(*  Title:      HOL/Tools/lin_arith.ML

    Author:     Tjark Weber and Tobias Nipkow, TU Muenchen

HOL setup for linear arithmetic (see Provers/Arith/fast_lin_arith.ML).

*)

signature LIN_ARITH =

sig

  val pre_tac: simpset -> int -> tactic

  val simple_tac: Proof.context -> int -> tactic

  val tac: Proof.context -> int -> tactic

  val simproc: simpset -> term -> thm option

  val add_inj_thms: thm list -> Context.generic -> Context.generic

  val add_lessD: thm -> Context.generic -> Context.generic

  val add_simps: thm list -> Context.generic -> Context.generic

  val add_simprocs: simproc list -> Context.generic -> Context.generic

  val add_inj_const: string * typ -> Context.generic -> Context.generic

  val add_discrete_type: string -> Context.generic -> Context.generic

  val set_number_of: (theory -> typ -> int -> cterm) -> Context.generic ->

    Context.generic

  val setup: Context.generic -> Context.generic

  val global_setup: theory -> theory

  val split_limit: int Config.T

  val neq_limit: int Config.T

  val warning_count: int Unsynchronized.ref

  val trace: bool Unsynchronized.ref

end;

structure Lin_Arith: LIN_ARITH =

struct

(* Parameters data for general linear arithmetic functor *)

structure LA_Logic: LIN_ARITH_LOGIC =

struct

val ccontr = ccontr;

val conjI = conjI;

val notI = notI;

val sym = sym;

val trueI = TrueI;

val not_lessD = @{thm linorder_not_less} RS iffD1;

val not_leD = @{thm linorder_not_le} RS iffD1;

fun mk_Eq thm = thm RS @{thm Eq_FalseI} handle THM _ => thm RS @{thm Eq_TrueI};

val mk_Trueprop = HOLogic.mk_Trueprop;

fun atomize thm = case Thm.prop_of thm of

    Const ("Trueprop", _) $ (Const (@{const_name "op &"}, _) $ _ $ _) =>

    atomize (thm RS conjunct1) @ atomize (thm RS conjunct2)

  | _ => [thm];

fun neg_prop ((TP as Const("Trueprop", _)) $ (Const (@{const_name Not}, _) $ t)) = TP $ t

  | neg_prop ((TP as Const("Trueprop", _)) $ t) = TP $ (HOLogic.Not $t)

  | neg_prop t = raise TERM ("neg_prop", [t]);

fun is_False thm =

  let val _ $ t = Thm.prop_of thm

  in t = HOLogic.false_const end;

fun is_nat t = (fastype_of1 t = HOLogic.natT);

fun mk_nat_thm thy t =

  let

    val cn = cterm_of thy (Var (("n", 0), HOLogic.natT))

    and ct = cterm_of thy t

  in instantiate ([], [(cn, ct)]) @{thm le0} end;

end;  (* LA_Logic *)

(* arith context data *)

structure Lin_Arith_Data = Generic_Data

(

  type T = {splits: thm list,

            inj_consts: (string * typ) list,

            discrete: string list};

  val empty = {splits = [], inj_consts = [], discrete = []};

  val extend = I;

  fun merge

   ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1},

    {splits= splits2, inj_consts= inj_consts2, discrete= discrete2}) : T =

   {splits = Thm.merge_thms (splits1, splits2),

    inj_consts = Library.merge (op =) (inj_consts1, inj_consts2),

    discrete = Library.merge (op =) (discrete1, discrete2)};

);

val get_arith_data = Lin_Arith_Data.get o Context.Proof;

fun add_split thm = Lin_Arith_Data.map (fn {splits, inj_consts, discrete} =>

  {splits = update Thm.eq_thm_prop thm splits,

   inj_consts = inj_consts, discrete = discrete});

fun add_discrete_type d = Lin_Arith_Data.map (fn {splits, inj_consts, discrete} =>

  {splits = splits, inj_consts = inj_consts,

   discrete = update (op =) d discrete});

fun add_inj_const c = Lin_Arith_Data.map (fn {splits, inj_consts, discrete} =>

  {splits = splits, inj_consts = update (op =) c inj_consts,

   discrete = discrete});

val (split_limit, setup_split_limit) = Attrib.config_int "linarith_split_limit" 9;

val (neq_limit, setup_neq_limit) = Attrib.config_int "linarith_neq_limit" 9;

structure LA_Data =

struct

val fast_arith_neq_limit = neq_limit;

(* Decomposition of terms *)

(*internal representation of linear (in-)equations*)

type decomp =

  ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool);

fun nT (Type ("fun", [N, _])) = (N = HOLogic.natT)

  | nT _                      = false;

fun add_atom (t : term) (m : Rat.rat) (p : (term * Rat.rat) list, i : Rat.rat) :

             (term * Rat.rat) list * Rat.rat =

  case AList.lookup Pattern.aeconv p t of

      NONE   => ((t, m) :: p, i)

    | SOME n => (AList.update Pattern.aeconv (t, Rat.add n m) p, i);

