(* Title: HOL/Tools/function_package/termination.ML
Author: Alexander Krauss, TU Muenchen
Context data for termination proofs
*)
signature TERMINATION =
sig
type data
datatype cell = Less of thm | LessEq of (thm * thm) | None of (thm * thm) | False of thm
val mk_sumcases : data -> typ -> term list -> term
val note_measure : int -> term -> data -> data
val note_chain : term -> term -> thm option -> data -> data
val note_descent : term -> term -> term -> cell -> data -> data
val get_num_points : data -> int
val get_types : data -> int -> typ
val get_measures : data -> int -> term list
(* read from cache *)
val get_chain : data -> term -> term -> thm option option
val get_descent : data -> term -> term -> term -> cell option
(* writes *)
val derive_descent : theory -> tactic -> term -> term -> term -> data -> data
val derive_descents : theory -> tactic -> term -> data -> data
val dest_call : data -> term -> ((string * typ) list * int * term * int * term * term)
val CALLS : (term list * int -> tactic) -> int -> tactic
(* Termination tactics. Sequential composition via continuations. (2nd argument is the error continuation) *)
type ttac = (data -> int -> tactic) -> (data -> int -> tactic) -> data -> int -> tactic
val TERMINATION : Proof.context -> (data -> int -> tactic) -> int -> tactic
val REPEAT : ttac -> ttac
val wf_union_tac : Proof.context -> tactic
end
structure Termination : TERMINATION =
struct
open FundefLib
val term2_ord = prod_ord TermOrd.fast_term_ord TermOrd.fast_term_ord
structure Term2tab = TableFun(type key = term * term val ord = term2_ord);
structure Term3tab = TableFun(type key = term * (term * term) val ord = prod_ord TermOrd.fast_term_ord term2_ord);
(** Analyzing binary trees **)
(* Skeleton of a tree structure *)
datatype skel =
SLeaf of int (* index *)
| SBranch of (skel * skel)
(* abstract make and dest functions *)
fun mk_tree leaf branch =
let fun mk (SLeaf i) = leaf i
| mk (SBranch (s, t)) = branch (mk s, mk t)
in mk end
fun dest_tree split =
let fun dest (SLeaf i) x = [(i, x)]
| dest (SBranch (s, t)) x =
let val (l, r) = split x
in dest s l @ dest t r end
in dest end
(* concrete versions for sum types *)
fun is_inj (Const ("Sum_Type.Inl", _) $ _) = true
| is_inj (Const ("Sum_Type.Inr", _) $ _) = true
| is_inj _ = false
fun dest_inl (Const ("Sum_Type.Inl", _) $ t) = SOME t
| dest_inl _ = NONE
fun dest_inr (Const ("Sum_Type.Inr", _) $ t) = SOME t
| dest_inr _ = NONE
fun mk_skel ps =
let
fun skel i ps =
if forall is_inj ps andalso not (null ps)
then let
val (j, s) = skel i (map_filter dest_inl ps)
val (k, t) = skel j (map_filter dest_inr ps)
in (k, SBranch (s, t)) end
else (i + 1, SLeaf i)
in
snd (skel 0 ps)
end
(* compute list of types for nodes *)
fun node_types sk T = dest_tree (fn Type ("+", [LT, RT]) => (LT, RT)) sk T |> map snd
(* find index and raw term *)
fun dest_inj (SLeaf i) trm = (i, trm)
| dest_inj (SBranch (s, t)) trm =
case dest_inl trm of
SOME trm' => dest_inj s trm'
| _ => dest_inj t (the (dest_inr trm))
(** Matrix cell datatype **)
datatype cell = Less of thm | LessEq of (thm * thm) | None of (thm * thm) | False of thm;
type data =
skel (* structure of the sum type encoding "program points" *)
* (int -> typ) (* types of program points *)
* (term list Inttab.table) (* measures for program points *)
* (thm option Term2tab.table) (* which calls form chains? *)
* (cell Term3tab.table) (* local descents *)
fun map_measures f (p, T, M, C, D) = (p, T, f M, C, D)
fun map_chains f (p, T, M, C, D) = (p, T, M, f C, D)
fun map_descent f (p, T, M, C, D) = (p, T, M, C, f D)
fun note_measure p m = map_measures (Inttab.insert_list (op aconv) (p, m))
fun note_chain c1 c2 res = map_chains (Term2tab.update ((c1, c2), res))
fun note_descent c m1 m2 res = map_descent (Term3tab.update ((c,(m1, m2)), res))
(* Build case expression *)
fun mk_sumcases (sk, _, _, _, _) T fs =
mk_tree (fn i => (nth fs i, domain_type (fastype_of (nth fs i))))
(fn ((f, fT), (g, gT)) => (SumTree.mk_sumcase fT gT T f g, SumTree.mk_sumT fT gT))
sk
|> fst
fun mk_sum_skel rel =
let
val cs = FundefLib.dest_binop_list @{const_name Un} rel
fun collect_pats (Const ("Collect", _) $ Abs (_, _, c)) =
let
val (Const ("op &", _) $ (Const ("op =", _) $ _ $ (Const ("Pair", _) $ r $ l)) $ Gam)
= Term.strip_qnt_body "Ex" c
in cons r o cons l end
in
mk_skel (fold collect_pats cs [])
end
fun create ctxt T rel =
let
val sk = mk_sum_skel rel
val Ts = node_types sk T
