(*  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 relT = HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT))

        val mlexT = (domT --> HOLogic.natT) --> relT --> relT

        fun mk_ms [] = Const (@{const_name "{}"}, relT)

          | mk_ms (f::fs) = 

            Const (@{const_name "Wellfounded_Relations.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 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 _) = " ? "

  | 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 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 = mk_goal (vars, prems, mfun $ lhs, mfun $ rhs)

      val less_thm = goals @{const_name HOL.less} |> prove thy solve_tac

    in

      if Thm.no_prems less_thm then

        Less (Goal.finish less_thm)

      else

        let

          val lesseq_thm = goals @{const_name HOL.less_eq} |> prove thy solve_tac

        in

          if Thm.no_prems lesseq_thm then

            LessEq (Goal.finish lesseq_thm, less_thm)

          else

            if prems_of lesseq_thm = [HOLogic.Trueprop $ HOLogic.false_const] then False lesseq_thm

            else None (lesseq_thm, less_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 "mlex_less"} 1)

    THEN PRIMITIVE (flip implies_elim less_thm)

  | prove_row (LessEq (lesseq_thm, _) :: tail) =

    (rtac @{thm "mlex_leq"} 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_goals ctxt st =

    Display.pretty_goals_aux (ProofContext.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_less, st_leq)) =

      "(" ^ 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 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 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 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