(* 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 INDUCTIVE =
sig
type 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 -> result -> result
type info = {names: string list, coind: bool} * result
val the_inductive: Proof.context -> term -> info
val the_inductive_global: Proof.context -> string -> info
val print_inductives: bool -> Proof.context -> unit
val get_monos: Proof.context -> thm list
val mono_add: attribute
val mono_del: attribute
val mk_cases_tac: Proof.context -> tactic
val mk_cases: Proof.context -> term -> thm
val inductive_forall_def: thm
val rulify: Proof.context -> thm -> thm
val inductive_cases: (Attrib.binding * term list) list -> local_theory ->
(string * thm list) list * local_theory
val inductive_cases_cmd: (Attrib.binding * string list) list -> local_theory ->
(string * thm list) list * local_theory
val ind_cases_rules: Proof.context ->
string list -> (binding * string option * mixfix) list -> thm list
val inductive_simps: (Attrib.binding * term list) list -> local_theory ->
(string * thm list) list * local_theory
val inductive_simps_cmd: (Attrib.binding * string list) list -> local_theory ->
(string * thm list) list * local_theory
type flags =
{quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
no_elim: bool, no_ind: bool, skip_mono: bool}
val add_inductive:
flags -> ((binding * typ) * mixfix) list ->
(string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
result * local_theory
val add_inductive_cmd: bool -> bool ->
(binding * string option * mixfix) list ->
(binding * string option * mixfix) list ->
Specification.multi_specs_cmd ->
(Facts.ref * Token.src list) list ->
local_theory -> result * local_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: theory -> thm -> int -> thm list -> term list list
val inductive_internals: bool Config.T
val select_disj_tac: Proof.context -> int -> int -> int -> tactic
type add_ind_def =
flags ->
term list -> (Attrib.binding * term) list -> thm list ->
term list -> (binding * mixfix) list ->
local_theory -> result * local_theory
val declare_rules: binding -> bool -> bool -> binding -> string list -> term list ->
thm list -> binding list -> Token.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: add_ind_def -> flags ->
((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
thm list -> local_theory -> result * local_theory
val gen_add_inductive_cmd: add_ind_def -> bool -> bool ->
(binding * string option * mixfix) list ->
(binding * string option * mixfix) list ->
Specification.multi_specs_cmd -> (Facts.ref * Token.src list) list ->
local_theory -> result * local_theory
val gen_ind_decl: add_ind_def -> bool -> (local_theory -> local_theory) parser
end;
structure Inductive: INDUCTIVE =
struct
(** theory context references **)
val inductive_forall_def = @{thm HOL.induct_forall_def};
val inductive_conj_def = @{thm HOL.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 simp_thms1 =
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_thms2 =
map mk_meta_eq [@{thm inf_fun_def}, @{thm inf_bool_def}] @ simp_thms1;
val simp_thms3 =
@{thms le_rel_bool_arg_iff if_False if_True conj_ac
le_fun_def le_bool_def sup_fun_def sup_bool_def simp_thms
if_bool_eq_disj all_simps ex_simps imp_conjL};
(** misc utilities **)
val inductive_internals = Attrib.setup_config_bool \<^binding>\<open>inductive_internals\<close> (K false);
fun message quiet_mode s = if quiet_mode then () else writeln s;
fun clean_message ctxt quiet_mode s =
if Config.get ctxt 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 \<^term>\<open>False\<close>) (K \<^term>\<open>True\<close>) 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>\<open>undefined\<close>, 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_tac ctxt =
