(* 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 inductive_forall_name = "HOL.induct_forall";
val inductive_forall_def = thm "induct_forall_def";
val inductive_atomize = thms "induct_atomize";
val inductive_rulify = thms "induct_rulify";
fun rulify_term thy = MetaSimplifier.rewrite_term thy inductive_rulify [];
val atomize_conv =
MetaSimplifier.rewrite_cterm (true, false, false) (K (K NONE))
(HOL_basic_ss addsimps inductive_atomize);
val atomize_intr = Conv.fconv_rule (Conv.prems_conv ~1 atomize_conv);
fun atomize_induct ctxt = Conv.fconv_rule (Conv.prems_conv ~1
(Conv.forall_conv ~1 (K (Conv.prems_conv ~1 atomize_conv)) ctxt));
val finite_Un = thm "finite_Un";
val supp_prod = thm "supp_prod";
val fresh_prod = thm "fresh_prod";
val perm_bool = mk_meta_eq (thm "perm_bool");
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))));
fun mk_perm_bool_simproc names = Simplifier.simproc_i
(theory_of_thm perm_bool) "perm_bool" [@{term "perm pi x"}] (fn thy => fn ss =>
fn Const ("Nominal.perm", _) $ _ $ t =>
if the_default "" (try (head_of #> dest_Const #> fst) t) mem names
then SOME perm_bool else NONE
| _ => NONE);
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 split_conj f names (Const ("op &", _) $ p $ q) _ = (case head_of p of
Const (name, _) =>
if name mem names then SOME (f p q) else NONE
| _ => NONE)
| split_conj _ _ _ _ = NONE;
fun strip_all [] t = t
| strip_all (_ :: xs) (Const ("All", _) $ Abs (s, T, t)) = strip_all xs t;
(*********************************************************************)
(* maps R ... & (ALL pi_1 ... pi_n z. P z (pi_1 o ... o pi_n o t)) *)
(* or ALL pi_1 ... pi_n. P (pi_1 o ... o pi_n o t) *)
(* to R ... & id (ALL z. (pi_1 o ... o pi_n o t)) *)
(* or id (ALL z. (pi_1 o ... o pi_n o t)) *)
(* *)
(* where "id" protects the subformula from simplification *)
(*********************************************************************)
fun inst_conj_all names ps pis (Const ("op &", _) $ p $ q) _ =
(case head_of p of
Const (name, _) =>
if name mem names then SOME (HOLogic.mk_conj (p,
Const ("Fun.id", HOLogic.boolT --> HOLogic.boolT) $
(subst_bounds (pis, strip_all pis q))))
else NONE
| _ => NONE)
| inst_conj_all names ps pis t u =
if member (op aconv) ps (head_of u) then
SOME (Const ("Fun.id", HOLogic.boolT --> HOLogic.boolT) $
(subst_bounds (pis, strip_all pis t)))
else NONE
| inst_conj_all _ _ _ _ _ = NONE;
fun inst_conj_all_tac k = EVERY
[TRY (EVERY [etac conjE 1, rtac conjI 1, atac 1]),
REPEAT_DETERM_N k (etac allE 1),
simp_tac (HOL_basic_ss addsimps [id_apply]) 1];
fun map_term f t u = (case f t u of
NONE => map_term' f t u | x => x)
and map_term' f (t $ u) (t' $ u') = (case (map_term f t t', map_term f u u') of
(NONE, NONE) => NONE
| (SOME t'', NONE) => SOME (t'' $ u)
| (NONE, SOME u'') => SOME (t $ u'')
| (SOME t'', SOME u'') => SOME (t'' $ u''))
| map_term' f (Abs (s, T, t)) (Abs (s', T', t')) = (case map_term f t t' of
NONE => NONE
| SOME t'' => SOME (Abs (s, T, t'')))
| map_term' _ _ _ = NONE;
(*********************************************************************)
(* Prove F[f t] from F[t], where F is monotone *)
(*********************************************************************)
fun map_thm ctxt f tac monos opt th =
let
val prop = prop_of th;
fun prove t =
Goal.prove ctxt [] [] t (fn _ =>
EVERY [cut_facts_tac [th] 1, etac rev_mp 1,
REPEAT_DETERM (FIRSTGOAL (resolve_tac monos)),
REPEAT_DETERM (rtac impI 1 THEN (atac 1 ORELSE tac))])
in Option.map prove (map_term f prop (the_default prop opt)) end;