(* decompose nested multiplications, bracketing them to the right and combining

   all their coefficients

   inj_consts: list of constants to be ignored when encountered

               (e.g. arithmetic type conversions that preserve value)

   m: multiplicity associated with the entire product

   returns either (SOME term, associated multiplicity) or (NONE, constant)

*)

fun demult (inj_consts : (string * typ) list) : term * Rat.rat -> term option * Rat.rat =

let

  fun demult ((mC as Const (@{const_name Groups.times}, _)) $ s $ t, m) =

      (case s of Const (@{const_name Groups.times}, _) $ s1 $ s2 =>

        (* bracketing to the right: '(s1 * s2) * t' becomes 's1 * (s2 * t)' *)

        demult (mC $ s1 $ (mC $ s2 $ t), m)

      | _ =>

        (* product 's * t', where either factor can be 'NONE' *)

        (case demult (s, m) of

          (SOME s', m') =>

            (case demult (t, m') of

              (SOME t', m'') => (SOME (mC $ s' $ t'), m'')

            | (NONE,    m'') => (SOME s', m''))

        | (NONE,    m') => demult (t, m')))

    | demult ((mC as Const (@{const_name Rings.divide}, _)) $ s $ t, m) =

      (* FIXME: Shouldn't we simplify nested quotients, e.g. '(s/t)/u' could

         become 's/(t*u)', and '(s*t)/u' could become 's*(t/u)' ?   Note that

         if we choose to do so here, the simpset used by arith must be able to

         perform the same simplifications. *)

      (* FIXME: Currently we treat the numerator as atomic unless the

         denominator can be reduced to a numeric constant.  It might be better

         to demult the numerator in any case, and invent a new term of the form

         '1 / t' if the numerator can be reduced, but the denominator cannot. *)

      (* FIXME: Currently we even treat the whole fraction as atomic unless the

         denominator can be reduced to a numeric constant.  It might be better

         to use the partially reduced denominator (i.e. 's / (2*t)' could be

         demult'ed to 's / t' with multiplicity .5).   This would require a

         very simple change only below, but it breaks existing proofs. *)

      (* quotient 's / t', where the denominator t can be NONE *)

      (* Note: will raise Rat.DIVZERO iff m' is Rat.zero *)

      (case demult (t, Rat.one) of

        (SOME _, _) => (SOME (mC $ s $ t), m)

      | (NONE,  m') => apsnd (Rat.mult (Rat.inv m')) (demult (s, m)))

    (* terms that evaluate to numeric constants *)

    | demult (Const (@{const_name Groups.uminus}, _) $ t, m) = demult (t, Rat.neg m)

    | demult (Const (@{const_name Groups.zero}, _), m) = (NONE, Rat.zero)

    | demult (Const (@{const_name Groups.one}, _), m) = (NONE, m)

    (*Warning: in rare cases number_of encloses a non-numeral,

      in which case dest_numeral raises TERM; hence all the handles below.

      Same for Suc-terms that turn out not to be numerals -

      although the simplifier should eliminate those anyway ...*)

    | demult (t as Const ("Int.number_class.number_of", _) $ n, m) =

      ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))

        handle TERM _ => (SOME t, m))

    | demult (t as Const (@{const_name Suc}, _) $ _, m) =

      ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat t)))

        handle TERM _ => (SOME t, m))

    (* injection constants are ignored *)

    | demult (t as Const f $ x, m) =

      if member (op =) inj_consts f then demult (x, m) else (SOME t, m)

    (* everything else is considered atomic *)

    | demult (atom, m) = (SOME atom, m)

in demult end;

fun decomp0 (inj_consts : (string * typ) list) (rel : string, lhs : term, rhs : term) :

            ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat) option =

let

  (* Turns a term 'all' and associated multiplicity 'm' into a list 'p' of

     summands and associated multiplicities, plus a constant 'i' (with implicit

     multiplicity 1) *)

  fun poly (Const (@{const_name Groups.plus}, _) $ s $ t,

        m : Rat.rat, pi : (term * Rat.rat) list * Rat.rat) = poly (s, m, poly (t, m, pi))

    | poly (all as Const (@{const_name Groups.minus}, T) $ s $ t, m, pi) =

        if nT T then add_atom all m pi else poly (s, m, poly (t, Rat.neg m, pi))

    | poly (all as Const (@{const_name Groups.uminus}, T) $ t, m, pi) =

        if nT T then add_atom all m pi else poly (t, Rat.neg m, pi)

    | poly (Const (@{const_name Groups.zero}, _), _, pi) =

        pi

    | poly (Const (@{const_name Groups.one}, _), m, (p, i)) =

        (p, Rat.add i m)

    | poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) =

        poly (t, m, (p, Rat.add i m))

    | poly (all as Const (@{const_name Groups.times}, _) $ _ $ _, m, pi as (p, i)) =

        (case demult inj_consts (all, m) of

           (NONE,   m') => (p, Rat.add i m')

         | (SOME u, m') => add_atom u m' pi)

    | poly (all as Const (@{const_name Rings.divide}, _) $ _ $ _, m, pi as (p, i)) =

        (case demult inj_consts (all, m) of

           (NONE,   m') => (p, Rat.add i m')

         | (SOME u, m') => add_atom u m' pi)

    | poly (all as Const ("Int.number_class.number_of", Type(_,[_,T])) $ t, m, pi as (p, i)) =

