src/HOL/Nominal/nominal_inductive.ML

chained facts are now included

(*  Title:      HOL/Nominal/nominal_inductive.ML
    ID:         $Id$
    Author:     Stefan Berghofer, TU Muenchen

Infrastructure for proving equivariance and strong induction theorems
for inductive predicates involving nominal datatypes.
*)

signature NOMINAL_INDUCTIVE =
sig
  val prove_strong_ind: string -> (string * string list) list -> theory -> Proof.state
  val prove_eqvt: string -> string list -> theory -> theory
end

structure NominalInductive : NOMINAL_INDUCTIVE =
struct

val finite_Un = thm "finite_Un";
val supp_prod = thm "supp_prod";
val fresh_prod = thm "fresh_prod";

val perm_boolI = thm "perm_boolI";
val (_, [perm_boolI_pi, _]) = Drule.strip_comb (snd (Thm.dest_comb
  (Drule.strip_imp_concl (cprop_of perm_boolI))));

val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE;

fun transp ([] :: _) = []
  | transp xs = map hd xs :: transp (map tl xs);

fun add_binders thy i (t as (_ $ _)) bs = (case strip_comb t of
      (Const (s, T), ts) => (case strip_type T of
        (Ts, Type (tname, _)) =>
          (case NominalPackage.get_nominal_datatype thy tname of
             NONE => fold (add_binders thy i) ts bs
           | SOME {descr, index, ...} => (case AList.lookup op =
                 (#3 (the (AList.lookup op = descr index))) s of
               NONE => fold (add_binders thy i) ts bs
             | SOME cargs => fst (fold (fn (xs, x) => fn (bs', cargs') =>
                 let val (cargs1, (u, _) :: cargs2) = chop (length xs) cargs'
                 in (add_binders thy i u
                   (fold (fn (u, T) =>
                      if exists (fn j => j < i) (loose_bnos u) then I
                      else insert (op aconv o pairself fst)
                        (incr_boundvars (~i) u, T)) cargs1 bs'), cargs2)
                 end) cargs (bs, ts ~~ Ts))))
      | _ => fold (add_binders thy i) ts bs)
    | (u, ts) => add_binders thy i u (fold (add_binders thy i) ts bs))
  | add_binders thy i (Abs (_, _, t)) bs = add_binders thy (i + 1) t bs
  | add_binders thy i _ bs = bs;

fun prove_strong_ind s avoids thy =
  let
    val ctxt = ProofContext.init thy;
    val ({names, ...}, {raw_induct, ...}) =
      InductivePackage.the_inductive ctxt (Sign.intern_const thy s);
    val eqvt_thms = NominalThmDecls.get_eqvt_thms thy;
    val _ = (case names \\ foldl (apfst prop_of #> add_term_consts) [] eqvt_thms of
        [] => ()
      | xs => error ("Missing equivariance theorem for predicate(s): " ^
          commas_quote xs));
    val induct_cases = map fst (fst (RuleCases.get (the
      (InductAttrib.lookup_inductS ctxt (hd names)))));
    val raw_induct' = Logic.unvarify (prop_of raw_induct);
    val concls = raw_induct' |> Logic.strip_imp_concl |> HOLogic.dest_Trueprop |>
      HOLogic.dest_conj |> map (HOLogic.dest_imp ##> strip_comb);
    val ps = map (fst o snd) concls;

    val _ = (case duplicates (op = o pairself fst) avoids of
        [] => ()
      | xs => error ("Duplicate case names: " ^ commas_quote (map fst xs)));
    val _ = assert_all (null o duplicates op = o snd) avoids
      (fn (a, _) => error ("Duplicate variable names for case " ^ quote a));
    val _ = (case map fst avoids \\ induct_cases of
        [] => ()
      | xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs));
    val avoids' = map (fn name =>
      (name, the_default [] (AList.lookup op = avoids name))) induct_cases;
    fun mk_avoids params (name, ps) =
      let val k = length params - 1
      in map (fn x => case find_index (equal x o fst) params of
          ~1 => error ("No such variable in case " ^ quote name ^
            " of inductive definition: " ^ quote x)
        | i => (Bound (k - i), snd (nth params i))) ps
      end;

