(* Title: HOL/Tools/function_package/lexicographic_order.ML
Author: Lukas Bulwahn, TU Muenchen
Method for termination proofs with lexicographic orderings.
*)
signature LEXICOGRAPHIC_ORDER =
sig
val lex_order_tac : Proof.context -> tactic -> tactic
val lexicographic_order_tac : Proof.context -> tactic
val lexicographic_order : Proof.context -> Proof.method
val setup: theory -> theory
end
structure LexicographicOrder : LEXICOGRAPHIC_ORDER =
struct
open FundefLib
(** General stuff **)
fun mk_measures domT mfuns =
let
val relT = HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT))
val mlexT = (domT --> HOLogic.natT) --> relT --> relT
fun mk_ms [] = Const (@{const_name Set.empty}, relT)
| mk_ms (f::fs) =
Const (@{const_name "mlex_prod"}, mlexT) $ f $ mk_ms fs
in
mk_ms mfuns
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 * thm) | None of (thm * thm) | False of thm;
fun is_Less (Less _) = true
| is_Less _ = false
fun is_LessEq (LessEq _) = true
| is_LessEq _ = false
fun pr_cell (Less _ ) = " < "
| pr_cell (LessEq _) = " <="
| pr_cell (None _) = " ? "
| pr_cell (False _) = " F "
(** Proof attempts to build the matrix **)
fun dest_term (t : term) =
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_goal (vars, prems, lhs, rhs) rel =
let
val concl = HOLogic.mk_binrel rel (lhs, rhs) |> HOLogic.mk_Trueprop
in
fold_rev Logic.all vars (Logic.list_implies (prems, concl))
end
fun prove thy solve_tac t =
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 = cterm_of thy o mk_goal (vars, prems, mfun $ lhs, mfun $ rhs)
in
case try_proof (goals @{const_name HOL.less}) solve_tac of
Solved thm => Less thm
| Stuck thm =>
(case try_proof (goals @{const_name HOL.less_eq}) solve_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
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
(** Proof Reconstruction **)
(* prove row :: cell list -> tactic *)
fun prove_row (Less less_thm :: _) =
(rtac @{thm "mlex_less"} 1)
THEN PRIMITIVE (Thm.elim_implies less_thm)
| prove_row (LessEq (lesseq_thm, _) :: tail) =
(rtac @{thm "mlex_leq"} 1)
THEN PRIMITIVE (Thm.elim_implies 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_goals ctxt st =
Display.pretty_goals_aux (Syntax.pp ctxt) Markup.none (true, false) (Thm.nprems_of st) st
|> Pretty.chunks
|> Pretty.string_of
fun row_index i = chr (i + 97)
fun col_index j = string_of_int (j + 1)
fun pr_unprovable_cell _ ((i,j), Less _) = ""
| pr_unprovable_cell ctxt ((i,j), LessEq (_, st)) =
"(" ^ row_index i ^ ", " ^ col_index j ^ ", <):\n" ^ pr_goals ctxt st
| pr_unprovable_cell ctxt ((i,j), None (st_leq, st_less)) =
"(" ^ row_index i ^ ", " ^ col_index j ^ ", <):\n" ^ pr_goals ctxt st_less
^ "\n(" ^ row_index i ^ ", " ^ col_index j ^ ", <=):\n" ^ pr_goals ctxt st_leq
| pr_unprovable_cell ctxt ((i,j), False st) =
"(" ^ row_index i ^ ", " ^ col_index j ^ ", <):\n" ^ pr_goals ctxt st
fun pr_unprovable_subgoals ctxt table =
table
|> map_index (fn (i,cs) => map_index (fn (j,x) => ((i,j), x)) cs)
|> flat
|> map (pr_unprovable_cell ctxt)
fun no_order_msg ctxt table tl measure_funs =
let
val prterm = Syntax.string_of_term ctxt
fun pr_fun t i = string_of_int i ^ ") " ^ prterm t
fun pr_goal t i =
let
val (_, _, lhs, rhs) = dest_term t
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 = "Result matrix:" :: (" " ^ 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 (prefix " " oo pr_goal) tl gc
val mstr = "Measures:" :: map2 (prefix " " oo pr_fun) measure_funs mc
val ustr = "Unfinished subgoals:" :: pr_unprovable_subgoals ctxt table
in
cat_lines (ustr @ gstr @ mstr @ tstr @ ["", "Could not find lexicographic termination order."])
end
(** The Main Function **)
fun lex_order_tac ctxt solve_tac (st: thm) =
let
val thy = ProofContext.theory_of ctxt
val ((trueprop $ (wf $ rel)) :: tl) = prems_of st
val (domT, _) = HOLogic.dest_prodT (HOLogic.dest_setT (fastype_of rel))
val measure_funs = MeasureFunctions.get_measure_functions ctxt 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 ctxt table 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 (Syntax.string_of_term ctxt relation))
in (* 4: proof reconstruction *)
st |> (PRIMITIVE (cterm_instantiate [(cterm_of thy rel, cterm_of thy relation)])
THEN (REPEAT (rtac @{thm "wf_mlex"} 1))
THEN (rtac @{thm "wf_empty"} 1)
THEN EVERY (map prove_row clean_table))
end
fun lexicographic_order_tac ctxt =
TRY (FundefCommon.apply_termination_rule ctxt 1)
THEN lex_order_tac ctxt (auto_tac (local_clasimpset_of ctxt addsimps2 FundefCommon.TerminationSimps.get ctxt))
val lexicographic_order = SIMPLE_METHOD o lexicographic_order_tac
val setup =
Method.setup @{binding lexicographic_order}
(Method.sections clasimp_modifiers >> (K lexicographic_order))
"termination prover for lexicographic orderings"
#> Context.theory_map (FundefCommon.set_termination_prover lexicographic_order)
end;