        (let val k = HOLogic.dest_numeral t

            val k2 = if k < 0 andalso T = HOLogic.natT then 0 else k

        in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k2))) end

        handle TERM _ => add_atom all m pi)

    | poly (all as Const f $ x, m, pi) =

        if f mem inj_consts then poly (x, m, pi) else add_atom all m pi

    | poly (all, m, pi) =

        add_atom all m pi

  val (p, i) = poly (lhs, Rat.one, ([], Rat.zero))

  val (q, j) = poly (rhs, Rat.one, ([], Rat.zero))

in

  case rel of

    @{const_name Orderings.less}    => SOME (p, i, "<", q, j)

  | @{const_name Orderings.less_eq} => SOME (p, i, "<=", q, j)

  | "op ="              => SOME (p, i, "=", q, j)

  | _                   => NONE

end handle Rat.DIVZERO => NONE;

fun of_lin_arith_sort thy U =

  Sign.of_sort thy (U, @{sort Rings.linordered_idom});

fun allows_lin_arith thy (discrete : string list) (U as Type (D, [])) : bool * bool =

      if of_lin_arith_sort thy U then (true, member (op =) discrete D)

      else if member (op =) discrete D then (true, true) else (false, false)

  | allows_lin_arith sg discrete U = (of_lin_arith_sort sg U, false);

fun decomp_typecheck (thy, discrete, inj_consts) (T : typ, xxx) : decomp option =

  case T of

    Type ("fun", [U, _]) =>

      (case allows_lin_arith thy discrete U of

        (true, d) =>

          (case decomp0 inj_consts xxx of

            NONE                   => NONE

          | SOME (p, i, rel, q, j) => SOME (p, i, rel, q, j, d))

      | (false, _) =>

          NONE)

  | _ => NONE;

fun negate (SOME (x, i, rel, y, j, d)) = SOME (x, i, "~" ^ rel, y, j, d)

  | negate NONE                        = NONE;

fun decomp_negation data

  ((Const ("Trueprop", _)) $ (Const (rel, T) $ lhs $ rhs)) : decomp option =

      decomp_typecheck data (T, (rel, lhs, rhs))

  | decomp_negation data ((Const ("Trueprop", _)) $

  (Const ("Not", _) $ (Const (rel, T) $ lhs $ rhs))) =

      negate (decomp_typecheck data (T, (rel, lhs, rhs)))

  | decomp_negation data _ =

      NONE;

fun decomp ctxt : term -> decomp option =

  let

    val thy = ProofContext.theory_of ctxt

    val {discrete, inj_consts, ...} = get_arith_data ctxt

  in decomp_negation (thy, discrete, inj_consts) end;

fun domain_is_nat (_ $ (Const (_, T) $ _ $ _))                      = nT T

  | domain_is_nat (_ $ (Const ("Not", _) $ (Const (_, T) $ _ $ _))) = nT T

  | domain_is_nat _                                                 = false;

(*---------------------------------------------------------------------------*)

(* the following code performs splitting of certain constants (e.g., min,    *)

(* max) in a linear arithmetic problem; similar to what split_tac later does *)

(* to the proof state                                                        *)

(*---------------------------------------------------------------------------*)

(* checks if splitting with 'thm' is implemented                             *)

fun is_split_thm thm =

  case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) => (

    (* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *)

    case head_of lhs of

      Const (a, _) => member (op =) [@{const_name Orderings.max},

                                    @{const_name Orderings.min},

                                    @{const_name Groups.abs},

                                    @{const_name Groups.minus},

                                    "Int.nat" (*DYNAMIC BINDING!*),

                                    "Divides.div_class.mod" (*DYNAMIC BINDING!*),

                                    "Divides.div_class.div" (*DYNAMIC BINDING!*)] a

    | _            => (warning ("Lin. Arith.: wrong format for split rule " ^

                                 Display.string_of_thm_without_context thm);

                       false))

  | _ => (warning ("Lin. Arith.: wrong format for split rule " ^

                   Display.string_of_thm_without_context thm);

          false);

(* substitute new for occurrences of old in a term, incrementing bound       *)

(* variables as needed when substituting inside an abstraction               *)

fun subst_term ([] : (term * term) list) (t : term) = t

  | subst_term pairs                     t          =

      (case AList.lookup Pattern.aeconv pairs t of

        SOME new =>

          new

      | NONE     =>

          (case t of Abs (a, T, body) =>

            let val pairs' = map (pairself (incr_boundvars 1)) pairs

            in  Abs (a, T, subst_term pairs' body)  end

          | t1 $ t2                   =>

            subst_term pairs t1 $ subst_term pairs t2

          | _ => t));

(* approximates the effect of one application of split_tac (followed by NNF  *)

(* normalization) on the subgoal represented by '(Ts, terms)'; returns a     *)

(* list of new subgoals (each again represented by a typ list for bound      *)

(* variables and a term list for premises), or NONE if split_tac would fail  *)

