(* Title: HOL/Tools/inductive.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
(Co)Inductive Definition module for HOL.
Features:
* least or greatest fixedpoints
* mutually recursive definitions
* definitions involving arbitrary monotone operators
* automatically proves introduction and elimination rules
Introduction rules have the form
[| M Pj ti, ..., Q x, ... |] ==> Pk t
where M is some monotone operator (usually the identity)
Q x is any side condition on the free variables
ti, t are any terms
Pj, Pk are two of the predicates being defined in mutual recursion
*)
signature BASIC_INDUCTIVE =
sig
type inductive_result =
{preds: term list, elims: thm list, raw_induct: thm,
induct: thm, inducts: thm list, intrs: thm list, eqs: thm list}
val transform_result: morphism -> inductive_result -> inductive_result
type inductive_info = {names: string list, coind: bool} * inductive_result
val the_inductive: Proof.context -> string -> inductive_info
val print_inductives: Proof.context -> unit
val mono_add: attribute
val mono_del: attribute
val get_monos: Proof.context -> thm list
val mk_cases: Proof.context -> term -> thm
val inductive_forall_def: thm
val rulify: thm -> thm
val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
thm list list * local_theory
val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
thm list list * local_theory
type inductive_flags =
{quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool}
val add_inductive_i:
inductive_flags -> ((binding * typ) * mixfix) list ->
(string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
inductive_result * local_theory
val add_inductive: bool -> bool ->
(binding * string option * mixfix) list ->
(binding * string option * mixfix) list ->
(Attrib.binding * string) list ->
(Facts.ref * Attrib.src list) list ->
bool -> local_theory -> inductive_result * local_theory
val add_inductive_global: inductive_flags ->
((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
thm list -> theory -> inductive_result * theory
val arities_of: thm -> (string * int) list
val params_of: thm -> term list
val partition_rules: thm -> thm list -> (string * thm list) list
val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
val infer_intro_vars: thm -> int -> thm list -> term list list
val setup: theory -> theory
end;
signature INDUCTIVE =
sig
include BASIC_INDUCTIVE
type add_ind_def =
inductive_flags ->
term list -> (Attrib.binding * term) list -> thm list ->
term list -> (binding * mixfix) list ->
local_theory -> inductive_result * local_theory
val declare_rules: binding -> bool -> bool -> string list -> term list ->
thm list -> binding list -> Attrib.src list list -> (thm * string list * int) list ->
thm list -> thm -> local_theory -> thm list * thm list * thm list * thm * thm list * local_theory
val add_ind_def: add_ind_def
val gen_add_inductive_i: add_ind_def -> inductive_flags ->
((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
thm list -> local_theory -> inductive_result * local_theory
val gen_add_inductive: add_ind_def -> bool -> bool ->
(binding * string option * mixfix) list ->
(binding * string option * mixfix) list ->
(Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
bool -> local_theory -> inductive_result * local_theory
val gen_ind_decl: add_ind_def -> bool -> (bool -> local_theory -> local_theory) parser
end;
structure Inductive: INDUCTIVE =
struct
(** theory context references **)
val inductive_forall_def = @{thm induct_forall_def};
val inductive_conj_name = "HOL.induct_conj";
val inductive_conj_def = @{thm induct_conj_def};
val inductive_conj = @{thms induct_conj};
val inductive_atomize = @{thms induct_atomize};
val inductive_rulify = @{thms induct_rulify};
val inductive_rulify_fallback = @{thms induct_rulify_fallback};
val notTrueE = TrueI RSN (2, notE);
val notFalseI = Seq.hd (atac 1 notI);
val simp_thms' = map mk_meta_eq
@{lemma "(~True) = False" "(~False) = True"
"(True --> P) = P" "(False --> P) = True"
"(P & True) = P" "(True & P) = P"
by (fact simp_thms)+};
val simp_thms'' = map mk_meta_eq [@{thm inf_fun_def}, @{thm inf_bool_def}] @ simp_thms';
val simp_thms''' = map mk_meta_eq
[@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_def}, @{thm sup_bool_def}];
(** context data **)
type inductive_result =
{preds: term list, elims: thm list, raw_induct: thm,
induct: thm, inducts: thm list, intrs: thm list, eqs: thm list};
