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(* Title: HOL/Nominal/nominal_inductive.ML
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 -> local_theory -> Proof.state
val prove_eqvt: string -> string list -> local_theory -> local_theory
end
structure NominalInductive : NOMINAL_INDUCTIVE =
struct
val inductive_forall_def = @{thm HOL.induct_forall_def};
val inductive_atomize = @{thms induct_atomize};
val inductive_rulify = @{thms induct_rulify};
fun rulify_term thy = Raw_Simplifier.rewrite_term thy inductive_rulify [];
fun atomize_conv ctxt =
Raw_Simplifier.rewrite_cterm (true, false, false) (K (K NONE))
(put_simpset HOL_basic_ss ctxt addsimps inductive_atomize);
fun atomize_intr ctxt = Conv.fconv_rule (Conv.prems_conv ~1 (atomize_conv ctxt));
fun atomize_induct ctxt = Conv.fconv_rule (Conv.prems_conv ~1
(Conv.params_conv ~1 (K (Conv.prems_conv ~1 (atomize_conv ctxt))) ctxt));
fun preds_of ps t = inter (op = o apsnd dest_Free) ps (Term.add_frees t []);
val fresh_prod = @{thm fresh_prod};
val perm_bool = mk_meta_eq @{thm perm_bool_def};
val perm_boolI = @{thm perm_boolI};
val (_, [perm_boolI_pi, _]) = Drule.strip_comb (snd (Thm.dest_comb
(Drule.strip_imp_concl (Thm.cprop_of perm_boolI))));
fun mk_perm_bool ctxt pi th =
th RS infer_instantiate ctxt [(#1 (dest_Var (Thm.term_of perm_boolI_pi)), pi)] perm_boolI;
fun mk_perm_bool_simproc names =
Simplifier.make_simproc \<^context> "perm_bool"
{lhss = [\<^term>\<open>perm pi x\<close>],
proc = fn _ => fn _ => fn ct =>
(case Thm.term_of ct of
Const (\<^const_name>\<open>Nominal.perm\<close>, _) $ _ $ 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 insert (op aconv o apply2 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 (\<^const_name>\<open>HOL.conj\<close>, _) $ 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 (\<^const_name>\<open>All\<close>, _) $ 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 (\<^const_name>\<open>HOL.conj\<close>, _) $ p $ q) _ =
(case head_of p of
Const (name, _) =>
if member (op =) names name then SOME (HOLogic.mk_conj (p,
Const (\<^const_name>\<open>Fun.id\<close>, 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 (\<^const_name>\<open>Fun.id\<close>, HOLogic.boolT --> HOLogic.boolT) $
(subst_bounds (pis, strip_all pis t)))
else NONE
| inst_conj_all _ _ _ _ _ = NONE;
fun inst_conj_all_tac ctxt k = EVERY
[TRY (EVERY [eresolve_tac ctxt [conjE] 1, resolve_tac ctxt [conjI] 1, assume_tac ctxt 1]),
REPEAT_DETERM_N k (eresolve_tac ctxt [allE] 1),
simp_tac (put_simpset HOL_basic_ss ctxt 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 = Thm.prop_of th;
fun prove t =
Goal.prove ctxt [] [] t (fn _ =>
EVERY [cut_facts_tac [th] 1, eresolve_tac ctxt [rev_mp] 1,
REPEAT_DETERM (FIRSTGOAL (resolve_tac ctxt monos)),
REPEAT_DETERM (resolve_tac ctxt [impI] 1 THEN (assume_tac ctxt 1 ORELSE tac))])
in Option.map prove (map_term f prop (the_default prop opt)) end;
val eta_contract_cterm = Thm.dest_arg o Thm.cprop_of o Thm.eta_conversion;
fun first_order_matchs pats objs = Thm.first_order_match
