src/HOL/Tools/coinduction.ML

coinduction method accepts a list of coinduction rules (takes the first matching one)

(*  Title:      HOL/Tools/coinduction.ML
    Author:     Johannes Hölzl, TU Muenchen
    Author:     Dmitriy Traytel, TU Muenchen
    Copyright   2013

Coinduction method that avoids some boilerplate compared to coinduct.
*)

signature COINDUCTION =
sig
  val coinduction_context_tactic: term list -> thm list option -> thm list -> int -> context_tactic
  val coinduction_tac: Proof.context -> term list -> thm list option -> thm list -> int -> tactic
end;

structure Coinduction : COINDUCTION =
struct

fun filter_in_out _ [] = ([], [])
  | filter_in_out P (x :: xs) =
      let
        val (ins, outs) = filter_in_out P xs;
      in
        if P x then (x :: ins, outs) else (ins, x :: outs)
      end;

fun ALLGOALS_SKIP skip tac st =
  let fun doall n = if n = skip then all_tac else tac n THEN doall (n - 1)
  in doall (Thm.nprems_of st) st  end;

fun THEN_ALL_NEW_SKIP skip tac1 tac2 i st =
  st |> (tac1 i THEN (fn st' =>
    Seq.INTERVAL tac2 (i + skip) (i + Thm.nprems_of st' - Thm.nprems_of st) st'));

fun DELETE_PREMS_AFTER skip tac i st =
  let
    val n = nth (Thm.prems_of st) (i - 1) |> Logic.strip_assums_hyp |> length;
  in
    (THEN_ALL_NEW_SKIP skip tac (REPEAT_DETERM_N n o eresolve0_tac [thin_rl])) i st
  end;

fun coinduction_context_tactic raw_vars opt_raw_thms prems =
  CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) =>
    let
      val lhs_of_eq = HOLogic.dest_Trueprop #> HOLogic.dest_eq #> fst;
      fun find_coinduct t =
        Induct.find_coinductP ctxt t @
        (try (Induct.find_coinductT ctxt o fastype_of o lhs_of_eq) t |> the_default []);
      val raw_thms =
        (case opt_raw_thms of
          SOME raw_thms => raw_thms
        | NONE => goal |> Logic.strip_assums_concl |> find_coinduct);
      val thy = Proof_Context.theory_of ctxt;
      val concl = Variable.focus NONE goal ctxt |> fst |> snd |> Logic.strip_imp_concl |> @{print};
      val raw_thm = (case find_first (fn thm =>
            Pattern.matches thy (Thm.concl_of thm, concl)) raw_thms of
          SOME thm => thm
        | NONE => error "No matching coinduction rule found")
      val skip = Integer.max 1 (Rule_Cases.get_consumes raw_thm) - 1;
      val cases = Rule_Cases.get raw_thm |> fst;
    in
      ((Object_Logic.rulify_tac ctxt THEN'
        Method.insert_tac ctxt prems THEN'
        Object_Logic.atomize_prems_tac ctxt THEN'
        DELETE_PREMS_AFTER skip (Subgoal.FOCUS (fn {concl, context = ctxt, params, prems, ...} =>
          let
            val vars = raw_vars @ map (Thm.term_of o snd) params;
            val names_ctxt = ctxt
              |> fold Variable.declare_names vars
              |> fold Variable.declare_thm (raw_thm :: prems);
            val thm_concl = Thm.cprop_of raw_thm |> strip_imp_concl;
            val (instT, inst) = Thm.match (thm_concl, concl);
            val rhoTs = map (fn (v, T) => (TVar v, Thm.typ_of T)) instT;
            val rhots = map (fn (v, t) => (Var v, Thm.term_of t)) inst;
            val xs = hd (Thm.prems_of raw_thm) |> HOLogic.dest_Trueprop |> strip_comb |> snd
              |> map (subst_atomic_types rhoTs);
            val raw_eqs = map (fn x => (x, AList.lookup op aconv rhots x |> the)) xs;
            val ((names, ctxt), Ts) = map_split (apfst fst o dest_Var o fst) raw_eqs
              |>> (fn names => Variable.variant_fixes names names_ctxt) ;
            val eqs =
              @{map 3} (fn name => fn T => fn (_, rhs) =>
                HOLogic.mk_eq (Free (name, T), rhs))
              names Ts raw_eqs;
            val phi = eqs @ map (HOLogic.dest_Trueprop o Thm.prop_of) prems
              |> try (Library.foldr1 HOLogic.mk_conj)
              |> the_default @{term True}
              |> Ctr_Sugar_Util.list_exists_free vars
              |> Term.map_abs_vars (Variable.revert_fixed ctxt)
              |> fold_rev Term.absfree (names ~~ Ts)
              |> Thm.cterm_of ctxt;
            val thm = infer_instantiate' ctxt [SOME phi] raw_thm;
            val e = length eqs;
            val p = length prems;
          in
            HEADGOAL (EVERY' [resolve_tac ctxt [thm],
              EVERY' (map (fn var =>
                resolve_tac ctxt
                  [infer_instantiate' ctxt [NONE, SOME (Thm.cterm_of ctxt var)] exI]) vars),
              if p = 0 then Ctr_Sugar_Util.CONJ_WRAP' (K (resolve_tac ctxt [refl])) eqs
              else
                REPEAT_DETERM_N e o (resolve_tac ctxt [conjI] THEN' resolve_tac ctxt [refl]) THEN'
                Ctr_Sugar_Util.CONJ_WRAP' (resolve_tac ctxt o single) prems,
              K (ALLGOALS_SKIP skip
                 (REPEAT_DETERM_N (length vars) o (eresolve_tac ctxt [exE] THEN' rotate_tac ~1) THEN'
                 DELETE_PREMS_AFTER 0 (Subgoal.FOCUS (fn {prems, params, context = ctxt, ...} =>
                   (case prems of
                     [] => all_tac
                   | inv :: case_prems =>
                       let
                         val (init, last) = funpow_yield (p + e - 1) (HOLogic.conj_elim ctxt) inv;
                         val inv_thms = init @ [last];
                         val eqs = take e inv_thms;
                         fun is_local_var t =
                           member (fn (t, (_, t')) => t aconv (Thm.term_of t')) params t;
                          val (eqs, assms') =
                            filter_in_out (is_local_var o lhs_of_eq o Thm.prop_of) eqs;
                          val assms = assms' @ drop e inv_thms
                        in
                          HEADGOAL (Method.insert_tac ctxt (assms @ case_prems)) THEN
                          Ctr_Sugar_General_Tactics.unfold_thms_tac ctxt eqs
                        end)) ctxt)))])
          end) ctxt) THEN'
        K (prune_params_tac ctxt)) i) st
      |> Seq.maps (fn st' =>
        CONTEXT_CASES (Rule_Cases.make_common ctxt (Thm.prop_of st') cases) all_tac (ctxt, st'))
    end);

