(*  Title:      HOL/Integ/int_arith1.ML

    ID:         $Id$

    Authors:    Larry Paulson and Tobias Nipkow

Simprocs and decision procedure for linear arithmetic.

*)

(*** Simprocs for numeric literals ***)

(** Combining of literal coefficients in sums of products **)

Goal "(x < y) = (x-y < (Numeral0::int))";

by (simp_tac (simpset() addsimps zcompare_rls) 1);

qed "zless_iff_zdiff_zless_0";

Goal "(x = y) = (x-y = (Numeral0::int))";

by (simp_tac (simpset() addsimps zcompare_rls) 1);

qed "eq_iff_zdiff_eq_0";

Goal "(x <= y) = (x-y <= (Numeral0::int))";

by (simp_tac (simpset() addsimps zcompare_rls) 1);

qed "zle_iff_zdiff_zle_0";

(** For combine_numerals **)

Goal "i*u + (j*u + k) = (i+j)*u + (k::int)";

by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib]) 1);

qed "left_zadd_zmult_distrib";

(** For cancel_numerals **)

val rel_iff_rel_0_rls = map (inst "y" "?u+?v")

                          [zless_iff_zdiff_zless_0, eq_iff_zdiff_eq_0, 

			   zle_iff_zdiff_zle_0] @

		        map (inst "y" "n")

                          [zless_iff_zdiff_zless_0, eq_iff_zdiff_eq_0, 

			   zle_iff_zdiff_zle_0];

Goal "!!i::int. (i*u + m = j*u + n) = ((i-j)*u + m = n)";

by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@

		                     zadd_ac@rel_iff_rel_0_rls) 1);

qed "eq_add_iff1";

Goal "!!i::int. (i*u + m = j*u + n) = (m = (j-i)*u + n)";

by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@

                                     zadd_ac@rel_iff_rel_0_rls) 1);

qed "eq_add_iff2";

Goal "!!i::int. (i*u + m < j*u + n) = ((i-j)*u + m < n)";

by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@

                                     zadd_ac@rel_iff_rel_0_rls) 1);

qed "less_add_iff1";

Goal "!!i::int. (i*u + m < j*u + n) = (m < (j-i)*u + n)";

by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@

                                     zadd_ac@rel_iff_rel_0_rls) 1);

qed "less_add_iff2";

Goal "!!i::int. (i*u + m <= j*u + n) = ((i-j)*u + m <= n)";

by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@

                                     zadd_ac@rel_iff_rel_0_rls) 1);

qed "le_add_iff1";

Goal "!!i::int. (i*u + m <= j*u + n) = (m <= (j-i)*u + n)";

by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]

                                     @zadd_ac@rel_iff_rel_0_rls) 1);

qed "le_add_iff2";

(*To tidy up the result of a simproc.  Only the RHS will be simplified.*)

Goal "u = u' ==> (t==u) == (t==u')";

by Auto_tac;

qed "eq_cong2";

structure Int_Numeral_Simprocs =

struct

(*Utilities*)

fun mk_numeral n = HOLogic.number_of_const HOLogic.intT $ HOLogic.mk_bin n;

(*Decodes a binary INTEGER*)

fun dest_numeral (Const("Numeral.number_of", _) $ w) = 

     (HOLogic.dest_binum w

      handle TERM _ => raise TERM("Int_Numeral_Simprocs.dest_numeral:1", [w]))

  | dest_numeral t = raise TERM("Int_Numeral_Simprocs.dest_numeral:2", [t]);

fun find_first_numeral past (t::terms) =

	((dest_numeral t, rev past @ terms)

	 handle TERM _ => find_first_numeral (t::past) terms)

  | find_first_numeral past [] = raise TERM("find_first_numeral", []);

val zero = mk_numeral 0;

val mk_plus = HOLogic.mk_binop "op +";

val uminus_const = Const ("uminus", HOLogic.intT --> HOLogic.intT);

(*Thus mk_sum[t] yields t+Numeral0; longer sums don't have a trailing zero*)

fun mk_sum []        = zero

  | mk_sum [t,u]     = mk_plus (t, u)