(eta_contract_cterm (Conjunction.mk_conjunction_balanced pats),
eta_contract_cterm (Conjunction.mk_conjunction_balanced objs));
fun first_order_mrs ths th = ths MRS
Thm.instantiate (first_order_matchs (cprems_of th) (map Thm.cprop_of ths)) th;
fun prove_strong_ind s avoids ctxt =
let
val thy = Proof_Context.theory_of ctxt;
val ({names, ...}, {raw_induct, intrs, elims, ...}) =
Inductive.the_inductive_global 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 o fst) (fst (Rule_Cases.get (the
(Induct.lookup_inductP ctxt (hd names)))));
val (raw_induct', ctxt') = ctxt
|> yield_singleton (Variable.import_terms false) (Thm.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 apply2 fst) avoids of
[] => ()
| xs => error ("Duplicate case names: " ^ commas_quote (map fst xs)));
val _ = avoids |> forall (fn (a, xs) => null (duplicates (op =) xs) orelse
error ("Duplicate variable names for case " ^ quote a));
val _ = (case subtract (op =) induct_cases (map fst avoids) of
[] => ()
| xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs));
val avoids' = if null induct_cases then replicate (length intrs) ("", [])
else 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 \<^sort>\<open>type\<close>
else Sign.minimize_sort thy (Sign.certify_sort thy (map (fn T => Sign.intern_class thy
("fs_" ^ Long_Name.base_name (fst (dest_Type T)))) atomTs));
val (fs_ctxt_tyname, _) = Name.variant "'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' = Thm.instantiate'
[SOME (Thm.global_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 HOLogic.mk_induct_forall 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) = map_filter (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
(NominalDatatype.fresh_const T fsT $ 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 (preds_of ps prem) then prem
else lift_prem prem) prems,
HOLogic.mk_Trueprop (lift_pred p ts));
val vs = map (Var o apfst (rpair 0)) (Term.rename_wrt_term prem params')
in
(Logic.list_all (params', prem), (rev vs, subst_bounds (vs, prem)))
end) prems);
val ind_vars =
(Old_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 = map (fn (params, bvars, prems, (p, ts)) =>
map (fn q => Logic.list_all (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))))
(mk_distinct bvars @
maps (fn (t, T) => map (fn (u, U) => HOLogic.mk_Trueprop
(NominalDatatype.fresh_const U T $ u $ t)) bvars)
(ts ~~ binder_types (fastype_of p)))) prems;
val perm_pi_simp = Global_Theory.get_thms thy "perm_pi_simp";
val pt2_atoms = map (fn aT => Global_Theory.get_thm thy
("pt_" ^ Long_Name.base_name (fst (dest_Type aT)) ^ "2")) atomTs;
val eqvt_ss = simpset_of (put_simpset HOL_basic_ss (Proof_Context.init_global thy)
addsimps (eqvt_thms @ perm_pi_simp @ pt2_atoms)
addsimprocs [mk_perm_bool_simproc [\<^const_name>\<open>Fun.id\<close>],
NominalPermeq.perm_simproc_app, NominalPermeq.perm_simproc_fun]);
val fresh_bij = Global_Theory.get_thms thy "fresh_bij";
val perm_bij = Global_Theory.get_thms thy "perm_bij";
val fs_atoms = map (fn aT => Global_Theory.get_thm thy
("fs_" ^ Long_Name.base_name (fst (dest_Type aT)) ^ "1")) atomTs;