fun coinduction_tac ctxt x1 x2 x3 x4 =
  Method.NO_CONTEXT_TACTIC ctxt (coinduction_context_tactic x1 x2 x3 x4);


local

val ruleN = "rule"
val arbitraryN = "arbitrary"

fun named_rule k arg get =
  Scan.lift (Args.$$$ k -- Args.colon) |-- Scan.repeat arg :|--
    (fn names => Scan.peek (fn context => Scan.succeed (names |> map (fn name =>
      (case get (Context.proof_of context) name of SOME x => x
      | NONE => error ("No rule for " ^ k ^ " " ^ quote name))))));

fun rule get_type get_pred =
  named_rule Induct.typeN (Args.type_name {proper = false, strict = false}) get_type ||
  named_rule Induct.predN (Args.const {proper = false, strict = false}) get_pred ||
  named_rule Induct.setN (Args.const {proper = false, strict = false}) get_pred ||
  Scan.lift (Args.$$$ ruleN -- Args.colon) |-- Attrib.thms;

val coinduct_rules =
  rule Induct.lookup_coinductT Induct.lookup_coinductP;

fun unless_more_args scan = Scan.unless (Scan.lift
  ((Args.$$$ arbitraryN || Args.$$$ Induct.typeN ||
    Args.$$$ Induct.predN || Args.$$$ Induct.setN || Args.$$$ ruleN) -- Args.colon)) scan;

val arbitrary = Scan.optional (Scan.lift (Args.$$$ arbitraryN -- Args.colon) |--
  Scan.repeat1 (unless_more_args Args.term)) [];

in

val _ =
  Theory.setup
    (Method.setup @{binding coinduction}
      (arbitrary -- Scan.option coinduct_rules >>
        (fn (arbitrary, opt_rules) => fn _ => fn facts =>
          Seq.DETERM (coinduction_context_tactic arbitrary opt_rules facts 1)))
      "coinduction on types or predicates/sets");

end;

end;