fun transform_result phi {preds, elims, raw_induct: thm, induct, inducts, intrs, eqs} =
let
val term = Morphism.term phi;
val thm = Morphism.thm phi;
val fact = Morphism.fact phi;
in
{preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
induct = thm induct, inducts = fact inducts, intrs = fact intrs, eqs = fact eqs}
end;
type inductive_info =
{names: string list, coind: bool} * inductive_result;
structure InductiveData = Generic_Data
(
type T = inductive_info Symtab.table * thm list;
val empty = (Symtab.empty, []);
val extend = I;
fun merge ((tab1, monos1), (tab2, monos2)) : T =
(Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
);
val get_inductives = InductiveData.get o Context.Proof;
fun print_inductives ctxt =
let
val (tab, monos) = get_inductives ctxt;
val space = Consts.space_of (Proof_Context.consts_of ctxt);
in
[Pretty.strs ("(co)inductives:" :: map #1 (Name_Space.extern_table ctxt (space, tab))),
Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm ctxt) monos)]
|> Pretty.chunks |> Pretty.writeln
end;
(* get and put data *)
fun the_inductive ctxt name =
(case Symtab.lookup (#1 (get_inductives ctxt)) name of
NONE => error ("Unknown (co)inductive predicate " ^ quote name)
| SOME info => info);
fun put_inductives names info = InductiveData.map
(apfst (fold (fn name => Symtab.update (name, info)) names));
(** monotonicity rules **)
val get_monos = #2 o get_inductives;
val map_monos = InductiveData.map o apsnd;
fun mk_mono ctxt thm =
let
fun eq2mono thm' = thm' RS (thm' RS eq_to_mono);
fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
handle THM _ => thm RS @{thm le_boolD}
in
case concl_of thm of
Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq)
| _ $ (Const (@{const_name HOL.eq}, _) $ _ $ _) => eq2mono thm
| _ $ (Const (@{const_name Orderings.less_eq}, _) $ _ $ _) =>
dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
(resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))
| _ => thm
end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm ctxt thm);
val mono_add =
Thm.declaration_attribute (fn thm => fn context =>
map_monos (Thm.add_thm (mk_mono (Context.proof_of context) thm)) context);
val mono_del =
Thm.declaration_attribute (fn thm => fn context =>
map_monos (Thm.del_thm (mk_mono (Context.proof_of context) thm)) context);
(** equations **)
structure Equation_Data = Generic_Data
(
type T = thm Item_Net.T;
val empty = Item_Net.init (op aconv o pairself Thm.prop_of)
(single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
val extend = I;
val merge = Item_Net.merge;
);
val add_equation = Thm.declaration_attribute (Equation_Data.map o Item_Net.update)
(** misc utilities **)
fun message quiet_mode s = if quiet_mode then () else writeln s;
fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
fun coind_prefix true = "co"
| coind_prefix false = "";
fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
fun make_bool_args f g [] i = []
| make_bool_args f g (x :: xs) i =
(if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
fun make_bool_args' xs =
make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
fun arg_types_of k c = drop k (binder_types (fastype_of c));
fun find_arg T x [] = raise Fail "find_arg"
| find_arg T x ((p as (_, (SOME _, _))) :: ps) =
apsnd (cons p) (find_arg T x ps)
| find_arg T x ((p as (U, (NONE, y))) :: ps) =
if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
else apsnd (cons p) (find_arg T x ps);
fun make_args Ts xs =
map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
(fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
fun make_args' Ts xs Us =
fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
fun dest_predicate cs params t =
let
val k = length params;
val (c, ts) = strip_comb t;
val (xs, ys) = chop k ts;
val i = find_index (fn c' => c' = c) cs;
in
if xs = params andalso i >= 0 then
SOME (c, i, ys, chop (length ys) (arg_types_of k c))
else NONE
end;
fun mk_names a 0 = []
| mk_names a 1 = [a]
| mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
fun select_disj 1 1 = []
| select_disj _ 1 = [rtac disjI1]
| select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
(** process rules **)
local
fun err_in_rule ctxt name t msg =
error (cat_lines ["Ill-formed introduction rule " ^ Binding.print name,
Syntax.string_of_term ctxt t, msg]);
fun err_in_prem ctxt name t p msg =