    val prems = map (fn (prem, avoid) =>
      let
        val prems = map (incr_boundvars 1) (Logic.strip_assums_hyp prem);
        val concl = incr_boundvars 1 (Logic.strip_assums_concl prem);
        val params = Logic.strip_params prem
      in
        (params,
         fold (add_binders thy 0) (prems @ [concl]) [] @
           map (apfst (incr_boundvars 1)) (mk_avoids params avoid),
         prems, strip_comb (HOLogic.dest_Trueprop concl))
      end) (Logic.strip_imp_prems raw_induct' ~~ avoids');

    val atomTs = distinct op = (maps (map snd o #2) prems);
    val ind_sort = if null atomTs then HOLogic.typeS
      else Sign.certify_sort thy (map (fn T => Sign.intern_class thy
        ("fs_" ^ Sign.base_name (fst (dest_Type T)))) atomTs);
    val fs_ctxt_tyname = Name.variant (map fst (term_tfrees raw_induct')) "'n";
    val fs_ctxt_name = Name.variant (add_term_names (raw_induct', [])) "z";
    val fsT = TFree (fs_ctxt_tyname, ind_sort);

    fun lift_pred' t (Free (s, T)) ts =
      list_comb (Free (s, fsT --> T), t :: ts);
    val lift_pred = lift_pred' (Bound 0);

    fun lift_prem (Const ("Trueprop", _) $ t) =
          let val (u, ts) = strip_comb t
          in
            if u mem ps then
              all fsT $ Abs ("z", fsT, HOLogic.mk_Trueprop
                (lift_pred u (map (incr_boundvars 1) ts)))
            else HOLogic.mk_Trueprop (lift_prem t)
          end
      | lift_prem (t as (f $ u)) =
          let val (p, ts) = strip_comb t
          in
            if p mem ps then
              HOLogic.all_const fsT $ Abs ("z", fsT,
                lift_pred p (map (incr_boundvars 1) ts))
            else lift_prem f $ lift_prem u
          end
      | lift_prem (Abs (s, T, t)) = Abs (s, T, lift_prem t)
      | lift_prem t = t;

    fun mk_distinct [] = []
      | mk_distinct ((x, T) :: xs) = List.mapPartial (fn (y, U) =>
          if T = U then SOME (HOLogic.mk_Trueprop
            (HOLogic.mk_not (HOLogic.eq_const T $ x $ y)))
          else NONE) xs @ mk_distinct xs;

    fun mk_fresh (x, T) = HOLogic.mk_Trueprop
      (Const ("Nominal.fresh", T --> fsT --> HOLogic.boolT) $ x $ Bound 0);

    val (prems', prems'') = split_list (map (fn (params, bvars, prems, (p, ts)) =>
      let
        val params' = params @ [("y", fsT)];
        val prem = Logic.list_implies
          (map mk_fresh bvars @ mk_distinct bvars @
           map (fn prem =>
             if null (term_frees prem inter ps) then prem
             else lift_prem prem) prems,
           HOLogic.mk_Trueprop (lift_pred p ts));
        val vs = map (Var o apfst (rpair 0)) (rename_wrt_term prem params')
      in
        (list_all (params', prem), (rev vs, subst_bounds (vs, prem)))
      end) prems);

    val ind_vars =
      (DatatypeProp.indexify_names (replicate (length atomTs) "pi") ~~
       map NominalAtoms.mk_permT atomTs) @ [("z", fsT)];
    val ind_Ts = rev (map snd ind_vars);

    val concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
      (map (fn (prem, (p, ts)) => HOLogic.mk_imp (prem,
        HOLogic.list_all (ind_vars, lift_pred p
          (map (fold_rev (NominalPackage.mk_perm ind_Ts)
            (map Bound (length atomTs downto 1))) ts)))) concls));

    val concl' = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
      (map (fn (prem, (p, ts)) => HOLogic.mk_imp (prem,
        lift_pred' (Free (fs_ctxt_name, fsT)) p ts)) concls));

    val vc_compat = map (fn (params, bvars, prems, (p, ts)) =>
      map (fn q => list_all (params, incr_boundvars ~1 (Logic.list_implies
          (filter (fn prem => null (ps inter term_frees prem)) prems, q))))
        (mk_distinct bvars @
         maps (fn (t, T) => map (fn (u, U) => HOLogic.mk_Trueprop
           (Const ("Nominal.fresh", U --> T --> HOLogic.boolT) $ u $ t)) bvars)
             (ts ~~ binder_types (fastype_of p)))) prems;