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 raw_induct = atomize_induct ctxt raw_induct;
val monos = InductivePackage.get_monos ctxt;
val eqvt_thms = NominalThmDecls.get_eqvt_thms ctxt;
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
(Induct.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);
val inductive_forall_def' = Drule.instantiate'
[SOME (ctyp_of thy fsT)] [] inductive_forall_def;
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 (t as (f $ u)) =
let val (p, ts) = strip_comb t
in
if p mem ps then
Const (inductive_forall_name,
(fsT --> HOLogic.boolT) --> HOLogic.boolT) $
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
(List.mapPartial (fn prem =>
if null (ps inter term_frees prem) then SOME prem
else map_term (split_conj (K o I) names) prem 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 perm_pi_simp = PureThy.get_thms thy (Name "perm_pi_simp");
val pt2_atoms = map (fn aT => PureThy.get_thm thy
(Name ("pt_" ^ Sign.base_name (fst (dest_Type aT)) ^ "2"))) atomTs;
val eqvt_ss = HOL_basic_ss addsimps (eqvt_thms @ perm_pi_simp @ pt2_atoms)
addsimprocs [mk_perm_bool_simproc ["Fun.id"]];
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");
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);
fun concat_perm pi1 pi2 =
let val T = fastype_of pi1
in if T = fastype_of pi2 then
Const ("List.append", T --> T --> T) $ pi1 $ pi2
else pi2
end;
val pis'' = fold (concat_perm #> map) pis' pis;
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;
fun mk_pi th =
Simplifier.simplify (HOL_basic_ss addsimps [id_apply]
addsimprocs [NominalPackage.perm_simproc])
(Simplifier.simplify eqvt_ss
(fold_rev (fn pi => fn th' => th' RS Drule.cterm_instantiate
[(perm_boolI_pi, cterm_of thy pi)] perm_boolI)
(rev pis' @ pis) th));
val (gprems1, gprems2) = split_list
(map (fn (th, t) =>
if null (term_frees t inter ps) then (SOME th, mk_pi th)
else
(map_thm ctxt (split_conj (K o I) names)
(etac conjunct1 1) monos NONE th,
mk_pi (the (map_thm ctxt (inst_conj_all names ps (rev pis''))
(inst_conj_all_tac (length pis'')) monos (SOME t) th))))
(gprems ~~ oprems)) |>> List.mapPartial I;
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';
(** 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)
(simp_tac (HOL_ss
addsimps inductive_forall_def' :: 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 (map (map atomize_intr) thss) |>
InductivePackage.rulify;
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 |>
Sign.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 |>
Sign.parent_path
end))
(map (map (rulify_term thy #> 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 raw_induct = atomize_induct ctxt raw_induct;
val elims = map (atomize_induct ctxt) elims;
val intrs = map atomize_intr intrs;
val monos = InductivePackage.get_monos ctxt;
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 perm_pi_simp = PureThy.get_thms thy (Name "perm_pi_simp");
val eqvt_ss = HOL_basic_ss addsimps
(NominalThmDecls.get_eqvt_thms ctxt @ perm_pi_simp) addsimprocs
[mk_perm_bool_simproc names];
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 => the_default th (map_thm ctxt
(split_conj (K I) names) (etac conjunct2 1) monos NONE th)) prems;
val prems'' = map (fn th' =>
Simplifier.simplify eqvt_ss (th' RS 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) =>
Sign.add_path (Sign.base_name name) #>
PureThy.add_thmss [(("eqvt", ths), [NominalThmDecls.eqvt_add])] #> snd #>
Sign.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