error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
"in introduction rule " ^ Binding.print name, Syntax.string_of_term ctxt t, msg]);
val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
val bad_app = "Inductive predicate must be applied to parameter(s) ";
fun atomize_term thy = Raw_Simplifier.rewrite_term thy inductive_atomize [];
in
fun check_rule ctxt cs params ((binding, att), rule) =
let
val params' = Term.variant_frees rule (Logic.strip_params rule);
val frees = rev (map Free params');
val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
val rule' = Logic.list_implies (prems, concl);
val aprems = map (atomize_term (Proof_Context.theory_of ctxt)) prems;
val arule = list_all_free (params', Logic.list_implies (aprems, concl));
fun check_ind err t = case dest_predicate cs params t of
NONE => err (bad_app ^
commas (map (Syntax.string_of_term ctxt) params))
| SOME (_, _, ys, _) =>
if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
then err bad_ind_occ else ();
fun check_prem' prem t =
if member (op =) cs (head_of t) then
check_ind (err_in_prem ctxt binding rule prem) t
else (case t of
Abs (_, _, t) => check_prem' prem t
| t $ u => (check_prem' prem t; check_prem' prem u)
| _ => ());
fun check_prem (prem, aprem) =
if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
else err_in_prem ctxt binding rule prem "Non-atomic premise";
in
(case concl of
Const (@{const_name Trueprop}, _) $ t =>
if member (op =) cs (head_of t) then
(check_ind (err_in_rule ctxt binding rule') t;
List.app check_prem (prems ~~ aprems))
else err_in_rule ctxt binding rule' bad_concl
| _ => err_in_rule ctxt binding rule' bad_concl);
((binding, att), arule)
end;
val rulify =
hol_simplify inductive_conj
#> hol_simplify inductive_rulify
#> hol_simplify inductive_rulify_fallback
#> Simplifier.norm_hhf;
end;
(** proofs for (co)inductive predicates **)
(* prove monotonicity *)
fun prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt =
(message (quiet_mode orelse skip_mono andalso !quick_and_dirty orelse fork_mono)
" Proving monotonicity ...";
(if skip_mono then Skip_Proof.prove else if fork_mono then Goal.prove_future else Goal.prove) ctxt
[] []
(HOLogic.mk_Trueprop
(Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
(fn _ => EVERY [rtac @{thm monoI} 1,
REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
REPEAT (FIRST
[atac 1,
resolve_tac (map (mk_mono ctxt) monos @ get_monos ctxt) 1,
etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
(* prove introduction rules *)
fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
let
val _ = clean_message quiet_mode " Proving the introduction rules ...";
val unfold = funpow k (fn th => th RS fun_cong)
(mono RS (fp_def RS
(if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
val rules = [refl, TrueI, notFalseI, exI, conjI];
val intrs = map_index (fn (i, intr) =>
Skip_Proof.prove ctxt [] [] intr (fn _ => EVERY
[rewrite_goals_tac rec_preds_defs,
rtac (unfold RS iffD2) 1,
EVERY1 (select_disj (length intr_ts) (i + 1)),
(*Not ares_tac, since refl must be tried before any equality assumptions;
backtracking may occur if the premises have extra variables!*)
DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)])
|> singleton (Proof_Context.export ctxt ctxt')) intr_ts
in (intrs, unfold) end;
(* prove elimination rules *)
fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
let
val _ = clean_message quiet_mode " Proving the elimination rules ...";
val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
fun dest_intr r =
(the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
Logic.strip_assums_hyp r, Logic.strip_params r);
val intrs = map dest_intr intr_ts ~~ intr_names;
val rules1 = [disjE, exE, FalseE];
val rules2 = [conjE, FalseE, notTrueE];
fun prove_elim c =
let
val Ts = arg_types_of (length params) c;
val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
val frees = map Free (anames ~~ Ts);
fun mk_elim_prem ((_, _, us, _), ts, params') =
list_all (params',
Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
(frees ~~ us) @ ts, P));
val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
map mk_elim_prem (map #1 c_intrs)
in
(Skip_Proof.prove ctxt'' [] prems P
(fn {prems, ...} => EVERY
[cut_facts_tac [hd prems] 1,
rewrite_goals_tac rec_preds_defs,
dtac (unfold RS iffD1) 1,
REPEAT (FIRSTGOAL (eresolve_tac rules1)),
REPEAT (FIRSTGOAL (eresolve_tac rules2)),