val exists_fresh' = Global_Theory.get_thms thy "exists_fresh'";
val fresh_atm = Global_Theory.get_thms thy "fresh_atm";
val swap_simps = Global_Theory.get_thms thy "swap_simps";
val perm_fresh_fresh = Global_Theory.get_thms thy "perm_fresh_fresh";
fun obtain_fresh_name ts T (freshs1, freshs2, ctxt) =
let
(** protect terms to avoid that fresh_prod interferes with **)
(** pairs used in introduction rules of inductive predicate **)
fun protect t =
let val T = fastype_of t in Const (\<^const_name>\<open>Fun.id\<close>, 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,
NominalDatatype.fresh_const T (fastype_of p) $
Bound 0 $ p)))
(fn _ => EVERY
[resolve_tac ctxt exists_fresh' 1,
resolve_tac ctxt fs_atoms 1]);
val (([(_, cx)], ths), ctxt') = Obtain.result
(fn ctxt' => EVERY
[eresolve_tac ctxt' [exE] 1,
full_simp_tac (put_simpset HOL_ss ctxt' addsimps (fresh_prod :: fresh_atm)) 1,
full_simp_tac (put_simpset HOL_basic_ss ctxt' addsimps [@{thm id_apply}]) 1,
REPEAT (eresolve_tac ctxt [conjE] 1)])
[ex] ctxt
in (freshs1 @ [Thm.term_of cx], freshs2 @ ths, 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, ...} =>
resolve_tac context [raw_induct] 1 THEN
EVERY (maps (fn ((((_, bvars, oprems, _), vc_compat_ths), ihyp), (vs, ihypt)) =>
[REPEAT (resolve_tac context [allI] 1),
simp_tac (put_simpset eqvt_ss context) 1,
SUBPROOF (fn {prems = gprems, params, concl, context = ctxt', ...} =>
let
val (params', (pis, z)) =
chop (length params - length atomTs - 1) (map (Thm.term_of o #2) 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 (NominalDatatype.mk_perm []) pis t) bvars';
val (P, ts) = strip_comb (HOLogic.dest_Trueprop (Thm.term_of concl));
val (freshs1, freshs2, ctxt'') = fold
(obtain_fresh_name (ts @ pi_bvars))
(map snd bvars') ([], [], ctxt');
val freshs2' = NominalDatatype.mk_not_sym freshs2;
val pis' = map NominalDatatype.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 (\<^const_name>\<open>append\<close>, T --> T --> T) $ pi1 $ pi2
else pi2
end;
val pis'' = fold (concat_perm #> map) pis' pis;
val env = Pattern.first_order_match thy (ihypt, Thm.prop_of ihyp)
(Vartab.empty, Vartab.empty);
val ihyp' = Thm.instantiate ([],
map (fn (v, t) => (dest_Var v, Thm.global_cterm_of thy t))
(map (Envir.subst_term env) vs ~~
map (fold_rev (NominalDatatype.mk_perm [])
(rev pis' @ pis)) params' @ [z])) ihyp;
fun mk_pi th =
Simplifier.simplify (put_simpset HOL_basic_ss ctxt' addsimps [@{thm id_apply}]
addsimprocs [NominalDatatype.perm_simproc])
(Simplifier.simplify (put_simpset eqvt_ss ctxt')
(fold_rev (mk_perm_bool ctxt' o Thm.cterm_of ctxt')
(rev pis' @ pis) th));
val (gprems1, gprems2) = split_list
(map (fn (th, t) =>
if null (preds_of ps t) then (SOME th, mk_pi th)
else
(map_thm ctxt' (split_conj (K o I) names)
(eresolve_tac ctxt' [conjunct1] 1) monos NONE th,
mk_pi (the (map_thm ctxt' (inst_conj_all names ps (rev pis''))
(inst_conj_all_tac ctxt' (length pis'')) monos (SOME t) th))))
(gprems ~~ oprems)) |>> map_filter I;
val vc_compat_ths' = map (fn th =>
let
val th' = first_order_mrs gprems1 th;