(* on the subgoal                                                            *)

(* FIXME: currently only the effect of certain split theorems is reproduced  *)

(*        (which is why we need 'is_split_thm').  A more canonical           *)

(*        implementation should analyze the right-hand side of the split     *)

(*        theorem that can be applied, and modify the subgoal accordingly.   *)

(*        Or even better, the splitter should be extended to provide         *)

(*        splitting on terms as well as splitting on theorems (where the     *)

(*        former can have a faster implementation as it does not need to be  *)

(*        proof-producing).                                                  *)

fun split_once_items ctxt (Ts : typ list, terms : term list) :

                     (typ list * term list) list option =

let

  val thy = ProofContext.theory_of ctxt

  (* takes a list  [t1, ..., tn]  to the term                                *)

  (*   tn' --> ... --> t1' --> False  ,                                      *)

  (* where ti' = HOLogic.dest_Trueprop ti                                    *)

  fun REPEAT_DETERM_etac_rev_mp tms =

    fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop tms)

      HOLogic.false_const

  val split_thms  = filter is_split_thm (#splits (get_arith_data ctxt))

  val cmap        = Splitter.cmap_of_split_thms split_thms

  val goal_tm     = REPEAT_DETERM_etac_rev_mp terms

  val splits      = Splitter.split_posns cmap thy Ts goal_tm

  val split_limit = Config.get ctxt split_limit

in

  if length splits > split_limit then (

    tracing ("linarith_split_limit exceeded (current value is " ^

      string_of_int split_limit ^ ")");

    NONE

  ) else case splits of

    [] =>

    (* split_tac would fail: no possible split *)

    NONE

  | (_, _::_, _, _, _) :: _ =>

    (* disallow a split that involves non-locally bound variables (except    *)

    (* when bound by outermost meta-quantifiers)                             *)

    NONE

  | (_, [], _, split_type, split_term) :: _ =>

    (* ignore all but the first possible split                               *)

    (case strip_comb split_term of

    (* ?P (max ?i ?j) = ((?i <= ?j --> ?P ?j) & (~ ?i <= ?j --> ?P ?i)) *)

      (Const (@{const_name Orderings.max}, _), [t1, t2]) =>

      let

        val rev_terms     = rev terms

        val terms1        = map (subst_term [(split_term, t1)]) rev_terms

        val terms2        = map (subst_term [(split_term, t2)]) rev_terms

        val t1_leq_t2     = Const (@{const_name Orderings.less_eq},

                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2

        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2

        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)

        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false]

        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false]

      in

        SOME [(Ts, subgoal1), (Ts, subgoal2)]

      end

    (* ?P (min ?i ?j) = ((?i <= ?j --> ?P ?i) & (~ ?i <= ?j --> ?P ?j)) *)

    | (Const (@{const_name Orderings.min}, _), [t1, t2]) =>

      let

        val rev_terms     = rev terms

        val terms1        = map (subst_term [(split_term, t1)]) rev_terms

        val terms2        = map (subst_term [(split_term, t2)]) rev_terms

        val t1_leq_t2     = Const (@{const_name Orderings.less_eq},

                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2

        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2

        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)

        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false]

        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false]

      in

        SOME [(Ts, subgoal1), (Ts, subgoal2)]

      end

    (* ?P (abs ?a) = ((0 <= ?a --> ?P ?a) & (?a < 0 --> ?P (- ?a))) *)

    | (Const (@{const_name Groups.abs}, _), [t1]) =>

      let

        val rev_terms   = rev terms

        val terms1      = map (subst_term [(split_term, t1)]) rev_terms

        val terms2      = map (subst_term [(split_term, Const (@{const_name Groups.uminus},

                            split_type --> split_type) $ t1)]) rev_terms

        val zero        = Const (@{const_name Groups.zero}, split_type)

        val zero_leq_t1 = Const (@{const_name Orderings.less_eq},

                            split_type --> split_type --> HOLogic.boolT) $ zero $ t1

        val t1_lt_zero  = Const (@{const_name Orderings.less},

                            split_type --> split_type --> HOLogic.boolT) $ t1 $ zero

        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)

        val subgoal1    = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false]

        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]

      in

        SOME [(Ts, subgoal1), (Ts, subgoal2)]

      end

    (* ?P (?a - ?b) = ((?a < ?b --> ?P 0) & (ALL d. ?a = ?b + d --> ?P d)) *)

    | (Const (@{const_name Groups.minus}, _), [t1, t2]) =>

      let

        (* "d" in the above theorem becomes a new bound variable after NNF   *)

        (* transformation, therefore some adjustment of indices is necessary *)

        val rev_terms       = rev terms

        val zero            = Const (@{const_name Groups.zero}, split_type)

        val d               = Bound 0

        val terms1          = map (subst_term [(split_term, zero)]) rev_terms

        val terms2          = map (subst_term [(incr_boundvars 1 split_term, d)])