EVERY (map (fn prem =>
DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
|> singleton (Proof_Context.export ctxt'' ctxt'''),
map #2 c_intrs, length Ts)
end
in map prove_elim cs end;
(* prove simplification equations *)
fun prove_eqs quiet_mode cs params intr_ts intrs (elims: (thm * bstring list * int) list) ctxt ctxt'' =
let
val _ = clean_message quiet_mode " Proving the simplification rules ...";
fun dest_intr r =
(the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
Logic.strip_assums_hyp r, Logic.strip_params r);
val intr_ts' = map dest_intr intr_ts;
fun prove_eq c (elim: thm * 'a * 'b) =
let
val Ts = arg_types_of (length params) c;
val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
val frees = map Free (anames ~~ Ts);
val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
let
fun list_ex ([], t) = t
| list_ex ((a,T)::vars, t) =
(HOLogic.exists_const T) $ (Abs(a, T, list_ex(vars,t)));
val conjs = map2 (curry HOLogic.mk_eq) frees us @ (map HOLogic.dest_Trueprop ts)
in
list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
end;
val lhs = list_comb (c, params @ frees)
val rhs =
if null c_intrs then @{term False} else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs)
val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
let
val (prems', last_prem) = split_last prems
in
EVERY1 (select_disj (length c_intrs) (i + 1))
THEN EVERY (replicate (length params) (rtac @{thm exI} 1))
THEN EVERY (map (fn prem => (rtac @{thm conjI} 1 THEN rtac prem 1)) prems')
THEN rtac last_prem 1
end) ctxt' 1
fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
EVERY (replicate (length params') (etac @{thm exE} 1))
THEN EVERY (replicate (length ts + length us - 1) (etac @{thm conjE} 1))
THEN Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
let
val (eqs, prems') = chop (length us) prems
val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs
in
rewrite_goal_tac rew_thms 1
THEN rtac intr 1
THEN (EVERY (map (fn p => rtac p 1) prems'))
end) ctxt' 1
in
Skip_Proof.prove ctxt' [] [] eq (fn {...} =>
rtac @{thm iffI} 1 THEN etac (#1 elim) 1
THEN EVERY (map_index prove_intr1 c_intrs)
THEN (if null c_intrs then etac @{thm FalseE} 1 else
let val (c_intrs', last_c_intr) = split_last c_intrs in
EVERY (map (fn ci => etac @{thm disjE} 1 THEN prove_intr2 ci) c_intrs')
THEN prove_intr2 last_c_intr
end))
|> rulify
|> singleton (Proof_Context.export ctxt' ctxt'')
end;
in
map2 prove_eq cs elims
end;
(* derivation of simplified elimination rules *)
local
(*delete needless equality assumptions*)
val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
(fn _ => assume_tac 1);
val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
fun simp_case_tac ss i =
EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
in
fun mk_cases ctxt prop =
let
val thy = Proof_Context.theory_of ctxt;
val ss = simpset_of ctxt;
fun err msg =
error (Pretty.string_of (Pretty.block
[Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
val elims = Induct.find_casesP ctxt prop;
val cprop = Thm.cterm_of thy prop;
val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
fun mk_elim rl =
Thm.implies_intr cprop (Tactic.rule_by_tactic ctxt tac (Thm.assume cprop RS rl))
|> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
in
(case get_first (try mk_elim) elims of
SOME r => r
| NONE => err "Proposition not an inductive predicate:")
end;
end;
(* inductive_cases *)
fun gen_inductive_cases prep_att prep_prop args lthy =
let
val thy = Proof_Context.theory_of lthy;
val facts = args |> Par_List.map (fn ((a, atts), props) =>
((a, map (prep_att thy) atts),
Par_List.map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
in lthy |> Local_Theory.notes facts |>> map snd end;
val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
val ind_cases_setup =
Method.setup @{binding ind_cases}
(Scan.lift (Scan.repeat1 Args.name_source --
Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.binding) []) >>
(fn (raw_props, fixes) => fn ctxt =>
let
val (_, ctxt') = Variable.add_fixes_binding fixes ctxt;
val props = Syntax.read_props ctxt' raw_props;
val ctxt'' = fold Variable.declare_term props ctxt';
val rules = Proof_Context.export ctxt'' ctxt (map (mk_cases ctxt'') props)
in Method.erule 0 rules end))
"dynamic case analysis on predicates";
(* derivation of simplified equation *)
fun mk_simp_eq ctxt prop =
let
val thy = Proof_Context.theory_of ctxt
