farewell to old-style mem infixes -- type inference in situations with mem_int and mem_string should provide enough information to resolve the type of (op =)
(* Title: HOL/Nominal/nominal_inductive2.ML
Author: Stefan Berghofer, TU Muenchen
Infrastructure for proving equivariance and strong induction theorems
for inductive predicates involving nominal datatypes.
Experimental version that allows to avoid lists of atoms.
*)
signature NOMINAL_INDUCTIVE2 =
sig
val prove_strong_ind: string -> string option -> (string * string list) list -> local_theory -> Proof.state
end
structure NominalInductive2 : NOMINAL_INDUCTIVE2 =
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.params_conv ~1 (K (Conv.prems_conv ~1 atomize_conv)) ctxt));
val fresh_postprocess =
Simplifier.full_simplify (HOL_basic_ss addsimps
[@{thm fresh_star_set_eq}, @{thm fresh_star_Un_elim},
@{thm fresh_star_insert_elim}, @{thm fresh_star_empty_elim}]);
fun preds_of ps t = inter (op = o apfst dest_Free) (Term.add_frees t []) ps;
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 pi th = th RS Drule.cterm_instantiate
[(perm_boolI_pi, pi)] 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 member (op =) names (the_default "" (try (head_of #> dest_Const #> fst) t))
then SOME perm_bool else NONE
| _ => NONE);
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 NominalDatatype.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 AList.map_default op = (T, [])
(insert op aconv (incr_boundvars (~i) u)))
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 member (op =) names name 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 z. P z (pi_1 o ... o pi_n o t) *)
(* to R ... & id (ALL z. P z (pi_1 o ... o pi_n o t)) *)
(* or id (ALL z. P 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 member (op =) names name 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 [@{thm 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 abs_params params t =
let val vs = map (Var o apfst (rpair 0)) (Term.rename_wrt_term t params)
in (list_all (params, t), (rev vs, subst_bounds (vs, t))) end;
fun inst_params thy (vs, p) th cts =
let val env = Pattern.first_order_match thy (p, prop_of th)
(Vartab.empty, Vartab.empty)
in Thm.instantiate ([],
map (Envir.subst_term env #> cterm_of thy) vs ~~ cts) th
end;
fun prove_strong_ind s alt_name avoids ctxt =
let
val thy = ProofContext.theory_of ctxt;
val ({names, ...}, {raw_induct, intrs, elims, ...}) =
Inductive.the_inductive ctxt (Sign.intern_const thy s);
val ind_params = Inductive.params_of raw_induct;
val raw_induct = atomize_induct ctxt raw_induct;
val elims = map (atomize_induct ctxt) elims;
val monos = Inductive.get_monos ctxt;
val eqvt_thms = NominalThmDecls.get_eqvt_thms ctxt;
val _ = (case subtract (op =) (fold (Term.add_const_names o Thm.prop_of) eqvt_thms []) names of
[] => ()
| xs => error ("Missing equivariance theorem for predicate(s): " ^
commas_quote xs));
val induct_cases = map fst (fst (Rule_Cases.get (the
(Induct.lookup_inductP ctxt (hd names)))));
val induct_cases' = if null induct_cases then replicate (length intrs) ""
else induct_cases;
val ([raw_induct'], ctxt') = Variable.import_terms false [prop_of raw_induct] ctxt;
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 _ = (case subtract (op =) induct_cases (map fst avoids) of
[] => ()
| xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs));
fun mk_avoids params name sets =