                                (map (incr_boundvars 1) rev_terms)

        val t1'             = incr_boundvars 1 t1

        val t2'             = incr_boundvars 1 t2

        val t1_lt_t2        = Const (@{const_name Orderings.less},

                                split_type --> split_type --> HOLogic.boolT) $ t1 $ t2

        val t1_eq_t2_plus_d = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $

                                (Const (@{const_name Groups.plus},

                                  split_type --> split_type --> split_type) $ t2' $ d)

        val not_false       = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)

        val subgoal1        = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false]

        val subgoal2        = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false]

      in

        SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)]

      end

    (* ?P (nat ?i) = ((ALL n. ?i = of_nat n --> ?P n) & (?i < 0 --> ?P 0)) *)

    | (Const ("Int.nat", _), [t1]) =>

      let

        val rev_terms   = rev terms

        val zero_int    = Const (@{const_name Groups.zero}, HOLogic.intT)

        val zero_nat    = Const (@{const_name Groups.zero}, HOLogic.natT)

        val n           = Bound 0

        val terms1      = map (subst_term [(incr_boundvars 1 split_term, n)])

                            (map (incr_boundvars 1) rev_terms)

        val terms2      = map (subst_term [(split_term, zero_nat)]) rev_terms

        val t1'         = incr_boundvars 1 t1

        val t1_eq_nat_n = Const ("op =", HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1' $

                            (Const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) $ n)

        val t1_lt_zero  = Const (@{const_name Orderings.less},

                            HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1 $ zero_int

        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)

        val subgoal1    = (HOLogic.mk_Trueprop t1_eq_nat_n) :: terms1 @ [not_false]

        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]

      in

        SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)]

      end

    (* ?P ((?n::nat) mod (number_of ?k)) =

         ((number_of ?k = 0 --> ?P ?n) & (~ (number_of ?k = 0) -->

           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P j))) *)

    | (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>

      let

        val rev_terms               = rev terms

        val zero                    = Const (@{const_name Groups.zero}, split_type)

        val i                       = Bound 1

        val j                       = Bound 0

        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms

        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, j)])

                                        (map (incr_boundvars 2) rev_terms)

        val t1'                     = incr_boundvars 2 t1

        val t2'                     = incr_boundvars 2 t2

        val t2_eq_zero              = Const ("op =",

                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero

        val t2_neq_zero             = HOLogic.mk_not (Const ("op =",

                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)

        val j_lt_t2                 = Const (@{const_name Orderings.less},

                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'

        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $

                                       (Const (@{const_name Groups.plus}, split_type --> split_type --> split_type) $

                                         (Const (@{const_name Groups.times},

                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)

        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)

        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]

        val subgoal2                = (map HOLogic.mk_Trueprop

                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])

                                          @ terms2 @ [not_false]

      in

        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]

      end

    (* ?P ((?n::nat) div (number_of ?k)) =

         ((number_of ?k = 0 --> ?P 0) & (~ (number_of ?k = 0) -->

           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P i))) *)

    | (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>

      let

        val rev_terms               = rev terms

        val zero                    = Const (@{const_name Groups.zero}, split_type)

        val i                       = Bound 1

        val j                       = Bound 0

        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms

        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, i)])

                                        (map (incr_boundvars 2) rev_terms)

        val t1'                     = incr_boundvars 2 t1

        val t2'                     = incr_boundvars 2 t2

        val t2_eq_zero              = Const ("op =",

                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero

        val t2_neq_zero             = HOLogic.mk_not (Const ("op =",

                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)

        val j_lt_t2                 = Const (@{const_name Orderings.less},

                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'

        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $

                                       (Const (@{const_name Groups.plus}, split_type --> split_type --> split_type) $

                                         (Const (@{const_name Groups.times},

                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)

        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)

        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]

        val subgoal2                = (map HOLogic.mk_Trueprop

                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])

                                          @ terms2 @ [not_false]

      in

        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]

      end

    (* ?P ((?n::int) mod (number_of ?k)) =

         ((number_of ?k = 0 --> ?P ?n) &

          (0 < number_of ?k -->

            (ALL i j.

              0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P j)) &

          (number_of ?k < 0 -->

            (ALL i j.

              number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P j))) *)

    | (Const ("Divides.div_class.mod",

        Type ("fun", [Type ("Int.int", []), _])), [t1, t2]) =>

      let

        val rev_terms               = rev terms

        val zero                    = Const (@{const_name Groups.zero}, split_type)

        val i                       = Bound 1

        val j                       = Bound 0

        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms

        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, j)])

                                        (map (incr_boundvars 2) rev_terms)

        val t1'                     = incr_boundvars 2 t1

        val t2'                     = incr_boundvars 2 t2

        val t2_eq_zero              = Const ("op =",

                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero

        val zero_lt_t2              = Const (@{const_name Orderings.less},

                                        split_type --> split_type --> HOLogic.boolT) $ zero $ t2'

        val t2_lt_zero              = Const (@{const_name Orderings.less},

                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero

        val zero_leq_j              = Const (@{const_name Orderings.less_eq},

                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j

        val j_leq_zero              = Const (@{const_name Orderings.less_eq},

                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero

        val j_lt_t2                 = Const (@{const_name Orderings.less},

                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'

        val t2_lt_j                 = Const (@{const_name Orderings.less},

                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j

        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $

                                       (Const (@{const_name Groups.plus}, split_type --> split_type --> split_type) $

                                         (Const (@{const_name Groups.times},

                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)

