(* Title: HOL/Tools/coinduction.ML
Author: Johannes Hölzl, TU Muenchen
Author: Dmitriy Traytel, TU Muenchen
Copyright 2013
Coinduction method that avoids some boilerplate compared to coinduct.
*)
signature COINDUCTION =
sig
val coinduction_tac: Proof.context -> term list -> thm option -> thm list -> cases_tactic
end;
structure Coinduction : COINDUCTION =
struct
fun filter_in_out _ [] = ([], [])
| filter_in_out P (x :: xs) =
let
val (ins, outs) = filter_in_out P xs;
in
if P x then (x :: ins, outs) else (ins, x :: outs)
end;
fun ALLGOALS_SKIP skip tac st =
let fun doall n = if n = skip then all_tac else tac n THEN doall (n - 1)
in doall (Thm.nprems_of st) st end;
fun THEN_ALL_NEW_SKIP skip tac1 tac2 i st =
st |> (tac1 i THEN (fn st' =>
Seq.INTERVAL tac2 (i + skip) (i + Thm.nprems_of st' - Thm.nprems_of st) st'));
fun DELETE_PREMS_AFTER skip tac i st =
let
val n = nth (Thm.prems_of st) (i - 1) |> Logic.strip_assums_hyp |> length;
in
(THEN_ALL_NEW_SKIP skip tac (REPEAT_DETERM_N n o eresolve0_tac [thin_rl])) i st
end;
fun coinduction_tac ctxt raw_vars opt_raw_thm prems = HEADGOAL (SUBGOAL_CASES (fn (goal, _, _) =>
let
val lhs_of_eq = HOLogic.dest_Trueprop #> HOLogic.dest_eq #> fst;
fun find_coinduct t =
Induct.find_coinductP ctxt t @
(try (Induct.find_coinductT ctxt o fastype_of o lhs_of_eq) t |> the_default [])
val raw_thm =
(case opt_raw_thm of
SOME raw_thm => raw_thm
| NONE => goal |> Logic.strip_assums_concl |> find_coinduct |> hd);
val skip = Integer.max 1 (Rule_Cases.get_consumes raw_thm) - 1
val cases = Rule_Cases.get raw_thm |> fst
in
HEADGOAL (
Object_Logic.rulify_tac ctxt THEN'
Method.insert_tac prems THEN'
Object_Logic.atomize_prems_tac ctxt THEN'
DELETE_PREMS_AFTER skip (Subgoal.FOCUS (fn {concl, context = ctxt, params, prems, ...} =>
let
val vars = raw_vars @ map (Thm.term_of o snd) params;
val names_ctxt = ctxt
|> fold Variable.declare_names vars
|> fold Variable.declare_thm (raw_thm :: prems);
val thm_concl = Thm.cprop_of raw_thm |> strip_imp_concl;
val (rhoTs, rhots) = Thm.match (thm_concl, concl)
|>> map (apply2 Thm.typ_of)
||> map (apply2 Thm.term_of);
val xs = hd (Thm.prems_of raw_thm) |> HOLogic.dest_Trueprop |> strip_comb |> snd
|> map (subst_atomic_types rhoTs);
val raw_eqs = map (fn x => (x, AList.lookup op aconv rhots x |> the)) xs;
val ((names, ctxt), Ts) = map_split (apfst fst o dest_Var o fst) raw_eqs
|>> (fn names => Variable.variant_fixes names names_ctxt) ;
val eqs =
@{map 3} (fn name => fn T => fn (_, rhs) =>
HOLogic.mk_eq (Free (name, T), rhs))
names Ts raw_eqs;
val phi = eqs @ map (HOLogic.dest_Trueprop o Thm.prop_of) prems
|> try (Library.foldr1 HOLogic.mk_conj)
|> the_default @{term True}
|> Ctr_Sugar_Util.list_exists_free vars
|> Term.map_abs_vars (Variable.revert_fixed ctxt)
|> fold_rev Term.absfree (names ~~ Ts)