let
val (_, ctxt') = ProofContext.add_fixes
(map (fn (s, T) => (Binding.name s, SOME T, NoSyn)) params) ctxt;
fun mk s =
let
val t = Syntax.read_term ctxt' s;
val t' = list_abs_free (params, t) |>
funpow (length params) (fn Abs (_, _, t) => t)
in (t', HOLogic.dest_setT (fastype_of t)) end
handle TERM _ =>
error ("Expression " ^ quote s ^ " to be avoided in case " ^
quote name ^ " is not a set type");
fun add_set p [] = [p]
| add_set (t, T) ((u, U) :: ps) =
if T = U then
let val S = HOLogic.mk_setT T
in (Const (@{const_name sup}, S --> S --> S) $ u $ t, T) :: ps
end
else (u, U) :: add_set (t, T) ps
in
fold (mk #> add_set) sets []
end;
val prems = map (fn (prem, name) =>
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,
if null avoids then
map (fn (T, ts) => (HOLogic.mk_set T ts, T))
(fold (add_binders thy 0) (prems @ [concl]) [])
else case AList.lookup op = avoids name of
NONE => []
| SOME sets =>
map (apfst (incr_boundvars 1)) (mk_avoids params name sets),
prems, strip_comb (HOLogic.dest_Trueprop concl))
end) (Logic.strip_imp_prems raw_induct' ~~ induct_cases');
val atomTs = distinct op = (maps (map snd o #2) prems);
val atoms = map (fst o dest_Type) atomTs;
val ind_sort = if null atomTs then HOLogic.typeS
else Sign.minimize_sort thy (Sign.certify_sort thy (map (fn a => Sign.intern_class thy
("fs_" ^ Long_Name.base_name a)) atoms));
val ([fs_ctxt_tyname], _) = Name.variants ["'n"] (Variable.names_of ctxt');
val ([fs_ctxt_name], ctxt'') = Variable.variant_fixes ["z"] ctxt';
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 member (op =) ps p 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_fresh (x, T) = HOLogic.mk_Trueprop
(NominalDatatype.fresh_star_const T fsT $ x $ Bound 0);
val (prems', prems'') = split_list (map (fn (params, sets, prems, (p, ts)) =>
let
val params' = params @ [("y", fsT)];
val prem = Logic.list_implies
(map mk_fresh sets @
map (fn prem =>
if null (preds_of ps prem) then prem
else lift_prem prem) prems,
HOLogic.mk_Trueprop (lift_pred p ts));
in abs_params params' prem end) prems);
val ind_vars =
(Datatype_Prop.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 (NominalDatatype.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, vc_compat') = map (fn (params, sets, prems, (p, ts)) =>
map (fn q => abs_params params (incr_boundvars ~1 (Logic.list_implies
(map_filter (fn prem =>
if null (preds_of ps prem) then SOME prem
else map_term (split_conj (K o I) names) prem prem) prems, q))))
(maps (fn (t, T) => map (fn (u, U) => HOLogic.mk_Trueprop
(NominalDatatype.fresh_star_const U T $ u $ t)) sets)
(ts ~~ binder_types (fastype_of p)) @
map (fn (u, U) => HOLogic.mk_Trueprop (Const (@{const_name finite},
HOLogic.mk_setT U --> HOLogic.boolT) $ u)) sets) |>
split_list) prems |> split_list;
val perm_pi_simp = PureThy.get_thms thy "perm_pi_simp";
val pt2_atoms = map (fn a => PureThy.get_thm thy
("pt_" ^ Long_Name.base_name a ^ "2")) atoms;
val eqvt_ss = Simplifier.global_context thy HOL_basic_ss
addsimps (eqvt_thms @ perm_pi_simp @ pt2_atoms)
addsimprocs [mk_perm_bool_simproc ["Fun.id"],
NominalPermeq.perm_simproc_app, NominalPermeq.perm_simproc_fun];
val fresh_star_bij = PureThy.get_thms thy "fresh_star_bij";
val pt_insts = map (NominalAtoms.pt_inst_of thy) atoms;
val at_insts = map (NominalAtoms.at_inst_of thy) atoms;
val dj_thms = maps (fn a =>
map (NominalAtoms.dj_thm_of thy a) (remove (op =) a atoms)) atoms;