        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)

        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]

        val subgoal2                = (map HOLogic.mk_Trueprop [zero_lt_t2, zero_leq_j])

                                        @ hd terms2_3

                                        :: (if tl terms2_3 = [] then [not_false] else [])

                                        @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])

                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])

        val subgoal3                = (map HOLogic.mk_Trueprop [t2_lt_zero, t2_lt_j])

                                        @ hd terms2_3

                                        :: (if tl terms2_3 = [] then [not_false] else [])

                                        @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])

                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])

        val Ts'                     = split_type :: split_type :: Ts

      in

        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]

      end

    (* ?P ((?n::int) div (number_of ?k)) =

         ((number_of ?k = 0 --> ?P 0) &

          (0 < number_of ?k -->

            (ALL i j.

              0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P i)) &

          (number_of ?k < 0 -->

            (ALL i j.

              number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P i))) *)

    | (Const ("Divides.div_class.div",

        Type ("fun", [Type ("Int.int", []), _])), [t1, t2]) =>

      let

        val rev_terms               = rev terms

        val zero                    = Const (@{const_name Groups.zero}, split_type)

        val i                       = Bound 1

        val j                       = Bound 0

        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms

        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, i)])

                                        (map (incr_boundvars 2) rev_terms)

        val t1'                     = incr_boundvars 2 t1

        val t2'                     = incr_boundvars 2 t2

        val t2_eq_zero              = Const ("op =",

                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero

        val zero_lt_t2              = Const (@{const_name Orderings.less},

                                        split_type --> split_type --> HOLogic.boolT) $ zero $ t2'

        val t2_lt_zero              = Const (@{const_name Orderings.less},

                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero

        val zero_leq_j              = Const (@{const_name Orderings.less_eq},

                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j

        val j_leq_zero              = Const (@{const_name Orderings.less_eq},

                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero

        val j_lt_t2                 = Const (@{const_name Orderings.less},

                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'

        val t2_lt_j                 = Const (@{const_name Orderings.less},

                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j

        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $

                                       (Const (@{const_name Groups.plus}, split_type --> split_type --> split_type) $

                                         (Const (@{const_name Groups.times},

                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)

        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)

        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]

        val subgoal2                = (map HOLogic.mk_Trueprop [zero_lt_t2, zero_leq_j])

                                        @ hd terms2_3

                                        :: (if tl terms2_3 = [] then [not_false] else [])

                                        @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])

                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])

        val subgoal3                = (map HOLogic.mk_Trueprop [t2_lt_zero, t2_lt_j])

                                        @ hd terms2_3

                                        :: (if tl terms2_3 = [] then [not_false] else [])

                                        @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])

                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])

        val Ts'                     = split_type :: split_type :: Ts

      in

        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]

      end

    (* this will only happen if a split theorem can be applied for which no  *)

    (* code exists above -- in which case either the split theorem should be *)

    (* implemented above, or 'is_split_thm' should be modified to filter it  *)

    (* out                                                                   *)

    | (t, ts) => (

      warning ("Lin. Arith.: split rule for " ^ Syntax.string_of_term ctxt t ^

        " (with " ^ string_of_int (length ts) ^

        " argument(s)) not implemented; proof reconstruction is likely to fail");

      NONE

    ))

end;  (* split_once_items *)

(* remove terms that do not satisfy 'p'; change the order of the remaining   *)

(* terms in the same way as filter_prems_tac does                            *)

fun filter_prems_tac_items (p : term -> bool) (terms : term list) : term list =

let

  fun filter_prems t (left, right) =

    if p t then (left, right @ [t]) else (left @ right, [])

  val (left, right) = fold filter_prems terms ([], [])

in

  right @ left

end;

(* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a     *)

(* subgoal that has 'terms' as premises                                      *)

fun negated_term_occurs_positively (terms : term list) : bool =

  List.exists

    (fn (Trueprop $ (Const ("Not", _) $ t)) =>

      member Pattern.aeconv terms (Trueprop $ t)

      | _ => false)

    terms;

fun pre_decomp ctxt (Ts : typ list, terms : term list) : (typ list * term list) list =

let

  (* repeatedly split (including newly emerging subgoals) until no further   *)

  (* splitting is possible                                                   *)

  fun split_loop ([] : (typ list * term list) list) =

      ([] : (typ list * term list) list)

    | split_loop (subgoal::subgoals) =

      (case split_once_items ctxt subgoal of

        SOME new_subgoals => split_loop (new_subgoals @ subgoals)

      | NONE              => subgoal :: split_loop subgoals)

  fun is_relevant t  = is_some (decomp ctxt t)

  (* filter_prems_tac is_relevant: *)

  val relevant_terms = filter_prems_tac_items is_relevant terms

  (* split_tac, NNF normalization: *)

  val split_goals    = split_loop [(Ts, relevant_terms)]