    val eqvt_ss = HOL_basic_ss addsimps eqvt_thms;
    val fresh_bij = PureThy.get_thms thy (Name "fresh_bij");
    val perm_bij = PureThy.get_thms thy (Name "perm_bij");
    val fs_atoms = map (fn aT => PureThy.get_thm thy
      (Name ("fs_" ^ Sign.base_name (fst (dest_Type aT)) ^ "1"))) atomTs;
    val exists_fresh' = PureThy.get_thms thy (Name "exists_fresh'");
    val fresh_atm = PureThy.get_thms thy (Name "fresh_atm");
    val calc_atm = PureThy.get_thms thy (Name "calc_atm");
    val perm_fresh_fresh = PureThy.get_thms thy (Name "perm_fresh_fresh");
    val pt2_atoms = map (fn aT => PureThy.get_thm thy
      (Name ("pt_" ^ Sign.base_name (fst (dest_Type aT)) ^ "2")) RS sym) atomTs;

    fun obtain_fresh_name ts T (freshs1, freshs2, ctxt) =
      let
        (** protect terms to avoid that supp_prod interferes with   **)
        (** pairs used in introduction rules of inductive predicate **)
        fun protect t =
          let val T = fastype_of t in Const ("Fun.id", T --> T) $ t end;
        val p = foldr1 HOLogic.mk_prod (map protect ts @ freshs1);
        val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
            (HOLogic.exists_const T $ Abs ("x", T,
              Const ("Nominal.fresh", T --> fastype_of p --> HOLogic.boolT) $
                Bound 0 $ p)))
          (fn _ => EVERY
            [resolve_tac exists_fresh' 1,
             simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);
        val (([cx], ths), ctxt') = Obtain.result
          (fn _ => EVERY
            [etac exE 1,
             full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,
             full_simp_tac (HOL_basic_ss addsimps [id_apply]) 1,
             REPEAT (etac conjE 1)])
          [ex] ctxt
      in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;

