(*  Title:      HOL/arith_data.ML

     ID:         $Id$

     Author:     Markus Wenzel, Stefan Berghofer and Tobias Nipkow

 Various arithmetic proof procedures.

 *)

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

 (* 1. Cancellation of common terms                                           *)

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

 structure NatArithUtils =

 struct

 (** abstract syntax of structure nat: 0, Suc, + **)

 (* mk_sum, mk_norm_sum *)

 val one = HOLogic.mk_nat 1;

 val mk_plus = HOLogic.mk_binop "op +";

 fun mk_sum [] = HOLogic.zero

   | mk_sum [t] = t

   | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);

 (*normal form of sums: Suc (... (Suc (a + (b + ...))))*)

 fun mk_norm_sum ts =

   let val (ones, sums) = partition (equal one) ts in

     funpow (length ones) HOLogic.mk_Suc (mk_sum sums)

   end;

 (* dest_sum *)

 val dest_plus = HOLogic.dest_bin "op +" HOLogic.natT;

 fun dest_sum tm =

   if HOLogic.is_zero tm then []

   else

     (case try HOLogic.dest_Suc tm of

       Some t => one :: dest_sum t

     | None =>

         (case try dest_plus tm of

           Some (t, u) => dest_sum t @ dest_sum u

         | None => [tm]));

 (** generic proof tools **)

 (* prove conversions *)

 val mk_eqv = HOLogic.mk_Trueprop o HOLogic.mk_eq;

 fun prove_conv expand_tac norm_tac sg tu =

   mk_meta_eq (prove_goalw_cterm_nocheck [] (cterm_of sg (mk_eqv tu))

     (K [expand_tac, norm_tac]))

   handle ERROR => error ("The error(s) above occurred while trying to prove " ^

     (string_of_cterm (cterm_of sg (mk_eqv tu))));

 val subst_equals = prove_goal HOL.thy "[| t = s; u = t |] ==> u = s"

   (fn prems => [cut_facts_tac prems 1, SIMPSET' asm_simp_tac 1]);

 (* rewriting *)

 fun simp_all rules = ALLGOALS (simp_tac (HOL_ss addsimps rules));

 val add_rules = [add_Suc, add_Suc_right, add_0, add_0_right];

 val mult_rules = [mult_Suc, mult_Suc_right, mult_0, mult_0_right];

 fun prep_simproc (name, pats, proc) =

   Simplifier.simproc (Theory.sign_of (the_context ())) name pats proc;

 end;

 signature ARITH_DATA =

 sig

   val nat_cancel_sums_add: simproc list

   val nat_cancel_sums: simproc list

 end;

 structure ArithData: ARITH_DATA =

 struct

 open NatArithUtils;

 (** cancel common summands **)

 structure Sum =

 struct

   val mk_sum = mk_norm_sum;

   val dest_sum = dest_sum;

   val prove_conv = prove_conv;

   val norm_tac = simp_all add_rules THEN simp_all add_ac;

 end;

 fun gen_uncancel_tac rule ct =

   rtac (instantiate' [] [None, Some ct] (rule RS subst_equals)) 1;

 (* nat eq *)

 structure EqCancelSums = CancelSumsFun

 (struct

   open Sum;

   val mk_bal = HOLogic.mk_eq;

   val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;

   val uncancel_tac = gen_uncancel_tac nat_add_left_cancel;

 end);

 (* nat less *)

 structure LessCancelSums = CancelSumsFun

 (struct

   open Sum;

   val mk_bal = HOLogic.mk_binrel "op <";

   val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT;

   val uncancel_tac = gen_uncancel_tac nat_add_left_cancel_less;

 end);

 (* nat le *)

 structure LeCancelSums = CancelSumsFun

 (struct

   open Sum;

   val mk_bal = HOLogic.mk_binrel "op <=";

   val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT;

   val uncancel_tac = gen_uncancel_tac nat_add_left_cancel_le;

 end);

 (* nat diff *)

 structure DiffCancelSums = CancelSumsFun

 (struct

   open Sum;

   val mk_bal = HOLogic.mk_binop "op -";

   val dest_bal = HOLogic.dest_bin "op -" HOLogic.natT;

   val uncancel_tac = gen_uncancel_tac diff_cancel;

 end);