let
fun tacs 1 1 = []
| tacs _ 1 = [resolve_tac ctxt @{thms disjI1}]
| tacs n i = resolve_tac ctxt @{thms disjI2} :: tacs (n - 1) (i - 1);
in fn n => fn i => EVERY' (tacs n i) end;
(** context data **)
type 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 info = {names: string list, coind: bool} * result;
val empty_infos =
Item_Net.init (op = o apply2 (#names o fst)) (#preds o snd)
val empty_equations =
Item_Net.init Thm.eq_thm_prop
(single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
datatype data = Data of
{infos: info Item_Net.T,
monos: thm list,
equations: thm Item_Net.T};
fun make_data (infos, monos, equations) =
Data {infos = infos, monos = monos, equations = equations};
structure Data = Generic_Data
(
type T = data;
val empty = make_data (empty_infos, [], empty_equations);
val extend = I;
fun merge (Data {infos = infos1, monos = monos1, equations = equations1},
Data {infos = infos2, monos = monos2, equations = equations2}) =
make_data (Item_Net.merge (infos1, infos2),
Thm.merge_thms (monos1, monos2),
Item_Net.merge (equations1, equations2));
);
fun map_data f =
Data.map (fn Data {infos, monos, equations} => make_data (f (infos, monos, equations)));
fun rep_data ctxt = Data.get (Context.Proof ctxt) |> (fn Data rep => rep);
fun print_inductives verbose ctxt =
let
val {infos, monos, ...} = rep_data ctxt;
val space = Consts.space_of (Proof_Context.consts_of ctxt);
val consts =
Item_Net.content infos
|> maps (fn ({names, ...}, result) => map (rpair result) names)
in
[Pretty.block
(Pretty.breaks
(Pretty.str "(co)inductives:" ::
map (Pretty.mark_str o #1)
(Name_Space.markup_entries verbose ctxt space consts))),
Pretty.big_list "monotonicity rules:" (map (Thm.pretty_thm_item ctxt) monos)]
end |> Pretty.writeln_chunks;
(* inductive info *)
fun the_inductive ctxt term =
Item_Net.retrieve (#infos (rep_data ctxt)) term
|> the_single
|> apsnd (transform_result (Morphism.transfer_morphism' ctxt))
fun the_inductive_global ctxt name =
#infos (rep_data ctxt)
|> Item_Net.content
|> filter (fn ({names, ...}, _) => member op = names name)
|> the_single
|> apsnd (transform_result (Morphism.transfer_morphism' ctxt))
fun put_inductives info =
map_data (fn (infos, monos, equations) =>
(Item_Net.update (apsnd (transform_result Morphism.trim_context_morphism) info) infos,
monos, equations));
(* monotonicity rules *)
fun get_monos ctxt =
#monos (rep_data ctxt)
|> map (Thm.transfer' ctxt);
fun mk_mono ctxt thm =
let
fun eq_to_mono thm' = thm' RS (thm' RS @{thm 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 Thm.concl_of thm of
Const (\<^const_name>\<open>Pure.eq\<close>, _) $ _ $ _ => eq_to_mono (HOLogic.mk_obj_eq thm)
| _ $ (Const (\<^const_name>\<open>HOL.eq\<close>, _) $ _ $ _) => eq_to_mono thm
| _ $ (Const (\<^const_name>\<open>Orderings.less_eq\<close>, _) $ _ $ _) =>
dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
(resolve_tac ctxt [@{thm le_funI}, @{thm le_boolI'}])) thm))
| _ => thm)
end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Thm.string_of_thm ctxt thm);
val mono_add =
Thm.declaration_attribute (fn thm => fn context =>
map_data (fn (infos, monos, equations) =>
(infos, Thm.add_thm (Thm.trim_context (mk_mono (Context.proof_of context) thm)) monos,
equations)) context);
val mono_del =
Thm.declaration_attribute (fn thm => fn context =>
map_data (fn (infos, monos, equations) =>
(infos, Thm.del_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
val _ =
Theory.setup
(Attrib.setup \<^binding>\<open>mono\<close> (Attrib.add_del mono_add mono_del)