    fun mk_proof thy thss =
      let val ctxt = ProofContext.init thy
      in Goal.prove_global thy [] prems' concl' (fn ihyps =>
        let val th = Goal.prove ctxt [] [] concl (fn {context, ...} =>
          rtac raw_induct 1 THEN
          EVERY (maps (fn ((((_, bvars, oprems, _), vc_compat_ths), ihyp), (vs, ihypt)) =>
            [REPEAT (rtac allI 1), simp_tac eqvt_ss 1,
             SUBPROOF (fn {prems = gprems, params, concl, context = ctxt', ...} =>
               let
                 val (params', (pis, z)) =
                   chop (length params - length atomTs - 1) (map term_of params) ||>
                   split_last;
                 val bvars' = map
                   (fn (Bound i, T) => (nth params' (length params' - i), T)
                     | (t, T) => (t, T)) bvars;
                 val pi_bvars = map (fn (t, _) =>
                   fold_rev (NominalPackage.mk_perm []) pis t) bvars';
                 val (P, ts) = strip_comb (HOLogic.dest_Trueprop (term_of concl));
                 val (freshs1, freshs2, ctxt'') = fold
                   (obtain_fresh_name (ts @ pi_bvars))
                   (map snd bvars') ([], [], ctxt');
                 val freshs2' = NominalPackage.mk_not_sym freshs2;
                 val pis' = map NominalPackage.perm_of_pair (pi_bvars ~~ freshs1);
                 val env = Pattern.first_order_match thy (ihypt, prop_of ihyp)
                   (Vartab.empty, Vartab.empty);
                 val ihyp' = Thm.instantiate ([], map (pairself (cterm_of thy))
                   (map (Envir.subst_vars env) vs ~~
                    map (fold_rev (NominalPackage.mk_perm [])
                      (rev pis' @ pis)) params' @ [z])) ihyp;
                 val (gprems1, gprems2) = pairself (map fst) (List.partition
                   (fn (th, t) => null (term_frees t inter ps)) (gprems ~~ oprems));
                 val vc_compat_ths' = map (fn th =>
                   let
                     val th' = gprems1 MRS
                       Thm.instantiate (Thm.first_order_match
                         (Conjunction.mk_conjunction_balanced (cprems_of th),
                          Conjunction.mk_conjunction_balanced (map cprop_of gprems1))) th;
                     val (bop, lhs, rhs) = (case concl_of th' of
                         _ $ (fresh $ lhs $ rhs) =>
                           (fn t => fn u => fresh $ t $ u, lhs, rhs)
                       | _ $ (_ $ (_ $ lhs $ rhs)) =>
                           (curry (HOLogic.mk_not o HOLogic.mk_eq), lhs, rhs));
                     val th'' = Goal.prove ctxt'' [] [] (HOLogic.mk_Trueprop
                         (bop (fold_rev (NominalPackage.mk_perm []) pis lhs)
                            (fold_rev (NominalPackage.mk_perm []) pis rhs)))
                       (fn _ => simp_tac (HOL_basic_ss addsimps
                          (fresh_bij @ perm_bij)) 1 THEN rtac th' 1)
                   in Simplifier.simplify (eqvt_ss addsimps fresh_atm) th'' end)
                     vc_compat_ths;
                 val vc_compat_ths'' = NominalPackage.mk_not_sym vc_compat_ths';
                 val gprems1' = map (fn th => fold_rev (fn pi => fn th' =>
                   Simplifier.simplify eqvt_ss (th' RS Drule.cterm_instantiate
                     [(perm_boolI_pi, cterm_of thy pi)] perm_boolI))
                       (rev pis' @ pis) th) gprems1;
                 val gprems2' = map (Simplifier.simplify eqvt_ss) gprems2;
                 (** Since calc_atm simplifies (pi :: 'a prm) o (x :: 'b) to x **)
                 (** we have to pre-simplify the rewrite rules                 **)
                 val calc_atm_ss = HOL_ss addsimps calc_atm @
                    map (Simplifier.simplify (HOL_ss addsimps calc_atm))
                      (vc_compat_ths'' @ freshs2');
                 val th = Goal.prove ctxt'' [] []
                   (HOLogic.mk_Trueprop (list_comb (P $ hd ts,
                     map (fold (NominalPackage.mk_perm []) pis') (tl ts))))
                   (fn _ => EVERY ([simp_tac eqvt_ss 1, rtac ihyp' 1,
                     REPEAT_DETERM_N (nprems_of ihyp - length gprems)
                       (simp_tac calc_atm_ss 1),
                     REPEAT_DETERM_N (length gprems)
                       (resolve_tac gprems1' 1 ORELSE
                        simp_tac (HOL_basic_ss addsimps pt2_atoms @ gprems2'
                          addsimprocs [NominalPackage.perm_simproc]) 1)]));
                 val final = Goal.prove ctxt'' [] [] (term_of concl)
                   (fn _ => cut_facts_tac [th] 1 THEN full_simp_tac (HOL_ss
                     addsimps vc_compat_ths'' @ freshs2' @
                       perm_fresh_fresh @ fresh_atm) 1);
                 val final' = ProofContext.export ctxt'' ctxt' [final];
               in resolve_tac final' 1 end) context 1])
                 (prems ~~ thss ~~ ihyps ~~ prems'')))
        in
          cut_facts_tac [th] 1 THEN REPEAT (etac conjE 1) THEN
          REPEAT (REPEAT (resolve_tac [conjI, impI] 1) THEN
            etac impE 1 THEN atac 1 THEN REPEAT (etac allE_Nil 1) THEN
            asm_full_simp_tac (simpset_of thy) 1)
        end)
      end;

  in
    thy |>
    ProofContext.init |>
    Proof.theorem_i NONE (fn thss => ProofContext.theory (fn thy =>
      let
        val ctxt = ProofContext.init thy;
        val rec_name = space_implode "_" (map Sign.base_name names);
        val ind_case_names = RuleCases.case_names induct_cases;
        val strong_raw_induct = mk_proof thy thss;
        val strong_induct =
          if length names > 1 then
            (strong_raw_induct, [ind_case_names, RuleCases.consumes 0])
          else (strong_raw_induct RSN (2, rev_mp),
            [ind_case_names, RuleCases.consumes 1]);
        val ([strong_induct'], thy') = thy |>
          Theory.add_path rec_name |>
          PureThy.add_thms [(("strong_induct", #1 strong_induct), #2 strong_induct)];
        val strong_inducts =
          ProjectRule.projects ctxt (1 upto length names) strong_induct'
      in
        thy' |>
        PureThy.add_thmss [(("strong_inducts", strong_inducts),
          [ind_case_names, RuleCases.consumes 1])] |> snd |>
        Theory.parent_path
      end))
      (map (map (rpair [])) vc_compat)
  end;

