk_Trueprop (HOLogic.mk_eq
|
|
778 |
(Free (s, mk_type thy ty), t)))) lthy;
|
|
779 |
val phi = ProofContext.export_morphism lthy' lthy
|
|
780 |
in
|
|
781 |
((id', Morphism.thm phi th),
|
|
782 |
((Symtab.update (s, (Morphism.term phi t', ty)) tab,
|
|
783 |
Name.declare s ctxt),
|
|
784 |
Local_Theory.exit_global lthy'))
|
|
785 |
end
|
|
786 |
| NONE => error ("Undeclared constant " ^ s));
|
|
787 |
|
|
788 |
fun add_var (s, ty) (ids, thy) =
|
|
789 |
let val ([Free p], ids') = mk_variables thy [s] ty ids
|
|
790 |
in (p, (ids', thy)) end;
|
|
791 |
|
|
792 |
fun add_init_vars vcs (ids_thy as ((tab, _), _)) =
|
|
793 |
fold_map add_var
|
|
794 |
(map_filter
|
|
795 |
(fn s => case try (unsuffix "~") s of
|
|
796 |
SOME s' => (case Symtab.lookup tab s' of
|
|
797 |
SOME (_, ty) => SOME (s, ty)
|
|
798 |
| NONE => error ("Undeclared identifier " ^ s'))
|
|
799 |
| NONE => NONE)
|
|
800 |
(fold_vcs (add_expr_idents o snd) vcs []))
|
|
801 |
ids_thy;
|
|
802 |
|
|
803 |
fun is_trivial_vc ([], [(_, Ident "true")]) = true
|
|
804 |
| is_trivial_vc _ = false;
|
|
805 |
|
|
806 |
fun rulenames rules = commas
|
|
807 |
(map (fn ((s, i), _) => s ^ "(" ^ string_of_int i ^ ")") rules);
|
|
808 |
|
|
809 |
(* sort definitions according to their dependency *)
|
|
810 |
fun sort_defs _ _ [] sdefs = rev sdefs
|
|
811 |
| sort_defs pfuns consts defs sdefs =
|
|
812 |
(case find_first (fn (_, (_, e)) =>
|
|
813 |
forall (Symtab.defined pfuns) (add_expr_pfuns e []) andalso
|
|
814 |
forall (fn id =>
|
|
815 |
member (fn (s, (_, (s', _))) => s = s') sdefs id orelse
|
|
816 |
member (fn (s, (s', _)) => s = s') consts id)
|
|
817 |
(add_expr_idents e [])) defs of
|
|
818 |
SOME d => sort_defs pfuns consts
|
|
819 |
(remove (op =) d defs) (d :: sdefs)
|
|
820 |
| NONE => error ("Bad definitions: " ^ rulenames defs));
|
|
821 |
|
|
822 |
fun set_vcs ({types, vars, consts, funs} : decls) (rules, _) vcs path thy =
|
|
823 |
let
|
|
824 |
val {pfuns, ...} = VCs.get thy;
|
|
825 |
val (defs', rules') = partition_opt dest_def rules;
|
|
826 |
val consts' =
|
|
827 |
subtract (fn ((_, (s, _)), (s', _)) => s = s') defs' (items consts);
|
|
828 |
val defs = sort_defs pfuns consts' defs' [];
|
|
829 |
(* ignore all complex rules in rls files *)
|
|
830 |
val (rules'', other_rules) =
|
|
831 |
List.partition (complex_rule o snd) rules';
|
|
832 |
val _ = if null rules'' then ()
|
|
833 |
else warning ("Ignoring rules: " ^ rulenames rules'');
|
|
834 |
|
|
835 |
val vcs' = VCtab.make (maps (fn (tr, vcs) =>
|
|
836 |
map (fn (s, (ps, cs)) => (s, (tr, false, ps, cs)))
|
|
837 |
(filter_out (is_trivial_vc o snd) vcs)) vcs);
|
|
838 |
|
|
839 |
val _ = (case filter_out (is_some o lookup funs)
|
|
840 |
(pfuns_of_vcs pfuns vcs') of
|
|
841 |
[] => ()
|
|
842 |
| fs => error ("Undeclared proof function(s) " ^ commas fs));
|
|
843 |
|
|
844 |
val (((defs', vars''), ivars), (ids, thy')) =
|
|
845 |
((Symtab.empty |>
|
|
846 |
Symtab.update ("false", (HOLogic.false_const, booleanN)) |>
|
|
847 |
Symtab.update ("true", (HOLogic.true_const, booleanN)),
|
|
848 |
Name.context), thy) |>
|
|
849 |
fold add_type_def (items types) |>
|
|
850 |
fold (snd oo add_const) consts' |>
|
|
851 |
fold_map (add_def types funs pfuns consts) defs ||>>
|
|
852 |
fold_map add_var (items vars) ||>>
|
|
853 |
add_init_vars vcs';
|
|
854 |
|
|
855 |
val ctxt =
|
|
856 |
[Element.Fixes (map (fn (s, T) =>
|
|
857 |
(Binding.name s, SOME T, NoSyn)) (vars'' @ ivars)),
|
|
858 |
Element.Assumes (map (fn (id, rl) =>
|
|
859 |
((mk_rulename id, []),
|
|
860 |
[(term_of_rule thy' types funs pfuns ids rl, [])]))
|
|
861 |
other_rules),
|
|
862 |
Element.Notes (Thm.definitionK,
|
|
863 |
[((Binding.name "defns", []), map (rpair [] o single o snd) defs')])]
|
|
864 |
|
|
865 |
in
|
|
866 |
set_env {ctxt = ctxt, defs = defs', types = types, funs = funs,
|
|
867 |
ids = ids, proving = false, vcs = vcs', path = path} thy'
|
|
868 |
end;
|
|
869 |
|
|
870 |
end;
|