"declaration of monotonicity rule");
(* equations *)
fun retrieve_equations ctxt =
Item_Net.retrieve (#equations (rep_data ctxt))
#> map (Thm.transfer' ctxt);
val equation_add_permissive =
Thm.declaration_attribute (fn thm =>
map_data (fn (infos, monos, equations) =>
(infos, monos, perhaps (try (Item_Net.update (Thm.trim_context thm))) equations)));
(** 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 = fold_rev (Logic.all o 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";
val _ =
(case concl of
Const (\<^const_name>\<open>Trueprop\<close>, _) $ 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);
in
((binding, att), arule)
end;
fun rulify ctxt =
hol_simplify ctxt inductive_conj
#> hol_simplify ctxt inductive_rulify
#> hol_simplify ctxt inductive_rulify_fallback
#> Simplifier.norm_hhf ctxt;
end;
(** proofs for (co)inductive predicates **)
(* prove monotonicity *)
fun prove_mono quiet_mode skip_mono predT fp_fun monos ctxt =
(message (quiet_mode orelse skip_mono andalso Config.get ctxt quick_and_dirty)
" Proving monotonicity ...";
(if skip_mono then Goal.prove_sorry else Goal.prove_future) ctxt
[] []
(HOLogic.mk_Trueprop
(Const (\<^const_name>\<open>Orderings.mono\<close>, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
(fn _ => EVERY [resolve_tac ctxt @{thms monoI} 1,
REPEAT (resolve_tac ctxt [@{thm le_funI}, @{thm le_boolI'}] 1),
REPEAT (FIRST
[assume_tac ctxt 1,
resolve_tac ctxt (map (mk_mono ctxt) monos @ get_monos ctxt) 1,
eresolve_tac ctxt @{thms le_funE} 1,
dresolve_tac ctxt @{thms 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 ctxt 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, @{lemma "\ False" by (rule notI)}, exI, conjI];
val intrs = map_index (fn (i, intr) =>
Goal.prove_sorry ctxt [] [] intr (fn _ => EVERY
[rewrite_goals_tac ctxt rec_preds_defs,
resolve_tac ctxt [unfold RS iffD2] 1,
select_disj_tac ctxt (length intr_ts) (i + 1) 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 ctxt rules 1 APPEND assume_tac ctxt 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 ctxt 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, @{lemma "\ True \ R" by (rule notE [OF _ TrueI])}];
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') =
Logic.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
(Goal.prove_sorry ctxt'' [] prems P
(fn {context = ctxt4, prems} => EVERY
[cut_tac (hd prems) 1,
rewrite_goals_tac ctxt4 rec_preds_defs,
dresolve_tac ctxt4 [unfold RS iffD1] 1,
REPEAT (FIRSTGOAL (eresolve_tac ctxt4 rules1)),
REPEAT (FIRSTGOAL (eresolve_tac ctxt4 rules2)),
EVERY (map (fn prem =>
DEPTH_SOLVE_1 (assume_tac ctxt4 1 ORELSE
resolve_tac ctxt [rewrite_rule ctxt4 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'' = (* FIXME ctxt'' ?? *)
let
val _ = clean_message ctxt 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>\<open>False\<close>
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 {context = ctxt'', params, prems, ...} =>
select_disj_tac ctxt'' (length c_intrs) (i + 1) 1 THEN
EVERY (replicate (length params) (resolve_tac ctxt'' @{thms exI} 1)) THEN
(if null prems then resolve_tac ctxt'' @{thms TrueI} 1
else
let
val (prems', last_prem) = split_last prems;
in
EVERY (map (fn prem =>
(resolve_tac ctxt'' @{thms conjI} 1 THEN resolve_tac ctxt'' [prem] 1)) prems')
THEN resolve_tac ctxt'' [last_prem] 1
end)) ctxt' 1;
fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
EVERY (replicate (length params') (eresolve_tac ctxt' @{thms exE} 1)) THEN
(if null ts andalso null us then resolve_tac ctxt' [intr] 1
else
EVERY (replicate (length ts + length us - 1) (eresolve_tac ctxt' @{thms conjE} 1)) THEN