val ctxt' = Variable.auto_fixes prop ctxt
val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of
val substs = Item_Net.retrieve (Equation_Data.get (Context.Proof ctxt)) (HOLogic.dest_Trueprop prop)
|> map_filter
(fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
(Vartab.empty, Vartab.empty), eq)
handle Pattern.MATCH => NONE)
val (subst, eq) = case substs of
[s] => s
| _ => error
("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique")
val inst = map (fn v => (cterm_of thy (Var v), cterm_of thy (Envir.subst_term subst (Var v))))
(Term.add_vars (lhs_of eq) [])
in
cterm_instantiate inst eq
|> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv
(Simplifier.full_rewrite (simpset_of ctxt))))
|> singleton (Variable.export ctxt' ctxt)
end
(* inductive simps *)
fun gen_inductive_simps prep_att prep_prop args lthy =
let
val thy = Proof_Context.theory_of lthy;
val facts = args |> map (fn ((a, atts), props) =>
((a, map (prep_att thy) atts),
map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
in lthy |> Local_Theory.notes facts |>> map snd end;
val inductive_simps = gen_inductive_simps Attrib.intern_src Syntax.read_prop;
val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
(* prove induction rule *)
fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
fp_def rec_preds_defs ctxt ctxt''' =
let
val _ = clean_message quiet_mode " Proving the induction rule ...";
val thy = Proof_Context.theory_of ctxt;
(* predicates for induction rule *)
val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
val preds = map2 (curry Free) pnames
(map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
(* transform an introduction rule into a premise for induction rule *)
fun mk_ind_prem r =
let
fun subst s =
(case dest_predicate cs params s of
SOME (_, i, ys, (_, Ts)) =>
let
val k = length Ts;
val bs = map Bound (k - 1 downto 0);
val P = list_comb (nth preds i, map (incr_boundvars k) ys @ bs);
val Q = list_abs (mk_names "x" k ~~ Ts,
HOLogic.mk_binop inductive_conj_name
(list_comb (incr_boundvars k s, bs), P))
in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
| NONE =>
(case s of
(t $ u) => (fst (subst t) $ fst (subst u), NONE)
| (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
| _ => (s, NONE)));
fun mk_prem s prems =
(case subst s of
(_, SOME (t, u)) => t :: u :: prems
| (t, _) => t :: prems);
val SOME (_, i, ys, _) = dest_predicate cs params
(HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
in
list_all_free (Logic.strip_params r,
Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
(map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
HOLogic.mk_Trueprop (list_comb (nth preds i, ys))))
end;
val ind_prems = map mk_ind_prem intr_ts;
(* make conclusions for induction rules *)
val Tss = map (binder_types o fastype_of) preds;
val (xnames, ctxt'') =
Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map (fn (((xnames, Ts), c), P) =>
let val frees = map Free (xnames ~~ Ts)
in HOLogic.mk_imp
(list_comb (c, params @ frees), list_comb (P, frees))
end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
(* make predicate for instantiation of abstract induction rule *)
val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
(map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
(make_bool_args HOLogic.mk_not I bs i)
(list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
val ind_concl = HOLogic.mk_Trueprop
(HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
val raw_fp_induct = (mono RS (fp_def RS @{thm def_lfp_induct}));
val induct = Skip_Proof.prove ctxt'' [] ind_prems ind_concl
(fn {prems, ...} => EVERY
[rewrite_goals_tac [inductive_conj_def],
DETERM (rtac raw_fp_induct 1),
REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
rewrite_goals_tac simp_thms'',
(*This disjE separates out the introduction rules*)
REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
(*Now break down the individual cases. No disjE here in case
some premise involves disjunction.*)
REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
REPEAT (FIRSTGOAL
(resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
(inductive_conj_def :: rec_preds_defs @ simp_thms'') prem,
conjI, refl] 1)) prems)]);
val lemma = Skip_Proof.prove ctxt'' [] []
(Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
[rewrite_goals_tac rec_preds_defs,
REPEAT (EVERY
[REPEAT (resolve_tac [conjI, impI] 1),
REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
atac 1,
rewrite_goals_tac simp_thms',
atac 1])])
in singleton (Proof_Context.export ctxt'' ctxt''') (induct RS lemma) end;
(** specification of (co)inductive predicates **)
fun mk_ind_def quiet_mode skip_mono fork_mono alt_name coind
cs intr_ts monos params cnames_syn lthy =
let
val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
val k = log 2 1 (length cs);
val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
val p :: xs = map Free (Variable.variant_frees lthy intr_ts
(("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
val bs = map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
(map (rpair HOLogic.boolT) (mk_names "b" k)));
fun subst t =
(case dest_predicate cs params t of
SOME (_, i, ts, (Ts, Us)) =>
let
val l = length Us;
val zs = map Bound (l - 1 downto 0);
in
list_abs (map (pair "z") Us, list_comb (p,
make_bool_args' bs i @ make_args argTs
((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
end
| NONE =>
(case t of
t1 $ t2 => subst t1 $ subst t2
| Abs (x, T, u) => Abs (x, T, subst u)
| _ => t));
(* transform an introduction rule into a conjunction *)
(* [| p_i t; ... |] ==> p_j u *)
(* is transformed into *)
(* b_j & x_j = u & p b_j t & ... *)
fun transform_rule r =
let
val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
(HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
val ps = make_bool_args HOLogic.mk_not I bs i @
map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
map (subst o HOLogic.dest_Trueprop)
(Logic.strip_assums_hyp r)
in
fold_rev (fn (x, T) => fn P => HOLogic.exists_const T $ Abs (x, T, P))
(Logic.strip_params r)
(if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
end
(* make a disjunction of all introduction rules *)
val fp_fun = fold_rev lambda (p :: bs @ xs)
(if null intr_ts then HOLogic.false_const
else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
(* add definiton of recursive predicates to theory *)
val rec_name =
if Binding.is_empty alt_name then
Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
else alt_name;
val ((rec_const, (_, fp_def)), lthy') = lthy
|> Local_Theory.conceal
|> Local_Theory.define
((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
((Binding.empty, [Attrib.internal (K Nitpick_Unfolds.add)]),
fold_rev lambda params
(Const (fp_name, (predT --> predT) --> predT) $ fp_fun)))
||> Local_Theory.restore_naming lthy;
val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
(cterm_of (Proof_Context.theory_of lthy') (list_comb (rec_const, params)));
val specs =
if length cs < 2 then []
else
map_index (fn (i, (name_mx, c)) =>
let
val Ts = arg_types_of (length params) c;
val xs = map Free (Variable.variant_frees lthy intr_ts
(mk_names "x" (length Ts) ~~ Ts))
in
(name_mx, (apfst Binding.conceal Attrib.empty_binding, fold_rev lambda (params @ xs)
(list_comb (rec_const, params @ make_bool_args' bs i @
make_args argTs (xs ~~ Ts)))))
end) (cnames_syn ~~ cs);
val (consts_defs, lthy'') = lthy'
|> fold_map Local_Theory.define specs;
val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
val mono = prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos lthy''';
val (_, lthy'''') =
Local_Theory.note (apfst Binding.conceal Attrib.empty_binding,
Proof_Context.export lthy''' lthy'' [mono]) lthy'';
in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
list_comb (rec_const, params), preds, argTs, bs, xs)
end;
fun declare_rules rec_binding coind no_ind cnames
preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
let
val rec_name = Binding.name_of rec_binding;
fun rec_qualified qualified = Binding.qualify qualified rec_name;
val intr_names = map Binding.name_of intr_bindings;
val ind_case_names = Rule_Cases.case_names intr_names;
val induct =
if coind then
(raw_induct, [Rule_Cases.case_names [rec_name],
Rule_Cases.case_conclusion (rec_name, intr_names),
Rule_Cases.consumes 1, Induct.coinduct_pred (hd cnames)])
else if no_ind orelse length cnames > 1 then
(raw_induct, [ind_case_names, Rule_Cases.consumes 0])
else (raw_induct RSN (2, rev_mp), [ind_case_names, Rule_Cases.consumes 1]);
val (intrs', lthy1) =
lthy |>
Spec_Rules.add
(if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
Local_Theory.notes
(map (rec_qualified false) intr_bindings ~~ intr_atts ~~
map (fn th => [([th],
[Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
map (hd o snd);
val (((_, elims'), (_, [induct'])), lthy2) =
