--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/function_package/termination.ML Tue Dec 16 08:46:07 2008 +0100
@@ -0,0 +1,324 @@
+(* Title: HOL/Tools/function_package/termination_data.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 : tactic
+end
+
+
+
+structure Termination : TERMINATION =
+struct
+
+open FundefLib
+
+val term2_ord = prod_ord Term.fast_term_ord Term.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 Term.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 "op 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 "op 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 st =
+ let
+ val thy = theory_of_thm st
+ 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 "{}"}, fastype_of rel)
+ else
+ foldr1 (HOLogic.mk_binop @{const_name "op 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' (CLASET' blast_tac))) (* 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