val (bop, lhs, rhs) = (case Thm.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 (NominalDatatype.mk_perm []) pis lhs)
(fold_rev (NominalDatatype.mk_perm []) pis rhs)))
(fn _ => simp_tac (put_simpset HOL_basic_ss ctxt'' addsimps
(fresh_bij @ perm_bij)) 1 THEN resolve_tac ctxt' [th'] 1)
in Simplifier.simplify (put_simpset eqvt_ss ctxt'' addsimps fresh_atm) th'' end)
vc_compat_ths;
val vc_compat_ths'' = NominalDatatype.mk_not_sym vc_compat_ths';
(** Since swap_simps simplifies (pi :: 'a prm) o (x :: 'b) to x **)
(** we have to pre-simplify the rewrite rules **)
val swap_simps_simpset = put_simpset HOL_ss ctxt'' addsimps swap_simps @
map (Simplifier.simplify (put_simpset HOL_ss ctxt'' addsimps swap_simps))
(vc_compat_ths'' @ freshs2');
val th = Goal.prove ctxt'' [] []
(HOLogic.mk_Trueprop (list_comb (P $ hd ts,
map (fold (NominalDatatype.mk_perm []) pis') (tl ts))))
(fn _ => EVERY ([simp_tac (put_simpset eqvt_ss ctxt'') 1,
resolve_tac ctxt'' [ihyp'] 1,
REPEAT_DETERM_N (Thm.nprems_of ihyp - length gprems)
(simp_tac swap_simps_simpset 1),
REPEAT_DETERM_N (length gprems)
(simp_tac (put_simpset HOL_basic_ss ctxt''
addsimps [inductive_forall_def']
addsimprocs [NominalDatatype.perm_simproc]) 1 THEN
resolve_tac ctxt'' gprems2 1)]));
val final = Goal.prove ctxt'' [] [] (Thm.term_of concl)
(fn _ => cut_facts_tac [th] 1 THEN full_simp_tac (put_simpset HOL_ss ctxt''
addsimps vc_compat_ths'' @ freshs2' @
perm_fresh_fresh @ fresh_atm) 1);
val final' = Proof_Context.export ctxt'' ctxt' [final];
in resolve_tac ctxt' final' 1 end) context 1])
(prems ~~ thss ~~ ihyps ~~ prems'')))
in
cut_facts_tac [th] 1 THEN REPEAT (eresolve_tac ctxt [conjE] 1) THEN
REPEAT (REPEAT (resolve_tac ctxt [conjI, impI] 1) THEN
eresolve_tac ctxt [impE] 1 THEN assume_tac ctxt 1 THEN
REPEAT (eresolve_tac ctxt @{thms allE_Nil} 1) THEN
asm_full_simp_tac ctxt 1)
end) |> singleton (Proof_Context.export ctxt' ctxt);
(** strong case analysis rule **)
val cases_prems = map (fn ((name, avoids), rule) =>
let
val ([rule'], ctxt') = Variable.import_terms false [Thm.prop_of rule] ctxt;
val prem :: prems = Logic.strip_imp_prems rule';
val concl = Logic.strip_imp_concl rule'
in
(prem,
List.drop (snd (strip_comb (HOLogic.dest_Trueprop prem)), length ind_params),
concl,
fold_map (fn (prem, (_, avoid)) => fn ctxt =>
let
val prems = Logic.strip_assums_hyp prem;
val params = Logic.strip_params prem;
val bnds = fold (add_binders thy 0) prems [] @ mk_avoids params avoid;
fun mk_subst (p as (s, T)) (i, j, ctxt, ps, qs, is, ts) =
if member (op = o apsnd fst) bnds (Bound i) then
let
val ([s'], ctxt') = Variable.variant_fixes [s] ctxt;
val t = Free (s', T)
in (i + 1, j, ctxt', ps, (t, T) :: qs, i :: is, t :: ts) end
else (i + 1, j + 1, ctxt, p :: ps, qs, is, Bound j :: ts);
val (_, _, ctxt', ps, qs, is, ts) = fold_rev mk_subst params
(0, 0, ctxt, [], [], [], [])
in
((ps, qs, is, map (curry subst_bounds (rev ts)) prems), ctxt')
end) (prems ~~ avoids) ctxt')
end)
(Inductive.partition_rules' raw_induct (intrs ~~ avoids') ~~
elims);
val cases_prems' =