  (* necessary because split_once_tac may normalize terms: *)

  val beta_eta_norm  = map (apsnd (map (Envir.eta_contract o Envir.beta_norm)))

    split_goals

  (* TRY (etac notE) THEN eq_assume_tac: *)

  val result         = filter_out (negated_term_occurs_positively o snd)

    beta_eta_norm

in

  result

end;

(* takes the i-th subgoal  [| A1; ...; An |] ==> B  to                       *)

(* An --> ... --> A1 --> B,  performs splitting with the given 'split_thms'  *)

(* (resulting in a different subgoal P), takes  P  to  ~P ==> False,         *)

(* performs NNF-normalization of ~P, and eliminates conjunctions,            *)

(* disjunctions and existential quantifiers from the premises, possibly (in  *)

(* the case of disjunctions) resulting in several new subgoals, each of the  *)

(* general form  [| Q1; ...; Qm |] ==> False.  Fails if more than            *)

(* !split_limit splits are possible.                              *)

local

  val nnf_simpset =

    empty_ss setmkeqTrue mk_eq_True

    setmksimps (mksimps mksimps_pairs)

    addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj},

      @{thm de_Morgan_conj}, not_all, not_ex, not_not]

  fun prem_nnf_tac ss = full_simp_tac (Simplifier.inherit_context ss nnf_simpset)

in

fun split_once_tac ss split_thms =

  let

    val ctxt = Simplifier.the_context ss

    val thy = ProofContext.theory_of ctxt

    val cond_split_tac = SUBGOAL (fn (subgoal, i) =>

      let

        val Ts = rev (map snd (Logic.strip_params subgoal))

        val concl = HOLogic.dest_Trueprop (Logic.strip_assums_concl subgoal)

        val cmap = Splitter.cmap_of_split_thms split_thms

        val splits = Splitter.split_posns cmap thy Ts concl

      in

        if null splits orelse length splits > Config.get ctxt split_limit then

          no_tac

        else if null (#2 (hd splits)) then

          split_tac split_thms i

        else

          (* disallow a split that involves non-locally bound variables      *)

          (* (except when bound by outermost meta-quantifiers)               *)

          no_tac

      end)

  in

    EVERY' [

      REPEAT_DETERM o etac rev_mp,

      cond_split_tac,

      rtac ccontr,

      prem_nnf_tac ss,

      TRY o REPEAT_ALL_NEW (DETERM o (eresolve_tac [conjE, exE] ORELSE' etac disjE))

    ]

  end;

end;  (* local *)

(* remove irrelevant premises, then split the i-th subgoal (and all new      *)

(* subgoals) by using 'split_once_tac' repeatedly.  Beta-eta-normalize new   *)

(* subgoals and finally attempt to solve them by finding an immediate        *)

(* contradiction (i.e., a term and its negation) in their premises.          *)

fun pre_tac ss i =

let

  val ctxt = Simplifier.the_context ss;

  val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))

  fun is_relevant t = is_some (decomp ctxt t)

in

  DETERM (

    TRY (filter_prems_tac is_relevant i)

      THEN (

        (TRY o REPEAT_ALL_NEW (split_once_tac ss split_thms))

          THEN_ALL_NEW

            (CONVERSION Drule.beta_eta_conversion

              THEN'

            (TRY o (etac notE THEN' eq_assume_tac)))

      ) i

  )

end;

end;  (* LA_Data *)

val pre_tac = LA_Data.pre_tac;

structure Fast_Arith = Fast_Lin_Arith(structure LA_Logic = LA_Logic and LA_Data = LA_Data);

fun map_inj_thms f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =

  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = f inj_thms,

    lessD = lessD, neqE = neqE, simpset = simpset, number_of = number_of};

fun map_lessD f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =

  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,

    lessD = f lessD, neqE = neqE, simpset = simpset, number_of = number_of};

fun map_simpset f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =

  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,

    lessD = lessD, neqE = neqE, simpset = f simpset, number_of = number_of};

fun map_number_of f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =

  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,

    lessD = lessD, neqE = neqE, simpset = simpset, number_of = f number_of};

fun add_inj_thms thms = Fast_Arith.map_data (map_inj_thms (append thms));

fun add_lessD thm = Fast_Arith.map_data (map_lessD (fn thms => thms @ [thm]));

fun add_simps thms = Fast_Arith.map_data (map_simpset (fn simpset => simpset addsimps thms));

fun add_simprocs procs = Fast_Arith.map_data (map_simpset (fn simpset => simpset addsimprocs procs));

fun set_number_of f = Fast_Arith.map_data (map_number_of (K (serial (), f)))

fun simple_tac ctxt = Fast_Arith.lin_arith_tac ctxt false;

val lin_arith_tac = Fast_Arith.lin_arith_tac;

val trace = Fast_Arith.trace;

val warning_count = Fast_Arith.warning_count;

(* reduce contradictory <= to False.