val finite_ineq = map2 (fn th => fn th' => th' RS (th RS
@{thm pt_set_finite_ineq})) pt_insts at_insts;
val perm_set_forget =
map (fn th => th RS @{thm dj_perm_set_forget}) dj_thms;
val perm_freshs_freshs = atomTs ~~ map2 (fn th => fn th' => th' RS (th RS
@{thm pt_freshs_freshs})) pt_insts at_insts;
fun obtain_fresh_name ts sets (T, fin) (freshs, ths1, ths2, ths3, ctxt) =
let
val thy = ProofContext.theory_of ctxt;
(** protect terms to avoid that fresh_star_prod_set 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);
val atom = fst (dest_Type T);
val {at_inst, ...} = NominalAtoms.the_atom_info thy atom;
val fs_atom = PureThy.get_thm thy
("fs_" ^ Long_Name.base_name atom ^ "1");
val avoid_th = Drule.instantiate'
[SOME (ctyp_of thy (fastype_of p))] [SOME (cterm_of thy p)]
([at_inst, fin, fs_atom] MRS @{thm at_set_avoiding});
val (([(_, cx)], th1 :: th2 :: ths), ctxt') = Obtain.result
(fn _ => EVERY
[rtac avoid_th 1,
full_simp_tac (HOL_ss addsimps [@{thm fresh_star_prod_set}]) 1,
full_simp_tac (HOL_basic_ss addsimps [@{thm id_apply}]) 1,
rotate_tac 1 1,
REPEAT (etac conjE 1)])
[] ctxt;
val (Ts1, _ :: Ts2) = take_prefix (not o equal T) (map snd sets);
val pTs = map NominalAtoms.mk_permT (Ts1 @ Ts2);
val (pis1, pis2) = chop (length Ts1)
(map Bound (length pTs - 1 downto 0));
val _ $ (f $ (_ $ pi $ l) $ r) = prop_of th2
val th2' =
Goal.prove ctxt [] []
(list_all (map (pair "pi") pTs, HOLogic.mk_Trueprop
(f $ fold_rev (NominalDatatype.mk_perm (rev pTs))
(pis1 @ pi :: pis2) l $ r)))
(fn _ => cut_facts_tac [th2] 1 THEN
full_simp_tac (HOL_basic_ss addsimps perm_set_forget) 1) |>
Simplifier.simplify eqvt_ss
in
(freshs @ [term_of cx],
ths1 @ ths, ths2 @ [th1], ths3 @ [th2'], ctxt')
end;
fun mk_ind_proof ctxt' thss =
Goal.prove ctxt' [] prems' concl' (fn {prems = ihyps, context = ctxt} =>
let val th = Goal.prove ctxt [] [] concl (fn {context, ...} =>
rtac raw_induct 1 THEN
EVERY (maps (fn (((((_, sets, oprems, _),
vc_compat_ths), vc_compat_vs), ihyp), vs_ihypt) =>
[REPEAT (rtac allI 1), simp_tac eqvt_ss 1,
SUBPROOF (fn {prems = gprems, params, concl, context = ctxt', ...} =>
let
val (cparams', (pis, z)) =
chop (length params - length atomTs - 1) (map #2 params) ||>
(map term_of #> split_last);
val params' = map term_of cparams'
val sets' = map (apfst (curry subst_bounds (rev params'))) sets;
val pi_sets = map (fn (t, _) =>
fold_rev (NominalDatatype.mk_perm []) pis t) sets';
val (P, ts) = strip_comb (HOLogic.dest_Trueprop (term_of concl));
val gprems1 = map_filter (fn (th, t) =>
if null (preds_of ps t) then SOME th
else
map_thm ctxt' (split_conj (K o I) names)
(etac conjunct1 1) monos NONE th)
(gprems ~~ oprems);
val vc_compat_ths' = map2 (fn th => fn p =>
let
val th' = gprems1 MRS inst_params thy p th cparams';
val (h, ts) =
strip_comb (HOLogic.dest_Trueprop (concl_of th'))
in
Goal.prove ctxt' [] []
(HOLogic.mk_Trueprop (list_comb (h,
map (fold_rev (NominalDatatype.mk_perm []) pis) ts)))
(fn _ => simp_tac (HOL_basic_ss addsimps
(fresh_star_bij @ finite_ineq)) 1 THEN rtac th' 1)
end) vc_compat_ths vc_compat_vs;
val (vc_compat_ths1, vc_compat_ths2) =
chop (length vc_compat_ths - length sets) vc_compat_ths';
val vc_compat_ths1' = map
(Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv
(Simplifier.rewrite eqvt_ss)))) vc_compat_ths1;
val (pis', fresh_ths1, fresh_ths2, fresh_ths3, ctxt'') = fold
(obtain_fresh_name ts sets)
(map snd sets' ~~ vc_compat_ths2) ([], [], [], [], ctxt');