 (** prepare nat_cancel simprocs **)

 val nat_cancel_sums_add = map prep_simproc

   [("nateq_cancel_sums",

      ["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"], EqCancelSums.proc),

    ("natless_cancel_sums",

      ["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"], LessCancelSums.proc),

    ("natle_cancel_sums",

      ["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"], LeCancelSums.proc)];

 val nat_cancel_sums = nat_cancel_sums_add @

   [prep_simproc ("natdiff_cancel_sums",

     ["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"], DiffCancelSums.proc)];

 end;

 open ArithData;

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

 (* 2. Linear arithmetic                                                      *)

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

 (* Parameters data for general linear arithmetic functor *)

 structure LA_Logic: LIN_ARITH_LOGIC =

 struct

 val ccontr = ccontr;

 val conjI = conjI;

 val neqE = linorder_neqE;

 val notI = notI;

 val sym = sym;

 val not_lessD = linorder_not_less RS iffD1;

 val not_leD = linorder_not_le RS iffD1;

 fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI);

 val mk_Trueprop = HOLogic.mk_Trueprop;

 fun neg_prop(TP$(Const("Not",_)$t)) = TP$t

   | neg_prop(TP$t) = TP $ (Const("Not",HOLogic.boolT-->HOLogic.boolT)$t);

 fun is_False thm =

   let val _ $ t = #prop(rep_thm thm)

   in t = Const("False",HOLogic.boolT) end;

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

 fun mk_nat_thm sg t =

   let val ct = cterm_of sg t  and cn = cterm_of sg (Var(("n",0),HOLogic.natT))

   in instantiate ([],[(cn,ct)]) le0 end;

 end;

 (* arith theory data *)

 structure ArithTheoryDataArgs =

 struct

   val name = "HOL/arith";

   type T = {splits: thm list, inj_consts: (string * typ)list, discrete: (string * bool) list, presburger: (int -> tactic) option};

   val empty = {splits = [], inj_consts = [], discrete = [], presburger = None};

   val copy = I;

   val prep_ext = I;

   fun merge ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, presburger= presburger1},

              {splits= splits2, inj_consts= inj_consts2, discrete= discrete2, presburger= presburger2}) =

    {splits = Drule.merge_rules (splits1, splits2),

     inj_consts = merge_lists inj_consts1 inj_consts2,

     discrete = merge_alists discrete1 discrete2,

     presburger = (case presburger1 of None => presburger2 | p => p)};

   fun print _ _ = ();

 end;

 structure ArithTheoryData = TheoryDataFun(ArithTheoryDataArgs);

 fun arith_split_add (thy, thm) = (ArithTheoryData.map (fn {splits,inj_consts,discrete,presburger} =>

   {splits= thm::splits, inj_consts= inj_consts, discrete= discrete, presburger= presburger}) thy, thm);

 fun arith_discrete d = ArithTheoryData.map (fn {splits,inj_consts,discrete,presburger} =>

   {splits = splits, inj_consts = inj_consts, discrete = d :: discrete, presburger= presburger});

 fun arith_inj_const c = ArithTheoryData.map (fn {splits,inj_consts,discrete,presburger} =>

   {splits = splits, inj_consts = c :: inj_consts, discrete = discrete, presburger = presburger});

 structure LA_Data_Ref: LIN_ARITH_DATA =

 struct

 (* Decomposition of terms *)

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

   | nT _ = false;

 fun add_atom(t,m,(p,i)) = (case assoc(p,t) of None => ((t,m)::p,i)

                            | Some n => (overwrite(p,(t,ratadd(n,m))), i));

 exception Zero;

 fun rat_of_term(numt,dent) =

   let val num = HOLogic.dest_binum numt and den = HOLogic.dest_binum dent

   in if den = 0 then raise Zero else int_ratdiv(num,den) end;

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

    in which case dest_binum 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...