Subgoal.FOCUS_PREMS (fn {context = ctxt'', 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 ctxt'' rew_thms 1 THEN
resolve_tac ctxt'' [intr] 1 THEN
EVERY (map (fn p => resolve_tac ctxt'' [p] 1) prems')
end) ctxt' 1);
in
Goal.prove_sorry ctxt' [] [] eq (fn _ =>
resolve_tac ctxt' @{thms iffI} 1 THEN
eresolve_tac ctxt' [#1 elim] 1 THEN
EVERY (map_index prove_intr1 c_intrs) THEN
(if null c_intrs then eresolve_tac ctxt' @{thms FalseE} 1
else
let val (c_intrs', last_c_intr) = split_last c_intrs in
EVERY (map (fn ci => eresolve_tac ctxt' @{thms disjE} 1 THEN prove_intr2 ci) c_intrs')
THEN prove_intr2 last_c_intr
end))
|> rulify ctxt'
|> 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>\<open>HOL\<close> [] [] \<^prop>\<open>\<And>P. a = a \<Longrightarrow> P \<Longrightarrow> P\<close>
(fn {context = ctxt, ...} => assume_tac ctxt 1);
val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
fun elim_tac ctxt = REPEAT o eresolve_tac ctxt elim_rls;
fun simp_case_tac ctxt i =
EVERY' [elim_tac ctxt,
asm_full_simp_tac ctxt,
elim_tac ctxt,
REPEAT o bound_hyp_subst_tac ctxt] i;
in
fun mk_cases_tac ctxt = ALLGOALS (simp_case_tac ctxt) THEN prune_params_tac ctxt;
fun mk_cases ctxt prop =
let
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 ctxt prop;
fun mk_elim rl =
Thm.implies_intr cprop
(Tactic.rule_by_tactic ctxt (mk_cases_tac ctxt) (Thm.assume cprop RS rl))
|> singleton (Proof_Context.export (Proof_Context.augment 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 thmss =
map snd args
|> burrow (grouped 10 Par_List.map_independent (mk_cases lthy o prep_prop lthy));
val facts =
map2 (fn ((a, atts), _) => fn thms => ((a, map (prep_att lthy) atts), [(thms, [])]))
args thmss;
in lthy |> Local_Theory.notes facts end;
val inductive_cases = gen_inductive_cases (K I) Syntax.check_prop;
val inductive_cases_cmd = gen_inductive_cases Attrib.check_src Syntax.read_prop;
(* ind_cases *)
fun ind_cases_rules ctxt raw_props raw_fixes =
let
val (props, ctxt') = Specification.read_props raw_props raw_fixes ctxt;
val rules = Proof_Context.export ctxt' ctxt (map (mk_cases ctxt') props);
in rules end;
val _ =
Theory.setup
(Method.setup \<^binding>\<open>ind_cases\<close>
(Scan.lift (Scan.repeat1 Parse.prop -- Parse.for_fixes) >>
(fn (props, fixes) => fn ctxt =>
Method.erule ctxt 0 (ind_cases_rules ctxt props fixes)))
"case analysis for inductive definitions, based on simplified elimination rule");
(* derivation of simplified equation *)
fun mk_simp_eq ctxt prop =
let
val thy = Proof_Context.theory_of ctxt;
val ctxt' = Proof_Context.augment prop ctxt;
val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of;
val substs =
retrieve_equations 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 => (fst v, Thm.cterm_of ctxt' (Envir.subst_term subst (Var v))))
(Term.add_vars (lhs_of eq) []);
in
infer_instantiate ctxt' inst eq
|> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv (Simplifier.full_rewrite ctxt')))
|> singleton (Proof_Context.export ctxt' ctxt)
end
(* inductive simps *)
fun gen_inductive_simps prep_att prep_prop args lthy =
let
val facts = args |> map (fn ((a, atts), props) =>
((a, map (prep_att lthy) atts),
map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
in lthy |> Local_Theory.notes facts end;
val inductive_simps = gen_inductive_simps (K I) Syntax.check_prop;
val inductive_simps_cmd = gen_inductive_simps Attrib.check_src Syntax.read_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''' = (* FIXME ctxt''' ?? *)
let
val _ = clean_message ctxt quiet_mode " Proving the induction rule ...";
(* 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 =
fold_rev Term.abs (mk_names "x" k ~~ Ts)