lthy1 |>
Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
fold_map (fn (name, (elim, cases, k)) =>
Local_Theory.note
((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
[Attrib.internal (K (Rule_Cases.case_names cases)),
Attrib.internal (K (Rule_Cases.consumes 1)),
Attrib.internal (K (Rule_Cases.constraints k)),
Attrib.internal (K (Induct.cases_pred name)),
Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
Local_Theory.note
((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
val (eqs', lthy3) = lthy2 |>
fold_map (fn (name, eq) => Local_Theory.note
((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
[Attrib.internal (K add_equation)]), [eq])
#> apfst (hd o snd))
(if null eqs then [] else (cnames ~~ eqs))
val (inducts, lthy4) =
if no_ind orelse coind then ([], lthy3)
else
let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
lthy3 |>
Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
inducts |> map (fn (name, th) => ([th],
[Attrib.internal (K ind_case_names),
Attrib.internal (K (Rule_Cases.consumes 1)),
Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
end;
in (intrs', elims', eqs', induct', inducts, lthy4) end;
type inductive_flags =
{quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool};
type add_ind_def =
inductive_flags ->
term list -> (Attrib.binding * term) list -> thm list ->
term list -> (binding * mixfix) list ->
local_theory -> inductive_result * local_theory;
fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}
cs intros monos params cnames_syn lthy =
let
val _ = null cnames_syn andalso error "No inductive predicates given";
val names = map (Binding.name_of o fst) cnames_syn;
val _ = message (quiet_mode andalso not verbose)
("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn; (* FIXME *)
val ((intr_names, intr_atts), intr_ts) =
apfst split_list (split_list (map (check_rule lthy cs params) intros));
val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
argTs, bs, xs) = mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts
monos params cnames_syn lthy;
val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
intr_ts rec_preds_defs lthy2 lthy1;
val elims =
if no_elim then []
else
prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
unfold rec_preds_defs lthy2 lthy1;
val raw_induct = zero_var_indexes
(if no_ind then Drule.asm_rl
else if coind then
singleton (Proof_Context.export lthy2 lthy1)
(rotate_prems ~1 (Object_Logic.rulify
(fold_rule rec_preds_defs
(rewrite_rule simp_thms'''
(mono RS (fp_def RS @{thm def_coinduct}))))))
else
prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
rec_preds_defs lthy2 lthy1);
val eqs =
if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1
val elims' = map (fn (th, ns, i) => (rulify th, ns, i)) elims
val intrs' = map rulify intrs
val (intrs'', elims'', eqs', induct, inducts, lthy3) = declare_rules rec_name coind no_ind
cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
val result =
{preds = preds,
intrs = intrs'',
elims = elims'',
raw_induct = rulify raw_induct,
induct = induct,
inducts = inducts,
eqs = eqs'};
val lthy4 = lthy3
|> Local_Theory.declaration false (fn phi =>
let val result' = transform_result phi result;
in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
in (result, lthy4) end;
(* external interfaces *)
fun gen_add_inductive_i mk_def
(flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono})
cnames_syn pnames spec monos lthy =
let
val thy = Proof_Context.theory_of lthy;
val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
(* abbrevs *)
val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
fun get_abbrev ((name, atts), t) =
if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
let
val _ = Binding.is_empty name andalso null atts orelse
error "Abbreviations may not have names or attributes";
val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
val var =
(case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
NONE => error ("Undeclared head of abbreviation " ^ quote x)
| SOME ((b, T'), mx) =>
if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
else (b, mx));
in SOME (var, rhs) end
else NONE;
val abbrevs = map_filter get_abbrev spec;
val bs = map (Binding.name_of o fst o fst) abbrevs;
(* predicates *)
val pre_intros = filter_out (is_some o get_abbrev) spec;
val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
val ps = map Free pnames;