map (fn (prem, args, concl, (prems, _)) =>
let
fun mk_prem (ps, [], _, prems) =
Logic.list_all (ps, Logic.list_implies (prems, concl))
| mk_prem (ps, qs, _, prems) =
Logic.list_all (ps, Logic.mk_implies
(Logic.list_implies
(mk_distinct qs @
maps (fn (t, T) => map (fn u => HOLogic.mk_Trueprop
(NominalDatatype.fresh_const T (fastype_of u) $ t $ u))
args) qs,
HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map HOLogic.dest_Trueprop prems))),
concl))
in map mk_prem prems end) cases_prems;
val cases_eqvt_simpset = put_simpset HOL_ss (Proof_Context.init_global thy)
addsimps eqvt_thms @ swap_simps @ perm_pi_simp
addsimprocs [NominalPermeq.perm_simproc_app,
NominalPermeq.perm_simproc_fun];
val simp_fresh_atm = map
(Simplifier.simplify (put_simpset HOL_basic_ss (Proof_Context.init_global thy)
addsimps fresh_atm));
fun mk_cases_proof ((((name, thss), elim), (prem, args, concl, (prems, ctxt'))),
prems') =
(name, Goal.prove ctxt' [] (prem :: prems') concl
(fn {prems = hyp :: hyps, context = ctxt1} =>
EVERY (resolve_tac ctxt1 [hyp RS elim] 1 ::
map (fn (((_, vc_compat_ths), case_hyp), (_, qs, is, _)) =>
SUBPROOF (fn {prems = case_hyps, params, context = ctxt2, concl, ...} =>
if null qs then
resolve_tac ctxt2 [first_order_mrs case_hyps case_hyp] 1
else
let
val params' = map (Thm.term_of o #2 o nth (rev params)) is;
val tab = params' ~~ map fst qs;
val (hyps1, hyps2) = chop (length args) case_hyps;
(* turns a = t and [x1 # t, ..., xn # t] *)
(* into [x1 # a, ..., xn # a] *)
fun inst_fresh th' ths =
let val (ths1, ths2) = chop (length qs) ths
in
(map (fn th =>
let
val (cf, ct) =
Thm.dest_comb (Thm.dest_arg (Thm.cprop_of th));
val arg_cong' = Thm.instantiate'
[SOME (Thm.ctyp_of_cterm ct)]
[NONE, SOME ct, SOME cf] (arg_cong RS iffD2);
val inst = Thm.first_order_match (ct,
Thm.dest_arg (Thm.dest_arg (Thm.cprop_of th')))
in [th', th] MRS Thm.instantiate inst arg_cong'
end) ths1,
ths2)
end;
val (vc_compat_ths1, vc_compat_ths2) =
chop (length vc_compat_ths - length args * length qs)
(map (first_order_mrs hyps2) vc_compat_ths);
val vc_compat_ths' =
NominalDatatype.mk_not_sym vc_compat_ths1 @
flat (fst (fold_map inst_fresh hyps1 vc_compat_ths2));
val (freshs1, freshs2, ctxt3) = fold
(obtain_fresh_name (args @ map fst qs @ params'))
(map snd qs) ([], [], ctxt2);
val freshs2' = NominalDatatype.mk_not_sym freshs2;
val pis = map (NominalDatatype.perm_of_pair)
((freshs1 ~~ map fst qs) @ (params' ~~ freshs1));
val mk_pis = fold_rev (mk_perm_bool ctxt3) (map (Thm.cterm_of ctxt3) pis);
val obj = Thm.global_cterm_of thy (foldr1 HOLogic.mk_conj (map (map_aterms
(fn x as Free _ =>
if member (op =) args x then x
else (case AList.lookup op = tab x of
SOME y => y
| NONE => fold_rev (NominalDatatype.mk_perm []) pis x)
| x => x) o HOLogic.dest_Trueprop o Thm.prop_of) case_hyps));
val inst = Thm.first_order_match (Thm.dest_arg
(Drule.strip_imp_concl (hd (cprems_of case_hyp))), obj);
val th = Goal.prove ctxt3 [] [] (Thm.term_of concl)
(fn {context = ctxt4, ...} =>
resolve_tac ctxt4 [Thm.instantiate inst case_hyp] 1 THEN