(HOLogic.mk_binop \<^const_name>\<open>HOL.induct_conj\<close>
(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
fold_rev (Logic.all o 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>\<open>Orderings.less_eq\<close> (rec_const, ind_pred));
val raw_fp_induct = mono RS (fp_def RS @{thm def_lfp_induct});
val induct = Goal.prove_sorry ctxt'' [] ind_prems ind_concl
(fn {context = ctxt3, prems} => EVERY
[rewrite_goals_tac ctxt3 [inductive_conj_def],
DETERM (resolve_tac ctxt3 [raw_fp_induct] 1),
REPEAT (resolve_tac ctxt3 [@{thm le_funI}, @{thm le_boolI}] 1),
rewrite_goals_tac ctxt3 simp_thms2,
(*This disjE separates out the introduction rules*)
REPEAT (FIRSTGOAL (eresolve_tac ctxt3 [disjE, exE, FalseE])),
(*Now break down the individual cases. No disjE here in case
some premise involves disjunction.*)
REPEAT (FIRSTGOAL (eresolve_tac ctxt3 [conjE] ORELSE' bound_hyp_subst_tac ctxt3)),
REPEAT (FIRSTGOAL
(resolve_tac ctxt3 [conjI, impI] ORELSE'
(eresolve_tac ctxt3 [notE] THEN' assume_tac ctxt3))),
EVERY (map (fn prem =>
DEPTH_SOLVE_1 (assume_tac ctxt3 1 ORELSE
resolve_tac ctxt3
[rewrite_rule ctxt3 (inductive_conj_def :: rec_preds_defs @ simp_thms2) prem,
conjI, refl] 1)) prems)]);
val lemma = Goal.prove_sorry ctxt'' [] []
(Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn {context = ctxt3, ...} => EVERY
[rewrite_goals_tac ctxt3 rec_preds_defs,
REPEAT (EVERY
[REPEAT (resolve_tac ctxt3 [conjI, impI] 1),
REPEAT (eresolve_tac ctxt3 [@{thm le_funE}, @{thm le_boolE}] 1),
assume_tac ctxt3 1,
rewrite_goals_tac ctxt3 simp_thms1,
assume_tac ctxt3 1])]);
in singleton (Proof_Context.export ctxt'' ctxt''') (induct RS lemma) end;
(* prove coinduction rule *)
fun If_const T = Const (\<^const_name>\<open>If\<close>, HOLogic.boolT --> T --> T --> T);
fun mk_If p t f = let val T = fastype_of t in If_const T $ p $ t $ f end;
fun prove_coindrule quiet_mode preds cs argTs bs xs params intr_ts mono
fp_def rec_preds_defs ctxt ctxt''' = (* FIXME ctxt''' ?? *)
let
val _ = clean_message ctxt quiet_mode " Proving the coinduction rule ...";
val n = length cs;
val (ns, xss) = map_split (fn pred =>
make_args' argTs xs (arg_types_of (length params) pred) |> `length) preds;
val xTss = map (map fastype_of) xss;
val (Rs_names, names_ctxt) = Variable.variant_fixes (mk_names "X" n) ctxt;
val Rs = map2 (fn name => fn Ts => Free (name, Ts ---> \<^typ>\<open>bool\<close>)) Rs_names xTss;
val Rs_applied = map2 (curry list_comb) Rs xss;
val preds_applied = map2 (curry list_comb) (map (fn p => list_comb (p, params)) preds) xss;
val abstract_list = fold_rev (absfree o dest_Free);
val bss = map (make_bool_args
(fn b => HOLogic.mk_eq (b, \<^term>\<open>False\<close>))
(fn b => HOLogic.mk_eq (b, \<^term>\<open>True\<close>)) bs) (0 upto n - 1);
val eq_undefinedss = map (fn ys => map (fn x =>
HOLogic.mk_eq (x, Const (\<^const_name>\<open>undefined\<close>, fastype_of x)))
(subtract (op =) ys xs)) xss;
val R =
@{fold 3} (fn bs => fn eqs => fn R => fn t => if null bs andalso null eqs then R else
mk_If (Library.foldr1 HOLogic.mk_conj (bs @ eqs)) R t)
bss eq_undefinedss Rs_applied \<^term>\<open>False\<close>
|> abstract_list (bs @ xs);
fun subst t =
(case dest_predicate cs params t of
SOME (_, i, ts, (_, Us)) =>
let
val l = length Us;
val bs = map Bound (l - 1 downto 0);
val args = map (incr_boundvars l) ts @ bs
in
HOLogic.mk_disj (list_comb (nth Rs i, args),
list_comb (nth preds i, params @ args))
|> fold_rev absdummy Us
end
| NONE =>
(case t of
t1 $ t2 => subst t1 $ subst t2
| Abs (x, T, u) => Abs (x, T, subst u)
| _ => t));
fun mk_coind_prem r =
let
val SOME (_, i, ts, (Ts, _)) =
dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
val ps =
map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
map (subst o HOLogic.dest_Trueprop) (Logic.strip_assums_hyp r);
in
(i, fold_rev (fn (x, T) => fn P => HOLogic.exists_const T $ Abs (x, T, P))
(Logic.strip_params r)
(if null ps then \<^term>\<open>True\<close> else foldr1 HOLogic.mk_conj ps))
end;
fun mk_prem i Ps = Logic.mk_implies
((nth Rs_applied i, Library.foldr1 HOLogic.mk_disj Ps) |> @{apply 2} HOLogic.mk_Trueprop)
|> fold_rev Logic.all (nth xss i);
val prems = map mk_coind_prem intr_ts |> AList.group (op =) |> sort (int_ord o apply2 fst)
|> map (uncurry mk_prem);
val concl = @{map 3} (fn xs =>
Ctr_Sugar_Util.list_all_free xs oo curry HOLogic.mk_imp) xss Rs_applied preds_applied
|> Library.foldr1 HOLogic.mk_conj |> HOLogic.mk_Trueprop;
val pred_defs_sym = if null rec_preds_defs then [] else map2 (fn n => fn thm =>
funpow n (fn thm => thm RS @{thm meta_fun_cong}) thm RS @{thm Pure.symmetric})
ns rec_preds_defs;
val simps = simp_thms3 @ pred_defs_sym;
val simprocs = [Simplifier.the_simproc ctxt "HOL.defined_All"];
val simplify = asm_full_simplify (Ctr_Sugar_Util.ss_only simps ctxt addsimprocs simprocs);
val coind = (mono RS (fp_def RS @{thm def_coinduct}))
|> infer_instantiate' ctxt [SOME (Thm.cterm_of ctxt R)]
|> simplify;
fun idx_of t = find_index (fn R =>
R = the_single (subtract (op =) (preds @ params) (map Free (Term.add_frees t [])))) Rs;
val coind_concls = HOLogic.dest_Trueprop (Thm.concl_of coind) |> HOLogic.dest_conj
|> map (fn t => (idx_of t, t)) |> sort (int_ord o @{apply 2} fst) |> map snd;
val reorder_bound_goals = map_filter (fn (t, u) => if t aconv u then NONE else
SOME (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u))))
((HOLogic.dest_Trueprop concl |> HOLogic.dest_conj) ~~ coind_concls);
val reorder_bound_thms = map (fn goal => Goal.prove_sorry ctxt [] [] goal
(fn {context = ctxt, prems = _} =>
HEADGOAL (EVERY' [resolve_tac ctxt [iffI],
REPEAT_DETERM o resolve_tac ctxt [allI, impI],
REPEAT_DETERM o dresolve_tac ctxt [spec], eresolve_tac ctxt [mp], assume_tac ctxt,
REPEAT_DETERM o resolve_tac ctxt [allI, impI],
REPEAT_DETERM o dresolve_tac ctxt [spec], eresolve_tac ctxt [mp], assume_tac ctxt])))
reorder_bound_goals;
val coinduction = Goal.prove_sorry ctxt [] prems concl (fn {context = ctxt, prems = CIH} =>
HEADGOAL (full_simp_tac
(Ctr_Sugar_Util.ss_only (simps @ reorder_bound_thms) ctxt addsimprocs simprocs) THEN'
resolve_tac ctxt [coind]) THEN
ALLGOALS (REPEAT_ALL_NEW (REPEAT_DETERM o resolve_tac ctxt [allI, impI, conjI] THEN'
REPEAT_DETERM o eresolve_tac ctxt [exE, conjE] THEN'
dresolve_tac ctxt (map simplify CIH) THEN'
REPEAT_DETERM o (assume_tac ctxt ORELSE'
eresolve_tac ctxt [conjE] ORELSE' dresolve_tac ctxt [spec, mp]))))
in
coinduction
|> length cs = 1 ? (Object_Logic.rulify ctxt #> rotate_prems ~1)
|> singleton (Proof_Context.export names_ctxt ctxt''')
end
(** specification of (co)inductive predicates **)
fun mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts monos params cnames_syn lthy =
let
val fp_name = if coind then \<^const_name>\<open>Inductive.gfp\<close> else \<^const_name>\<open>Inductive.lfp\<close>;
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
fold_rev (Term.abs o 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 \<^term>\<open>True\<close> 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 \<^term>\<open>False\<close>
else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
(* add definition of recursive predicates to theory *)
val is_auxiliary = length cs > 1;
val rec_binding =
if Binding.is_empty alt_name then Binding.conglomerate (map #1 cnames_syn) else alt_name;
