(* Title: HOL/Tools/function_package/lexicographic_order.ML
ID: $Id$
Author: Lukas Bulwahn, TU Muenchen
Method for termination proofs with lexicographic orderings.
*)
signature LEXICOGRAPHIC_ORDER =
sig
val lexicographic_order : thm list -> Proof.context -> Method.method
(* exported for use by size-change termination prototype.
FIXME: provide a common interface later *)
val mk_base_funs : theory -> typ -> term list
(* exported for debugging *)
val setup: theory -> theory
end
structure LexicographicOrder : LEXICOGRAPHIC_ORDER =
struct
(** General stuff **)
fun mk_measures domT mfuns =
let val list = HOLogic.mk_list (domT --> HOLogic.natT) mfuns
in
Const (@{const_name "List.measures"}, fastype_of list --> (HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT)))) $ list
end
fun del_index n [] = []
| del_index n (x :: xs) =
if n > 0 then x :: del_index (n - 1) xs else xs
fun transpose ([]::_) = []
| transpose xss = map hd xss :: transpose (map tl xss)
(** Matrix cell datatype **)
datatype cell = Less of thm | LessEq of thm | None of thm | False of thm;
fun is_Less (Less _) = true
| is_Less _ = false
fun is_LessEq (LessEq _) = true
| is_LessEq _ = false
fun thm_of_cell (Less thm) = thm
| thm_of_cell (LessEq thm) = thm
| thm_of_cell (False thm) = thm
| thm_of_cell (None thm) = thm
fun pr_cell (Less _ ) = " < "
| pr_cell (LessEq _) = " <= "
| pr_cell (None _) = " N "
| pr_cell (False _) = " F "
(** Generating Measure Functions **)
fun mk_comp g f =
let
val fT = fastype_of f
val gT as (Type ("fun", [xT, _])) = fastype_of g
val comp = Abs ("f", fT, Abs ("g", gT, Abs ("x", xT, Bound 2 $ (Bound 1 $ Bound 0))))
in
Envir.beta_norm (comp $ f $ g)
end
fun mk_base_funs thy (T as Type("*", [fT, sT])) = (* products *)
map (mk_comp (Const ("fst", T --> fT))) (mk_base_funs thy fT)
@ map (mk_comp (Const ("snd", T --> sT))) (mk_base_funs thy sT)
| mk_base_funs thy T = (* default: size function, if available *)
if Sorts.of_sort (Sign.classes_of thy) (T, [HOLogic.class_size])
then [HOLogic.size_const T]
else []
fun mk_sum_case f1 f2 =
let
val Type ("fun", [fT, Q]) = fastype_of f1
val Type ("fun", [sT, _]) = fastype_of f2
in
Const (@{const_name "Sum_Type.sum_case"}, (fT --> Q) --> (sT --> Q) --> Type("+", [fT, sT]) --> Q) $ f1 $ f2
end
fun constant_0 T = Abs ("x", T, HOLogic.zero)
fun constant_1 T = Abs ("x", T, HOLogic.Suc_zero)
fun mk_funorder_funs (Type ("+", [fT, sT])) =
map (fn m => mk_sum_case m (constant_0 sT)) (mk_funorder_funs fT)
@ map (fn m => mk_sum_case (constant_0 fT) m) (mk_funorder_funs sT)
| mk_funorder_funs T = [ constant_1 T ]
fun mk_ext_base_funs thy (Type("+", [fT, sT])) =
product (mk_ext_base_funs thy fT) (mk_ext_base_funs thy sT)
|> map (uncurry mk_sum_case)
| mk_ext_base_funs thy T = mk_base_funs thy T
fun mk_all_measure_funs thy (T as Type ("+", _)) =
mk_ext_base_funs thy T @ mk_funorder_funs T
| mk_all_measure_funs thy T = mk_base_funs thy T
(** Proof attempts to build the matrix **)
fun dest_term (t : term) =
let
val (vars, prop) = FundefLib.dest_all_all t
val prems = Logic.strip_imp_prems prop
val (lhs, rhs) = Logic.strip_imp_concl prop
|> HOLogic.dest_Trueprop
|> HOLogic.dest_mem |> fst
|> HOLogic.dest_prod
in
(vars, prems, lhs, rhs)
end
fun mk_goal (vars, prems, lhs, rhs) rel =
let
val concl = HOLogic.mk_binrel rel (lhs, rhs) |> HOLogic.mk_Trueprop
in
Logic.list_implies (prems, concl)
|> fold_rev FundefLib.mk_forall vars
end
fun prove (thy: theory) solve_tac (t: term) =
cterm_of thy t |> Goal.init
|> SINGLE solve_tac |> the
fun mk_cell (thy : theory) solve_tac (vars, prems, lhs, rhs) mfun =
let
val goals = mk_goal (vars, prems, mfun $ lhs, mfun $ rhs)
val less_thm = goals "Orderings.ord_class.less" |> prove thy solve_tac
in
if Thm.no_prems less_thm then
Less (Goal.finish less_thm)
else
let
val lesseq_thm = goals "Orderings.ord_class.less_eq" |> prove thy solve_tac
in
if Thm.no_prems lesseq_thm then
LessEq (Goal.finish lesseq_thm)
else
if prems_of lesseq_thm = [HOLogic.Trueprop $ HOLogic.false_const] then False lesseq_thm
else None lesseq_thm
end
end
(** Search algorithms **)
fun check_col ls = forall (fn c => is_Less c orelse is_LessEq c) ls andalso not (forall (is_LessEq) ls)