val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
val expand = Assumption.export_term ctxt3 lthy #> Proof_Context.cert_term lthy;
fun close_rule r = list_all_free (rev (fold_aterms
(fn t as Free (v as (s, _)) =>
if Variable.is_fixed ctxt1 s orelse
member (op =) ps t then I else insert (op =) v
| _ => I) r []), r);
val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
in
lthy
|> mk_def flags cs intros monos ps preds
||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
end;
fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos int lthy =
let
val ((vars, intrs), _) = lthy
|> Proof_Context.set_mode Proof_Context.mode_abbrev
|> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
val (cs, ps) = chop (length cnames_syn) vars;
val monos = Attrib.eval_thms lthy raw_monos;
val flags = {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
coind = coind, no_elim = false, no_ind = false, skip_mono = false, fork_mono = not int};
in
lthy
|> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
end;
val add_inductive_i = gen_add_inductive_i add_ind_def;
val add_inductive = gen_add_inductive add_ind_def;
fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
let
val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
val ctxt' = thy
|> Named_Target.theory_init
|> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
|> Local_Theory.exit;
val info = #2 (the_inductive ctxt' name);
in (info, Proof_Context.theory_of ctxt') end;
(* read off arities of inductive predicates from raw induction rule *)
fun arities_of induct =
map (fn (_ $ t $ u) =>
(fst (dest_Const (head_of t)), length (snd (strip_comb u))))
(HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
(* read off parameters of inductive predicate from raw induction rule *)
fun params_of induct =
let
val (_ $ t $ u :: _) =
HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
val (_, ts) = strip_comb t;
val (_, us) = strip_comb u
in
List.take (ts, length ts - length us)
end;
val pname_of_intr =
concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
(* partition introduction rules according to predicate name *)
fun gen_partition_rules f induct intros =
fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
(map (rpair [] o fst) (arities_of induct));
val partition_rules = gen_partition_rules I;
fun partition_rules' induct = gen_partition_rules fst induct;
fun unpartition_rules intros xs =
fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
(fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
(* infer order of variables in intro rules from order of quantifiers in elim rule *)
fun infer_intro_vars elim arity intros =
let
val thy = theory_of_thm elim;
val _ :: cases = prems_of elim;
val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
fun mtch (t, u) =
let
val params = Logic.strip_params t;
val vars = map (Var o apfst (rpair 0))
(Name.variant_list used (map fst params) ~~ map snd params);
val ts = map (curry subst_bounds (rev vars))
(List.drop (Logic.strip_assums_hyp t, arity));
val us = Logic.strip_imp_prems u;
val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
(Vartab.empty, Vartab.empty);
in
map (Envir.subst_term tab) vars
end
in
map (mtch o apsnd prop_of) (cases ~~ intros)
end;
(** package setup **)
(* setup theory *)
val setup =
ind_cases_setup #>
Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
"declaration of monotonicity rule";
(* outer syntax *)
val _ = Keyword.keyword "monos";
fun gen_ind_decl mk_def coind =
Parse.fixes -- Parse.for_fixes --
Scan.optional Parse_Spec.where_alt_specs [] --
Scan.optional (Parse.$$$ "monos" |-- Parse.!!! Parse_Spec.xthms1) []
>> (fn (((preds, params), specs), monos) =>
(snd oo gen_add_inductive mk_def true coind preds params specs monos));
val ind_decl = gen_ind_decl add_ind_def;
val _ =
Outer_Syntax.local_theory' "inductive" "define inductive predicates" Keyword.thy_decl
(ind_decl false);
val _ =
Outer_Syntax.local_theory' "coinductive" "define coinductive predicates" Keyword.thy_decl
(ind_decl true);
val _ =
Outer_Syntax.local_theory "inductive_cases"
"create simplified instances of elimination rules (improper)" Keyword.thy_script
(Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
val _ =
Outer_Syntax.local_theory "inductive_simps"
"create simplification rules for inductive predicates" Keyword.thy_script
(Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
end;