val M = Inttab.make (map_index (apsnd (MeasureFunctions.get_measure_functions ctxt)) Ts)
in
(sk, nth Ts, M, Term2tab.empty, Term3tab.empty)
end
fun get_num_points (sk, _, _, _, _) =
let
fun num (SLeaf i) = i + 1
| num (SBranch (s, t)) = num t
in num sk end
fun get_types (_, T, _, _, _) = T
fun get_measures (_, _, M, _, _) = Inttab.lookup_list M
fun get_chain (_, _, _, C, _) c1 c2 =
Term2tab.lookup C (c1, c2)
fun get_descent (_, _, _, _, D) c m1 m2 =
Term3tab.lookup D (c, (m1, m2))
fun dest_call D (Const ("Collect", _) $ Abs (_, _, c)) =
let
val n = get_num_points D
val (sk, _, _, _, _) = D
val vs = Term.strip_qnt_vars "Ex" c
(* FIXME: throw error "dest_call" for malformed terms *)
val (Const ("op &", _) $ (Const ("op =", _) $ _ $ (Const ("Pair", _) $ r $ l)) $ Gam)
= Term.strip_qnt_body "Ex" c
val (p, l') = dest_inj sk l
val (q, r') = dest_inj sk r
in
(vs, p, l', q, r', Gam)
end
| dest_call D t = error "dest_call"
fun derive_desc_aux thy tac c (vs, p, l', q, r', Gam) m1 m2 D =
case get_descent D c m1 m2 of
SOME _ => D
| NONE => let
fun cgoal rel =
Term.list_all (vs,
Logic.mk_implies (HOLogic.mk_Trueprop Gam,
HOLogic.mk_Trueprop (Const (rel, @{typ "nat => nat => bool"})
$ (m2 $ r') $ (m1 $ l'))))
|> cterm_of thy
in
note_descent c m1 m2
(case try_proof (cgoal @{const_name HOL.less}) tac of
Solved thm => Less thm
| Stuck thm =>
(case try_proof (cgoal @{const_name HOL.less_eq}) tac of
Solved thm2 => LessEq (thm2, thm)
| Stuck thm2 =>
if prems_of thm2 = [HOLogic.Trueprop $ HOLogic.false_const]
then False thm2 else None (thm2, thm)
| _ => raise Match) (* FIXME *)
| _ => raise Match) D
end
fun derive_descent thy tac c m1 m2 D =
derive_desc_aux thy tac c (dest_call D c) m1 m2 D
(* all descents in one go *)
fun derive_descents thy tac c D =
let val cdesc as (vs, p, l', q, r', Gam) = dest_call D c
in fold_product (derive_desc_aux thy tac c cdesc)
(get_measures D p) (get_measures D q) D
end
fun CALLS tac i st =
if Thm.no_prems st then all_tac st
else case Thm.term_of (Thm.cprem_of st i) of
(_ $ (_ $ rel)) => tac (FundefLib.dest_binop_list @{const_name Un} rel, i) st
|_ => no_tac st
type ttac = (data -> int -> tactic) -> (data -> int -> tactic) -> data -> int -> tactic
fun TERMINATION ctxt tac =
SUBGOAL (fn (_ $ (Const (@{const_name "wf"}, wfT) $ rel), i) =>
let
val (T, _) = HOLogic.dest_prodT (HOLogic.dest_setT (domain_type wfT))
in
tac (create ctxt T rel) i
end)
(* A tactic to convert open to closed termination goals *)
local
fun dest_term (t : term) = (* FIXME, cf. Lexicographic order *)
let
val (vars, prop) = FundefLib.dest_all_all t
val (prems, concl) = Logic.strip_horn prop
val (lhs, rhs) = concl
|> HOLogic.dest_Trueprop
|> HOLogic.dest_mem |> fst
|> HOLogic.dest_prod
in
(vars, prems, lhs, rhs)
end
fun mk_pair_compr (T, qs, l, r, conds) =
let
val pT = HOLogic.mk_prodT (T, T)
val n = length qs
val peq = HOLogic.eq_const pT $ Bound n $ (HOLogic.pair_const T T $ l $ r)
val conds' = if null conds then [HOLogic.true_const] else conds
in
HOLogic.Collect_const pT $
Abs ("uu_", pT,
(foldr1 HOLogic.mk_conj (peq :: conds')
|> fold_rev (fn v => fn t => HOLogic.exists_const (fastype_of v) $ lambda v t) qs))
end
in
fun wf_union_tac ctxt st =
let
val thy = ProofContext.theory_of ctxt
val cert = cterm_of (theory_of_thm st)
val ((trueprop $ (wf $ rel)) :: ineqs) = prems_of st
fun mk_compr ineq =
let
val (vars, prems, lhs, rhs) = dest_term ineq
in
mk_pair_compr (fastype_of lhs, vars, lhs, rhs, map (ObjectLogic.atomize_term thy) prems)
end
val relation =
if null ineqs then
Const (@{const_name Set.empty}, fastype_of rel)
else
foldr1 (HOLogic.mk_binop @{const_name Un}) (map mk_compr ineqs)
fun solve_membership_tac i =
(EVERY' (replicate (i - 2) (rtac @{thm UnI2})) (* pick the right component of the union *)
THEN' (fn j => TRY (rtac @{thm UnI1} j))
THEN' (rtac @{thm CollectI}) (* unfold comprehension *)
THEN' (fn i => REPEAT (rtac @{thm exI} i)) (* Turn existentials into schematic Vars *)
THEN' ((rtac @{thm refl}) (* unification instantiates all Vars *)
ORELSE' ((rtac @{thm conjI})
THEN' (rtac @{thm refl})
THEN' (blast_tac (local_claset_of ctxt)))) (* Solve rest of context... not very elegant *)
) i
in
((PRIMITIVE (Drule.cterm_instantiate [(cert rel, cert relation)])
THEN ALLGOALS (fn i => if i = 1 then all_tac else solve_membership_tac i))) st
end
end
(* continuation passing repeat combinator *)
fun REPEAT ttac cont err_cont =
ttac (fn D => fn i => (REPEAT ttac cont cont D i)) err_cont
end