|> Thm.cterm_of ctxt;
val thm = Ctr_Sugar_Util.cterm_instantiate_pos [SOME phi] raw_thm;
val e = length eqs;
val p = length prems;
in
HEADGOAL (EVERY' [resolve_tac ctxt [thm],
EVERY' (map (fn var =>
resolve_tac ctxt
[Ctr_Sugar_Util.cterm_instantiate_pos
[NONE, SOME (Thm.cterm_of ctxt var)] exI]) vars),
if p = 0 then Ctr_Sugar_Util.CONJ_WRAP' (K (resolve_tac ctxt [refl])) eqs
else
REPEAT_DETERM_N e o (resolve_tac ctxt [conjI] THEN' resolve_tac ctxt [refl]) THEN'
Ctr_Sugar_Util.CONJ_WRAP' (resolve_tac ctxt o single) prems,
K (ALLGOALS_SKIP skip
(REPEAT_DETERM_N (length vars) o (eresolve_tac ctxt [exE] THEN' rotate_tac ~1) THEN'
DELETE_PREMS_AFTER 0 (Subgoal.FOCUS (fn {prems, params, context = ctxt, ...} =>
(case prems of
[] => all_tac
| inv :: case_prems =>
let
val (init, last) = funpow_yield (p + e - 1) (HOLogic.conj_elim ctxt) inv;
val inv_thms = init @ [last];
val eqs = take e inv_thms;
fun is_local_var t =
member (fn (t, (_, t')) => t aconv (Thm.term_of t')) params t;
val (eqs, assms') =
filter_in_out (is_local_var o lhs_of_eq o Thm.prop_of) eqs;
val assms = assms' @ drop e inv_thms
in
HEADGOAL (Method.insert_tac (assms @ case_prems)) THEN
Ctr_Sugar_General_Tactics.unfold_thms_tac ctxt eqs
end)) ctxt)))])
end) ctxt) THEN'
K (prune_params_tac ctxt))
#> Seq.maps (fn st =>
CASES (Rule_Cases.make_common ctxt (Thm.prop_of st) cases) all_tac st)
end));
local
val ruleN = "rule"
val arbitraryN = "arbitrary"
fun single_rule [rule] = rule
| single_rule _ = error "Single rule expected";
fun named_rule k arg get =
Scan.lift (Args.$$$ k -- Args.colon) |-- Scan.repeat arg :|--
(fn names => Scan.peek (fn context => Scan.succeed (names |> map (fn name =>
(case get (Context.proof_of context) name of SOME x => x
| NONE => error ("No rule for " ^ k ^ " " ^ quote name))))));
fun rule get_type get_pred =
named_rule Induct.typeN (Args.type_name {proper = false, strict = false}) get_type ||
named_rule Induct.predN (Args.const {proper = false, strict = false}) get_pred ||
named_rule Induct.setN (Args.const {proper = false, strict = false}) get_pred ||
Scan.lift (Args.$$$ ruleN -- Args.colon) |-- Attrib.thms;
val coinduct_rule =
rule Induct.lookup_coinductT Induct.lookup_coinductP >> single_rule;
fun unless_more_args scan = Scan.unless (Scan.lift
((Args.$$$ arbitraryN || Args.$$$ Induct.typeN ||
Args.$$$ Induct.predN || Args.$$$ Induct.setN || Args.$$$ ruleN) -- Args.colon)) scan;
val arbitrary = Scan.optional (Scan.lift (Args.$$$ arbitraryN -- Args.colon) |--
Scan.repeat1 (unless_more_args Args.term)) [];
in
val _ =
Theory.setup
(Method.setup @{binding coinduction}
(arbitrary -- Scan.option coinduct_rule >>
(fn (arbitrary, opt_rule) => fn ctxt => fn facts =>
Seq.DETERM (coinduction_tac ctxt arbitrary opt_rule facts)))
"coinduction on types or predicates/sets");
end;
end;