SUBPROOF (fn {context = ctxt5, prems = fresh_hyps, ...} =>
let
val fresh_hyps' = NominalDatatype.mk_not_sym fresh_hyps;
val case_simpset = cases_eqvt_simpset addsimps freshs2' @
simp_fresh_atm (vc_compat_ths' @ fresh_hyps');
val fresh_fresh_simpset = case_simpset addsimps perm_fresh_fresh;
val hyps1' = map
(mk_pis #> Simplifier.simplify fresh_fresh_simpset) hyps1;
val hyps2' = map
(mk_pis #> Simplifier.simplify case_simpset) hyps2;
val case_hyps' = hyps1' @ hyps2'
in
simp_tac case_simpset 1 THEN
REPEAT_DETERM (TRY (resolve_tac ctxt5 [conjI] 1) THEN
resolve_tac ctxt5 case_hyps' 1)
end) ctxt4 1)
val final = Proof_Context.export ctxt3 ctxt2 [th]
in resolve_tac ctxt2 final 1 end) ctxt1 1)
(thss ~~ hyps ~~ prems))) |>
singleton (Proof_Context.export ctxt' ctxt))
in
ctxt'' |>
Proof.theorem NONE (fn thss => fn ctxt => (* FIXME ctxt/ctxt' should be called lthy/lthy' *)
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 ctxt)) thss;
val thsss = Inductive.partition_rules' raw_induct (intrs ~~ thss');
val strong_raw_induct =
mk_ind_proof ctxt thss' |> Inductive.rulify ctxt;
val strong_cases = map (mk_cases_proof ##> Inductive.rulify ctxt)
(thsss ~~ elims ~~ cases_prems ~~ cases_prems');
val strong_induct_atts =
map (Attrib.internal o K)
[ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of strong_raw_induct))];
val strong_induct =
if length names > 1 then strong_raw_induct
else strong_raw_induct RSN (2, rev_mp);
val ((_, [strong_induct']), ctxt') = ctxt |> Local_Theory.note
((rec_qualified (Binding.name "strong_induct"), strong_induct_atts), [strong_induct]);
val strong_inducts =
Project_Rule.projects ctxt (1 upto length names) strong_induct';
in
ctxt' |>
Local_Theory.notes
[((rec_qualified (Binding.name "strong_inducts"), []),
strong_inducts |> map (fn th => ([th],
[Attrib.internal (K ind_case_names),
Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of th)))])))] |> snd |>
Local_Theory.notes (map (fn ((name, elim), (_, cases)) =>
((Binding.qualified_name (Long_Name.qualify (Long_Name.base_name name) "strong_cases"),
[Attrib.internal (K (Rule_Cases.case_names (map snd cases))),
Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of elim)))]), [([elim], [])]))
(strong_cases ~~ induct_cases')) |> snd
end)
(map (map (rulify_term thy #> rpair [])) vc_compat)
end;
fun prove_eqvt s xatoms ctxt = (* FIXME ctxt should be called lthy *)
let
val thy = Proof_Context.theory_of ctxt;
val ({names, ...}, {raw_induct, intrs, elims, ...}) =
Inductive.the_inductive_global 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 ctxt) intrs;
val monos = Inductive.get_monos ctxt;
val intrs' = Inductive.unpartition_rules intrs
(map (fn (((s, ths), (_, k)), th) =>
(s, ths ~~ Inductive.infer_intro_vars thy th k ths))
(Inductive.partition_rules raw_induct intrs ~~
Inductive.arities_of raw_induct ~~ elims));
val k = length (Inductive.params_of raw_induct);
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 subtract (op =) atoms' atoms of
[] => ()
| xs => error ("No such atoms: " ^ commas xs);
atoms)
end;
val perm_pi_simp = Global_Theory.get_thms thy "perm_pi_simp";