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_rev (concat_perm #> map) pis' pis;
val ihyp' = inst_params thy vs_ihypt ihyp
(map (fold_rev (NominalDatatype.mk_perm [])
(pis' @ pis) #> cterm_of thy) params' @ [cterm_of thy z]);
fun mk_pi th =
Simplifier.simplify (HOL_basic_ss addsimps [@{thm id_apply}]
addsimprocs [NominalDatatype.perm_simproc])
(Simplifier.simplify eqvt_ss
(fold_rev (mk_perm_bool o cterm_of thy)
(pis' @ pis) th));
val gprems2 = map (fn (th, t) =>
if null (preds_of ps t) then mk_pi th
else
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);
val perm_freshs_freshs' = map (fn (th, (_, T)) =>
th RS the (AList.lookup op = perm_freshs_freshs T))
(fresh_ths2 ~~ sets);
val th = Goal.prove ctxt'' [] []
(HOLogic.mk_Trueprop (list_comb (P $ hd ts,
map (fold_rev (NominalDatatype.mk_perm []) pis') (tl ts))))
(fn _ => EVERY ([simp_tac eqvt_ss 1, rtac ihyp' 1] @
map (fn th => rtac th 1) fresh_ths3 @
[REPEAT_DETERM_N (length gprems)
(simp_tac (HOL_basic_ss
addsimps [inductive_forall_def']
addsimprocs [NominalDatatype.perm_simproc]) 1 THEN
resolve_tac gprems2 1)]));
val final = Goal.prove ctxt'' [] [] (term_of concl)
(fn _ => cut_facts_tac [th] 1 THEN full_simp_tac (HOL_ss
addsimps vc_compat_ths1' @ fresh_ths1 @
perm_freshs_freshs') 1);
val final' = ProofContext.export ctxt'' ctxt' [final];
in resolve_tac final' 1 end) context 1])
(prems ~~ thss ~~ vc_compat' ~~ 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 @{thm allE_Nil} 1) THEN
asm_full_simp_tac (simpset_of ctxt) 1)
end) |>
fresh_postprocess |>
singleton (ProofContext.export ctxt' ctxt);
in
ctxt'' |>
Proof.theorem NONE (fn thss => fn ctxt =>
let
val rec_name = space_implode "_" (map Long_Name.base_name names);
val rec_qualified = Binding.qualify false rec_name;
val ind_case_names = Rule_Cases.case_names induct_cases;
val induct_cases' = Inductive.partition_rules' raw_induct
(intrs ~~ induct_cases);
val thss' = map (map atomize_intr) thss;
val thsss = Inductive.partition_rules' raw_induct (intrs ~~ thss');
val strong_raw_induct =
mk_ind_proof ctxt thss' |> Inductive.rulify;
val strong_induct =
if length names > 1 then
(strong_raw_induct, [ind_case_names, Rule_Cases.consumes 0])
else (strong_raw_induct RSN (2, rev_mp),
[ind_case_names, Rule_Cases.consumes 1]);
val (induct_name, inducts_name) =
case alt_name of
NONE => (rec_qualified (Binding.name "strong_induct"),
rec_qualified (Binding.name "strong_inducts"))
| SOME s => (Binding.name s, Binding.name (s ^ "s"));
val ((_, [strong_induct']), ctxt') = ctxt |> Local_Theory.note
((induct_name,
map (Attrib.internal o K) (#2 strong_induct)), [#1 strong_induct]);
val strong_inducts =
Project_Rule.projects ctxt' (1 upto length names) strong_induct'
in
ctxt' |>
Local_Theory.note
((inducts_name,
[Attrib.internal (K ind_case_names),
Attrib.internal (K (Rule_Cases.consumes 1))]),
strong_inducts) |> snd
end)
(map (map (rulify_term thy #> rpair [])) vc_compat)
end;
(* outer syntax *)
local structure P = OuterParse and K = OuterKeyword in
val _ =
OuterSyntax.local_theory_to_proof "nominal_inductive2"
"prove strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal
(P.xname --
Scan.option (P.$$$ "(" |-- P.!!! (P.name --| P.$$$ ")")) --
(Scan.optional (P.$$$ "avoids" |-- P.enum1 "|" (P.name --
(P.$$$ ":" |-- P.and_list1 P.term))) []) >> (fn ((name, rule_name), avoids) =>
prove_strong_ind name rule_name avoids));
end;
end