fun prove_eqvt s xatoms thy =
  let
    val ctxt = ProofContext.init thy;
    val ({names, ...}, {raw_induct, intrs, elims, ...}) =
      InductivePackage.the_inductive ctxt (Sign.intern_const thy s);
    val intrs' = InductivePackage.unpartition_rules intrs
      (map (fn (((s, ths), (_, k)), th) =>
           (s, ths ~~ InductivePackage.infer_intro_vars th k ths))
         (InductivePackage.partition_rules raw_induct intrs ~~
          InductivePackage.arities_of raw_induct ~~ elims));
    val atoms' = NominalAtoms.atoms_of thy;
    val atoms =
      if null xatoms then atoms' else
      let val atoms = map (Sign.intern_type thy) xatoms
      in
        (case duplicates op = atoms of
             [] => ()
           | xs => error ("Duplicate atoms: " ^ commas xs);
         case atoms \\ atoms' of
             [] => ()
           | xs => error ("No such atoms: " ^ commas xs);
         atoms)
      end;
    val eqvt_ss = HOL_basic_ss addsimps NominalThmDecls.get_eqvt_thms thy;
    val t = Logic.unvarify (concl_of raw_induct);
    val pi = Name.variant (add_term_names (t, [])) "pi";
    val ps = map (fst o HOLogic.dest_imp)
      (HOLogic.dest_conj (HOLogic.dest_Trueprop t));
    fun eqvt_tac th pi (intr, vs) st =
      let
        fun eqvt_err s = error
          ("Could not prove equivariance for introduction rule\n" ^
           Sign.string_of_term (theory_of_thm intr)
             (Logic.unvarify (prop_of intr)) ^ "\n" ^ s);
        val res = SUBPROOF (fn {prems, params, ...} =>
          let
            val prems' = map (fn th' => Simplifier.simplify eqvt_ss
              (if null (names inter term_consts (prop_of th')) then th' RS th
               else th')) prems;
            val intr' = Drule.cterm_instantiate (map (cterm_of thy) vs ~~
               map (cterm_of thy o NominalPackage.mk_perm [] pi o term_of) params)
               intr
          in (rtac intr' THEN_ALL_NEW (TRY o resolve_tac prems')) 1
          end) ctxt 1 st
      in
        case (Seq.pull res handle THM (s, _, _) => eqvt_err s) of
          NONE => eqvt_err ("Rule does not match goal\n" ^
            Sign.string_of_term (theory_of_thm st) (hd (prems_of st)))
        | SOME (th, _) => Seq.single th
      end;
    val thss = map (fn atom =>
      let
        val pi' = Free (pi, NominalAtoms.mk_permT (Type (atom, [])));
        val perm_boolI' = Drule.cterm_instantiate
          [(perm_boolI_pi, cterm_of thy pi')] perm_boolI
      in map (fn th => zero_var_indexes (th RS mp))
        (DatatypeAux.split_conj_thm (Goal.prove_global thy [] []
          (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn p =>
            HOLogic.mk_imp (p, list_comb
             (apsnd (map (NominalPackage.mk_perm [] pi')) (strip_comb p)))) ps)))
          (fn _ => EVERY (rtac raw_induct 1 :: map (fn intr_vs =>
              full_simp_tac eqvt_ss 1 THEN
              eqvt_tac perm_boolI' pi' intr_vs) intrs'))))
      end) atoms
  in
    fold (fn (name, ths) =>
      Theory.add_path (Sign.base_name name) #>
      PureThy.add_thmss [(("eqvt", ths), [NominalThmDecls.eqvt_add])] #> snd #>
      Theory.parent_path) (names ~~ transp thss) thy
  end;


(* outer syntax *)

local structure P = OuterParse and K = OuterKeyword in

val nominal_inductiveP =
  OuterSyntax.command "nominal_inductive"
    "prove equivariance and strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal
    (P.name -- Scan.optional (P.$$$ "avoids" |-- P.and_list1 (P.name --
      (P.$$$ ":" |-- Scan.repeat1 P.name))) [] >> (fn (name, avoids) =>
        Toplevel.print o Toplevel.theory_to_proof (prove_strong_ind name avoids)));

val equivarianceP =
  OuterSyntax.command "equivariance"
    "prove equivariance for inductive predicate involving nominal datatypes" K.thy_decl
    (P.name -- Scan.optional (P.$$$ "[" |-- P.list1 P.name --| P.$$$ "]") [] >>
      (fn (name, atoms) => Toplevel.theory (prove_eqvt name atoms)));

val _ = OuterSyntax.add_keywords ["avoids"];
val _ = OuterSyntax.add_parsers [nominal_inductiveP, equivarianceP];

end;

end