   Most of the work is done by the cancel tactics. *)

val init_arith_data =

  Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, number_of, ...} =>

   {add_mono_thms = @{thms add_mono_thms_linordered_semiring} @ @{thms add_mono_thms_linordered_field} @ add_mono_thms,

    mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} ::

      @{lemma "a = b ==> c*a = c*b" by (rule arg_cong)} :: mult_mono_thms,

    inj_thms = inj_thms,

    lessD = lessD @ [@{thm "Suc_leI"}],

    neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_linordered_idom}],

    simpset = HOL_basic_ss

      addsimps @{thms ring_distribs}

      addsimps [@{thm if_True}, @{thm if_False}]

      addsimps

       [@{thm add_0_left},

        @{thm add_0_right},

        @{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"},

        @{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"},

        @{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"},

        @{thm "not_one_less_zero"}]

      addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]

       (*abel_cancel helps it work in abstract algebraic domains*)

      addsimprocs Nat_Arith.nat_cancel_sums_add

      addcongs [@{thm if_weak_cong}],

    number_of = number_of}) #>

  add_discrete_type @{type_name nat};

fun add_arith_facts ss =

  add_prems (Arith_Data.get_arith_facts (MetaSimplifier.the_context ss)) ss;

val simproc = add_arith_facts #> Fast_Arith.lin_arith_simproc;

(* generic refutation procedure *)

(* parameters:

   test: term -> bool

   tests if a term is at all relevant to the refutation proof;

   if not, then it can be discarded. Can improve performance,

   esp. if disjunctions can be discarded (no case distinction needed!).

   prep_tac: int -> tactic

   A preparation tactic to be applied to the goal once all relevant premises

   have been moved to the conclusion.

   ref_tac: int -> tactic

   the actual refutation tactic. Should be able to deal with goals

   [| A1; ...; An |] ==> False

   where the Ai are atomic, i.e. no top-level &, | or EX

*)

local

  val nnf_simpset =

    empty_ss setmkeqTrue mk_eq_True

    setmksimps (mksimps mksimps_pairs)

    addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj},

      @{thm de_Morgan_conj}, @{thm not_all}, @{thm not_ex}, @{thm not_not}];

  fun prem_nnf_tac i st =

    full_simp_tac (Simplifier.global_context (Thm.theory_of_thm st) nnf_simpset) i st;

in

fun refute_tac test prep_tac ref_tac =

  let val refute_prems_tac =

        REPEAT_DETERM

              (eresolve_tac [@{thm conjE}, @{thm exE}] 1 ORELSE

               filter_prems_tac test 1 ORELSE

               etac @{thm disjE} 1) THEN

        (DETERM (etac @{thm notE} 1 THEN eq_assume_tac 1) ORELSE

         ref_tac 1);

  in EVERY'[TRY o filter_prems_tac test,

            REPEAT_DETERM o etac @{thm rev_mp}, prep_tac, rtac @{thm ccontr}, prem_nnf_tac,

            SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]

  end;

end;

(* arith proof method *)

local

fun raw_tac ctxt ex =

  (* FIXME: K true should be replaced by a sensible test (perhaps "is_some o

     decomp sg"? -- but note that the test is applied to terms already before

     they are split/normalized) to speed things up in case there are lots of

     irrelevant terms involved; elimination of min/max can be optimized:

     (max m n + k <= r) = (m+k <= r & n+k <= r)

     (l <= min m n + k) = (l <= m+k & l <= n+k)

  *)

  refute_tac (K true)

    (* Splitting is also done inside simple_tac, but not completely --    *)

    (* split_tac may use split theorems that have not been implemented in *)

    (* simple_tac (cf. pre_decomp and split_once_items above), and        *)

    (* split_limit may trigger.                                           *)

    (* Therefore splitting outside of simple_tac may allow us to prove    *)

    (* some goals that simple_tac alone would fail on.                    *)

    (REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))

    (lin_arith_tac ctxt ex);

in

fun gen_tac ex ctxt = FIRST' [simple_tac ctxt,

  Object_Logic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_tac ctxt ex];

val tac = gen_tac true;

end;

(* context setup *)

val setup =

  init_arith_data #>

  Simplifier.map_ss (fn ss => ss addsimprocs [Simplifier.simproc (@{theory}) "fast_nat_arith"

    ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] (K simproc)]

    (* Because of fast_nat_arith_simproc, the arithmetic solver is really only

    useful to detect inconsistencies among the premises for subgoals which are

    *not* themselves (in)equalities, because the latter activate

    fast_nat_arith_simproc anyway. However, it seems cheaper to activate the

    solver all the time rather than add the additional check. *)

    addSolver (mk_solver' "lin_arith"

      (add_arith_facts #> Fast_Arith.cut_lin_arith_tac)))

val global_setup =

  setup_split_limit #> setup_neq_limit #>

  Attrib.setup @{binding arith_split} (Scan.succeed (Thm.declaration_attribute add_split))

    "declaration of split rules for arithmetic procedure" #>

  Method.setup @{binding linarith}

    (Scan.succeed (fn ctxt =>

      METHOD (fn facts =>

        HEADGOAL (Method.insert_tac (Arith_Data.get_arith_facts ctxt @ facts)

          THEN' tac ctxt)))) "linear arithmetic" #>

  Arith_Data.add_tactic "linear arithmetic" gen_tac;

end;