 *)

 fun number_of_Sucs (Const("Suc",_) $ n) = number_of_Sucs n + 1

   | number_of_Sucs t = if HOLogic.is_zero t then 0

                        else raise TERM("number_of_Sucs",[])

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

    their coefficients

 *)

 fun demult inj_consts =

 let

 fun demult((mC as Const("op *",_)) $ s $ t,m) = ((case s of

         Const("Numeral.number_of",_)$n

         => demult(t,ratmul(m,rat_of_int(HOLogic.dest_binum n)))

       | Const("uminus",_)$(Const("Numeral.number_of",_)$n)

         => demult(t,ratmul(m,rat_of_int(~(HOLogic.dest_binum n))))

       | Const("Suc",_) $ _

         => demult(t,ratmul(m,rat_of_int(number_of_Sucs s)))

       | Const("op *",_) $ s1 $ s2 => demult(mC $ s1 $ (mC $ s2 $ t),m)

       | Const("HOL.divide",_) $ numt $ (Const("Numeral.number_of",_)$dent) =>

           let val den = HOLogic.dest_binum dent

           in if den = 0 then raise Zero

              else demult(mC $ numt $ t,ratmul(m, ratinv(rat_of_int den)))

           end

       | _ => atomult(mC,s,t,m)

       ) handle TERM _ => atomult(mC,s,t,m))

   | demult(atom as Const("HOL.divide",_) $ t $ (Const("Numeral.number_of",_)$dent), m) =

       (let val den = HOLogic.dest_binum dent

        in if den = 0 then raise Zero else demult(t,ratmul(m, ratinv(rat_of_int den))) end

        handle TERM _ => (Some atom,m))

   | demult(Const("0",_),m) = (None, rat_of_int 0)

   | demult(Const("1",_),m) = (None, m)

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

       ((None,ratmul(m,rat_of_int(HOLogic.dest_binum n)))

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

   | demult(Const("uminus",_)$t, m) = demult(t,ratmul(m,rat_of_int(~1)))

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

       (if f mem inj_consts then Some x else Some t,m)

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

 and atomult(mC,atom,t,m) = (case demult(t,m) of (None,m') => (Some atom,m')

                             | (Some t',m') => (Some(mC $ atom $ t'),m'))

 in demult end;

 fun decomp2 inj_consts (rel,lhs,rhs) =

 let

 (* Turn term into list of summand * multiplicity plus a constant *)

 fun poly(Const("op +",_) $ s $ t, m, pi) = poly(s,m,poly(t,m,pi))

   | poly(all as Const("op -",T) $ s $ t, m, pi) =

       if nT T then add_atom(all,m,pi)

       else poly(s,m,poly(t,ratneg m,pi))

   | poly(Const("uminus",_) $ t, m, pi) = poly(t,ratneg m,pi)

   | poly(Const("0",_), _, pi) = pi

   | poly(Const("1",_), m, (p,i)) = (p,ratadd(i,m))

   | poly(Const("Suc",_)$t, m, (p,i)) = poly(t, m, (p,ratadd(i,m)))

   | poly(t as Const("op *",_) $ _ $ _, m, pi as (p,i)) =

       (case demult inj_consts (t,m) of

          (None,m') => (p,ratadd(i,m))

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

   | poly(t as Const("HOL.divide",_) $ _ $ _, m, pi as (p,i)) =

       (case demult inj_consts (t,m) of

          (None,m') => (p,ratadd(i,m'))

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

   | poly(all as (Const("Numeral.number_of",_)$t,m,(p,i))) =

       ((p,ratadd(i,ratmul(m,rat_of_int(HOLogic.dest_binum t))))

        handle TERM _ => add_atom all)

   | 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 x  = add_atom x;

 val (p,i) = poly(lhs,rat_of_int 1,([],rat_of_int 0))

 and (q,j) = poly(rhs,rat_of_int 1,([],rat_of_int 0))

   in case rel of

        "op <"  => Some(p,i,"<",q,j)

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

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

      | _       => None

   end handle Zero => None;

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

   | negate None = None;

 fun decomp1 (discrete,inj_consts) (T,xxx) =

   (case T of

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

        (case assoc(discrete,D) of

           None => None

         | Some d => (case decomp2 inj_consts xxx of

                        None => None

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

    | _ => None);

 fun decomp2 data (_$(Const(rel,T)$lhs$rhs)) = decomp1 data (T,(rel,lhs,rhs))

   | decomp2 data (_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) =

       negate(decomp1 data (T,(rel,lhs,rhs)))

   | decomp2 data _ = None

 fun decomp sg =

   let val {discrete, inj_consts, ...} = ArithTheoryData.get_sg sg

   in decomp2 (discrete,inj_consts) end

 fun number_of(n,T) = HOLogic.number_of_const T $ (HOLogic.mk_bin n)

 end;

 structure Fast_Arith =

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

 val fast_arith_tac    = Fast_Arith.lin_arith_tac false

 and fast_ex_arith_tac = Fast_Arith.lin_arith_tac

 and trace_arith    = Fast_Arith.trace

 and fast_arith_neq_limit = Fast_Arith.fast_arith_neq_limit;

 local

 val isolateSuc =

   let val thy = theory "Nat"

   in prove_goal thy "Suc(i+j) = i+j + Suc 0"