fun transform_table table col = table |> filter_out (fn x => is_Less (nth x col)) |> map (del_index col)
fun transform_order col order = map (fn x => if x >= col then x + 1 else x) order
(* simple depth-first search algorithm for the table *)
fun search_table table =
case table of
[] => SOME []
| _ =>
let
val col = find_index (check_col) (transpose table)
in case col of
~1 => NONE
| _ =>
let
val order_opt = (table, col) |-> transform_table |> search_table
in case order_opt of
NONE => NONE
| SOME order =>SOME (col :: transform_order col order)
end
end
(* find all positions of elements in a list *)
fun find_index_list P =
let fun find _ [] = []
| find n (x :: xs) = if P x then n :: find (n + 1) xs else find (n + 1) xs
in find 0 end
(* simple breadth-first search algorithm for the table *)
fun bfs_search_table nodes =
case nodes of
[] => sys_error "INTERNAL ERROR IN lexicographic order termination tactic - fun search_table (breadth search finished)"
| (node::rnodes) => let
val (order, table) = node
in
case table of
[] => SOME (foldr (fn (c, order) => c :: transform_order c order) [] (rev order))
| _ => let
val cols = find_index_list (check_col) (transpose table)
in
case cols of
[] => NONE
| _ => let
val newtables = map (transform_table table) cols
val neworders = map (fn c => c :: order) cols
val newnodes = neworders ~~ newtables
in
bfs_search_table (rnodes @ newnodes)
end
end
end
fun nsearch_table table = bfs_search_table [([], table)]
(** Proof Reconstruction **)
(* prove row :: cell list -> tactic *)
fun prove_row (Less less_thm :: _) =
(rtac @{thm "measures_less"} 1)
THEN PRIMITIVE (flip implies_elim less_thm)
| prove_row (LessEq lesseq_thm :: tail) =
(rtac @{thm "measures_lesseq"} 1)
THEN PRIMITIVE (flip implies_elim lesseq_thm)
THEN prove_row tail
| prove_row _ = sys_error "lexicographic_order"
(** Error reporting **)
fun pr_table table = writeln (cat_lines (map (fn r => concat (map pr_cell r)) table))
fun pr_unprovable_subgoals table =
filter_out (fn x => is_Less x orelse is_LessEq x) (flat table)
|> map ((fn th => Pretty.string_of (Pretty.chunks (Display.pretty_goals (Thm.nprems_of th) th))) o thm_of_cell)
fun no_order_msg table thy tl measure_funs =
let
fun pr_fun t i = string_of_int i ^ ") " ^ string_of_cterm (cterm_of thy t)
fun pr_goal t i =
let
val (_, _, lhs, rhs) = dest_term t
val prterm = string_of_cterm o (cterm_of thy)
in (* also show prems? *)
i ^ ") " ^ prterm rhs ^ " ~> " ^ prterm lhs
end
val gc = map (fn i => chr (i + 96)) (1 upto length table)
val mc = 1 upto length measure_funs
val tstr = " " ^ concat (map (enclose " " " " o string_of_int) mc)
:: map2 (fn r => fn i => i ^ ": " ^ concat (map pr_cell r)) table gc
val gstr = "Calls:" :: map2 pr_goal tl gc
val mstr = "Measures:" :: map2 pr_fun measure_funs mc
val ustr = "Unfinished subgoals:" :: pr_unprovable_subgoals table
in
cat_lines ("Could not find lexicographic termination order:" :: tstr @ gstr @ mstr @ ustr)
end
(** The Main Function **)
fun lexicographic_order_tac ctxt solve_tac (st: thm) =
let
val thy = theory_of_thm st
val ((trueprop $ (wf $ rel)) :: tl) = prems_of st
val (domT, _) = HOLogic.dest_prodT (HOLogic.dest_setT (fastype_of rel))
val measure_funs = mk_all_measure_funs thy domT (* 1: generate measures *)
(* 2: create table *)
val table = map (fn t => map (mk_cell thy solve_tac (dest_term t)) measure_funs) tl
val order = the (search_table table) (* 3: search table *)
handle Option => error (no_order_msg table thy tl measure_funs)
val clean_table = map (fn x => map (nth x) order) table
val relation = mk_measures domT (map (nth measure_funs) order)
val _ = writeln ("Found termination order: " ^ quote (ProofContext.string_of_term ctxt relation))
in (* 4: proof reconstruction *)
st |> (PRIMITIVE (cterm_instantiate [(cterm_of thy rel, cterm_of thy relation)])
THEN rtac @{thm "wf_measures"} 1
THEN EVERY (map prove_row clean_table))
end
fun lexicographic_order thms ctxt = Method.SIMPLE_METHOD (FundefCommon.apply_termination_rule ctxt 1
THEN lexicographic_order_tac ctxt (auto_tac (local_clasimpset_of ctxt)))
val setup = Method.add_methods [("lexicographic_order", Method.bang_sectioned_args clasimp_modifiers lexicographic_order,
"termination prover for lexicographic orderings")]
end