val rec_name = Binding.name_of rec_binding;
val internals = Config.get lthy inductive_internals;
val ((rec_const, (_, fp_def)), lthy') = lthy
|> is_auxiliary ? Proof_Context.concealed
|> Local_Theory.define
((rec_binding, case cnames_syn of [(_, mx)] => mx | _ => NoSyn),
((Thm.make_def_binding internals rec_binding, @{attributes [nitpick_unfold]}),
fold_rev lambda params
(Const (fp_name, (predT --> predT) --> predT) $ fp_fun)))
||> Proof_Context.restore_naming lthy;
val fp_def' =
Simplifier.rewrite (put_simpset HOL_basic_ss lthy' addsimps [fp_def])
(Thm.cterm_of lthy' (list_comb (rec_const, params)));
val specs =
if is_auxiliary then
map_index (fn (i, ((b, 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
((b, mx),
((Thm.make_def_binding internals b, []), fold_rev lambda (params @ xs)
(list_comb (rec_const, params @ make_bool_args' bs i @
make_args argTs (xs ~~ Ts)))))
end) (cnames_syn ~~ cs)
else [];
val (consts_defs, lthy'') = lthy'
|> fold_map Local_Theory.define specs;
val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
val (_, ctxt'') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
val mono = prove_mono quiet_mode skip_mono predT fp_fun monos ctxt'';
val (_, lthy''') = lthy''
|> Local_Theory.note
((if internals
then Binding.qualify true rec_name (Binding.name "mono")
else Binding.empty, []),
Proof_Context.export ctxt'' lthy'' [mono]);
in
(lthy''', Proof_Context.transfer (Proof_Context.theory_of lthy''') ctxt'',
rec_binding, 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 spec_name 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 =
if forall (equal "") intr_names then []
else [Attrib.case_names intr_names];
val induct =
if coind then
(raw_induct,
[Attrib.case_names [rec_name],
Attrib.case_conclusion (rec_name, intr_names),
Attrib.consumes (1 - Thm.nprems_of raw_induct),
Attrib.internal (K (Induct.coinduct_pred (hd cnames)))])
else if no_ind orelse length cnames > 1 then
(raw_induct, ind_case_names @ [Attrib.consumes (~ (Thm.nprems_of raw_induct))])
else
(raw_induct RSN (2, rev_mp),
ind_case_names @ [Attrib.consumes (~ (Thm.nprems_of raw_induct))]);
val (intrs', lthy1) =
lthy |>
Spec_Rules.add spec_name
(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], @{attributes [Pure.intro?]})]) 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"),
((if forall (equal "") cases then [] else [Attrib.case_names cases]) @
[Attrib.consumes (1 - Thm.nprems_of elim), Attrib.constraints k,
Attrib.internal (K (Induct.cases_pred name))] @ @{attributes [Pure.elim?]})),
[elim]) #>
apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
Local_Theory.note
((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")), #2 induct),
[rulify lthy1 (#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 equation_add_permissive)]), [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],
ind_case_names @
[Attrib.consumes (1 - Thm.nprems_of th),
Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
end;
in (intrs', elims', eqs', induct', inducts, lthy4) end;
type flags =
{quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
no_elim: bool, no_ind: bool, skip_mono: bool};
type add_ind_def =
flags ->
term list -> (Attrib.binding * term) list -> thm list ->
term list -> (binding * mixfix) list ->
local_theory -> result * local_theory;
fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_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 spec_name = Binding.conglomerate (map #1 cnames_syn);
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_binding, mono, fp_def, rec_preds_defs, rec_const, preds,