      (fn _ => [simp_tac (simpset_of thy) 1])

   end;

 (* reduce contradictory <= to False.

    Most of the work is done by the cancel tactics.

 *)

 val add_rules =

  [add_zero_left,add_zero_right,Zero_not_Suc,Suc_not_Zero,le_0_eq,

   One_nat_def,isolateSuc];

 val add_mono_thms_ordered_semiring = map (fn s => prove_goal (the_context ()) s

  (fn prems => [cut_facts_tac prems 1,

                blast_tac (claset() addIs [add_mono]) 1]))

 ["(i <= j) & (k <= l) ==> i + k <= j + (l::'a::ordered_semidom)",

  "(i  = j) & (k <= l) ==> i + k <= j + (l::'a::ordered_semidom)",

  "(i <= j) & (k  = l) ==> i + k <= j + (l::'a::ordered_semidom)",

  "(i  = j) & (k  = l) ==> i + k  = j + (l::'a::ordered_semidom)"

 ];

 in

 val init_lin_arith_data =

  Fast_Arith.setup @

  [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset = _} =>

    {add_mono_thms = add_mono_thms @ add_mono_thms_ordered_semiring,

     mult_mono_thms = mult_mono_thms,

     inj_thms = inj_thms,

     lessD = lessD @ [Suc_leI],

     simpset = HOL_basic_ss addsimps add_rules addsimprocs nat_cancel_sums_add}),

   ArithTheoryData.init, arith_discrete ("nat", true)];

 end;

 val fast_nat_arith_simproc =

   Simplifier.simproc (Theory.sign_of (the_context ())) "fast_nat_arith"

     ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] Fast_Arith.lin_arith_prover;

 (* 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. *)

 (* arith proof method *)

 (* FIXME: K true should be replaced by a sensible test 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)

 *)

 local

 fun raw_arith_tac ex i st =

   refute_tac (K true)

    (REPEAT o split_tac (#splits (ArithTheoryData.get_sg (Thm.sign_of_thm st))))

    ((REPEAT_DETERM o etac linorder_neqE) THEN' fast_ex_arith_tac ex)

    i st;

 fun presburger_tac i st =

   (case ArithTheoryData.get_sg (sign_of_thm st) of

      {presburger = Some tac, ...} =>

        (tracing "Simple arithmetic decision procedure failed.\nNow trying full Presburger arithmetic..."; tac i st)

    | _ => no_tac st);

 in

 val simple_arith_tac = FIRST' [fast_arith_tac,

   ObjectLogic.atomize_tac THEN' raw_arith_tac true];

 val arith_tac = FIRST' [fast_arith_tac,

   ObjectLogic.atomize_tac THEN' raw_arith_tac true,

   presburger_tac];

 val silent_arith_tac = FIRST' [fast_arith_tac,

   ObjectLogic.atomize_tac THEN' raw_arith_tac false,

   presburger_tac];

 fun arith_method prems =

   Method.METHOD (fn facts => HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac));

 end;

 (* theory setup *)

 val arith_setup =

  [Simplifier.change_simpset_of (op addsimprocs) nat_cancel_sums] @

   init_lin_arith_data @

   [Simplifier.change_simpset_of (op addSolver)

    (mk_solver "lin. arith." Fast_Arith.cut_lin_arith_tac),

   Simplifier.change_simpset_of (op addsimprocs) [fast_nat_arith_simproc],

   Method.add_methods [("arith", (arith_method o #2) oo Method.syntax Args.bang_facts,

     "decide linear arithmethic")],

   Attrib.add_attributes [("arith_split",

     (Attrib.no_args arith_split_add, Attrib.no_args Attrib.undef_local_attribute),

     "declaration of split rules for arithmetic procedure")]];