val (([t], [pi]), ctxt') = ctxt |>
Variable.import_terms false [Thm.concl_of raw_induct] ||>>
Variable.variant_fixes ["pi"];
val eqvt_simpset = put_simpset HOL_basic_ss ctxt' addsimps
(NominalThmDecls.get_eqvt_thms ctxt' @ perm_pi_simp) addsimprocs
[mk_perm_bool_simproc names,
NominalPermeq.perm_simproc_app, NominalPermeq.perm_simproc_fun];
val ps = map (fst o HOLogic.dest_imp)
(HOLogic.dest_conj (HOLogic.dest_Trueprop t));
fun eqvt_tac pi (intr, vs) st =
let
fun eqvt_err s =
let val ([t], ctxt'') = Variable.import_terms true [Thm.prop_of intr] ctxt'
in error ("Could not prove equivariance for introduction rule\n" ^
Syntax.string_of_term ctxt'' t ^ "\n" ^ s)
end;
val res = SUBPROOF (fn {context = ctxt'', prems, params, ...} =>
let
val prems' = map (fn th => the_default th (map_thm ctxt''
(split_conj (K I) names) (eresolve_tac ctxt'' [conjunct2] 1) monos NONE th)) prems;
val prems'' = map (fn th => Simplifier.simplify eqvt_simpset
(mk_perm_bool ctxt'' (Thm.cterm_of ctxt'' pi) th)) prems';
val intr' = infer_instantiate ctxt'' (map (#1 o dest_Var) vs ~~
map (Thm.cterm_of ctxt'' o NominalDatatype.mk_perm [] pi o Thm.term_of o #2) params)
intr
in (resolve_tac ctxt'' [intr'] THEN_ALL_NEW (TRY o resolve_tac ctxt'' 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" ^
Syntax.string_of_term ctxt' (hd (Thm.prems_of st)))
| SOME (th, _) => Seq.single th
end;
val thss = map (fn atom =>
let val pi' = Free (pi, NominalAtoms.mk_permT (Type (atom, [])))
in map (fn th => zero_var_indexes (th RS mp))
(Old_Datatype_Aux.split_conj_thm (Goal.prove ctxt' [] []
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn p =>
let
val (h, ts) = strip_comb p;
val (ts1, ts2) = chop k ts
in
HOLogic.mk_imp (p, list_comb (h, ts1 @
map (NominalDatatype.mk_perm [] pi') ts2))
end) ps)))
(fn _ => EVERY (resolve_tac ctxt' [raw_induct] 1 :: map (fn intr_vs =>
full_simp_tac eqvt_simpset 1 THEN
eqvt_tac pi' intr_vs) intrs')) |>
singleton (Proof_Context.export ctxt' ctxt)))
end) atoms
in
ctxt |>
Local_Theory.notes (map (fn (name, ths) =>
((Binding.qualified_name (Long_Name.qualify (Long_Name.base_name name) "eqvt"),
[Attrib.internal (K NominalThmDecls.eqvt_add)]), [(ths, [])]))
(names ~~ transp thss)) |> snd
end;
(* outer syntax *)
val _ =
Outer_Syntax.local_theory_to_proof \<^command_keyword>\<open>nominal_inductive\<close>
"prove equivariance and strong induction theorem for inductive predicate involving nominal datatypes"
(Parse.name -- Scan.optional (\<^keyword>\<open>avoids\<close> |-- Parse.and_list1 (Parse.name --
(\<^keyword>\<open>:\<close> |-- Scan.repeat1 Parse.name))) [] >> (fn (name, avoids) =>
prove_strong_ind name avoids));
val _ =
Outer_Syntax.local_theory \<^command_keyword>\<open>equivariance\<close>
"prove equivariance for inductive predicate involving nominal datatypes"
(Parse.name -- Scan.optional (\<^keyword>\<open>[\<close> |-- Parse.list1 Parse.name --| \<^keyword>\<open>]\<close>) [] >>
(fn (name, atoms) => prove_eqvt name atoms));
end
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