argTs, bs, xs) = mk_ind_def quiet_mode skip_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
prove_coindrule quiet_mode preds cs argTs bs xs params intr_ts mono fp_def
rec_preds_defs lthy2 lthy1
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 lthy1 th, ns, i)) elims;
val intrs' = map (rulify lthy1) intrs;
val (intrs'', elims'', eqs', induct, inducts, lthy3) =
declare_rules rec_binding coind no_ind
spec_name cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
val result =
{preds = preds,
intrs = intrs'',
elims = elims'',
raw_induct = rulify lthy3 raw_induct,
induct = induct,
inducts = inducts,
eqs = eqs'};
val lthy4 = lthy3
|> Local_Theory.declaration {syntax = false, pervasive = false} (fn phi =>
let val result' = transform_result phi result;
in put_inductives ({names = cnames, coind = coind}, result') end);
in (result, lthy4) end;
(* external interfaces *)
fun gen_add_inductive mk_def
flags cnames_syn pnames spec monos lthy =
let
(* 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 (K []) 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 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 =
fold (Logic.all o Free) (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_cmd mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
let
val ((vars, intrs), _) = lthy
|> Proof_Context.set_mode Proof_Context.mode_abbrev
|> Specification.read_multi_specs (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};
in
lthy
|> gen_add_inductive mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
end;
val add_inductive = gen_add_inductive add_ind_def;
val add_inductive_cmd = gen_add_inductive_cmd add_ind_def;
(* 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 (Thm.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 (Thm.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 =
Thm.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 thy elim arity intros =
let
val _ :: cases = Thm.prems_of elim;
val used = map (fst o fst) (Term.add_vars (Thm.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 Thm.prop_of) (cases ~~ intros)
end;
(** outer syntax **)
fun gen_ind_decl mk_def coind =
Parse.vars -- Parse.for_fixes --
Scan.optional Parse_Spec.where_multi_specs [] --
Scan.optional (\<^keyword>\<open>monos\<close> |-- Parse.!!! Parse.thms1) []
>> (fn (((preds, params), specs), monos) =>
(snd o gen_add_inductive_cmd mk_def true coind preds params specs monos));
val ind_decl = gen_ind_decl add_ind_def;
val _ =
Outer_Syntax.local_theory \<^command_keyword>\<open>inductive\<close> "define inductive predicates"
(ind_decl false);
val _ =
Outer_Syntax.local_theory \<^command_keyword>\<open>coinductive\<close> "define coinductive predicates"
(ind_decl true);
val _ =
Outer_Syntax.local_theory \<^command_keyword>\<open>inductive_cases\<close>
"create simplified instances of elimination rules"
(Parse.and_list1 Parse_Spec.simple_specs >> (snd oo inductive_cases_cmd));
val _ =
Outer_Syntax.local_theory \<^command_keyword>\<open>inductive_simps\<close>
"create simplification rules for inductive predicates"
(Parse.and_list1 Parse_Spec.simple_specs >> (snd oo inductive_simps_cmd));
val _ =
Outer_Syntax.command \<^command_keyword>\<open>print_inductives\<close>
"print (co)inductive definitions and monotonicity rules"
(Parse.opt_bang >> (fn b => Toplevel.keep (print_inductives b o Toplevel.context_of)));
end;
¤ Diese beiden folgenden Angebotsgruppen bietet das Unternehmen0.63Angebot
Wie Sie bei der Firma Beratungs- und Dienstleistungen beauftragen können
¤
|
Lebenszyklus
Die hierunter aufgelisteten Ziele sind für diese Firma wichtig
Ziele
Entwicklung einer Software für die statische Quellcodeanalyse
|