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

(*this version ALWAYS includes a trailing zero*)

fun long_mk_sum []        = zero

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

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

(*decompose additions AND subtractions as a sum*)

fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =

        dest_summing (pos, t, dest_summing (pos, u, ts))

  | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =

        dest_summing (pos, t, dest_summing (not pos, u, ts))

  | dest_summing (pos, t, ts) =

	if pos then t::ts else uminus_const$t :: ts;

fun dest_sum t = dest_summing (true, t, []);

val mk_diff = HOLogic.mk_binop "op -";

val dest_diff = HOLogic.dest_bin "op -" HOLogic.intT;

val one = mk_numeral 1;

val mk_times = HOLogic.mk_binop "op *";

fun mk_prod [] = one

  | mk_prod [t] = t

  | mk_prod (t :: ts) = if t = one then mk_prod ts

                        else mk_times (t, mk_prod ts);

val dest_times = HOLogic.dest_bin "op *" HOLogic.intT;

fun dest_prod t =

      let val (t,u) = dest_times t 

      in  dest_prod t @ dest_prod u  end

      handle TERM _ => [t];

(*DON'T do the obvious simplifications; that would create special cases*) 

fun mk_coeff (k, ts) = mk_times (mk_numeral k, ts);

(*Express t as a product of (possibly) a numeral with other sorted terms*)

fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t

  | dest_coeff sign t =

    let val ts = sort Term.term_ord (dest_prod t)

	val (n, ts') = find_first_numeral [] ts

                          handle TERM _ => (1, ts)

    in (sign*n, mk_prod ts') end;

(*Find first coefficient-term THAT MATCHES u*)

fun find_first_coeff past u [] = raise TERM("find_first_coeff", []) 

  | find_first_coeff past u (t::terms) =

	let val (n,u') = dest_coeff 1 t

	in  if u aconv u' then (n, rev past @ terms)

			  else find_first_coeff (t::past) u terms

	end

	handle TERM _ => find_first_coeff (t::past) u terms;

(*Simplify Numeral1*n and n*Numeral1 to n*)

val add_0s = [zadd_0, zadd_0_right];

val mult_1s = [zmult_1, zmult_1_right, zmult_minus1, zmult_minus1_right];

(*To perform binary arithmetic*)

val bin_simps = [add_number_of_left] @ bin_arith_simps @ bin_rel_simps;

(*To evaluate binary negations of coefficients*)

val zminus_simps = NCons_simps @

                   [number_of_minus RS sym, 

		    bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,

		    bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];

(*To let us treat subtraction as addition*)

val diff_simps = [zdiff_def, zminus_zadd_distrib, zminus_zminus];

(*push the unary minus down: - x * y = x * - y *)

val int_minus_mult_eq_1_to_2 = 

    [zmult_zminus, zmult_zminus_right RS sym] MRS trans |> standard;

(*to extract again any uncancelled minuses*)

val int_minus_from_mult_simps = 

    [zminus_zminus, zmult_zminus, zmult_zminus_right];

(*combine unary minus with numeric literals, however nested within a product*)

val int_mult_minus_simps =

    [zmult_assoc, zmult_zminus RS sym, int_minus_mult_eq_1_to_2];

(*Apply the given rewrite (if present) just once*)

fun trans_tac None      = all_tac

  | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));

fun prove_conv name tacs sg (hyps: thm list) (t,u) =

  if t aconv u then None

  else

  let val ct = cterm_of sg (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u)))

  in Some

     (prove_goalw_cterm [] ct (K tacs)

      handle ERROR => error 

	  ("The error(s) above occurred while trying to prove " ^

	   string_of_cterm ct ^ "\nInternal failure of simproc " ^ name))

  end;

(*version without the hyps argument*)

fun prove_conv_nohyps name tacs sg = prove_conv name tacs sg [];

fun simplify_meta_eq rules =

    mk_meta_eq o

    simplify (HOL_basic_ss addeqcongs[eq_cong2] addsimps rules)

fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc;

fun prep_pat s = Thm.read_cterm (Theory.sign_of (the_context())) (s, HOLogic.termT);

val prep_pats = map prep_pat;

structure CancelNumeralsCommon =

  struct

  val mk_sum    	= mk_sum

  val dest_sum		= dest_sum

  val mk_coeff		= mk_coeff

  val dest_coeff	= dest_coeff 1

  val find_first_coeff	= find_first_coeff []

  val trans_tac         = trans_tac

  val norm_tac = 

     ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@

                                         zminus_simps@zadd_ac))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@int_mult_minus_simps))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps int_minus_from_mult_simps@

                                              zadd_ac@zmult_ac))

  val numeral_simp_tac	= ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))

  val simplify_meta_eq  = simplify_meta_eq (add_0s@mult_1s)

  end;

structure EqCancelNumerals = CancelNumeralsFun

 (open CancelNumeralsCommon

  val prove_conv = prove_conv "inteq_cancel_numerals"

  val mk_bal   = HOLogic.mk_eq

  val dest_bal = HOLogic.dest_bin "op =" HOLogic.intT

  val bal_add1 = eq_add_iff1 RS trans

  val bal_add2 = eq_add_iff2 RS trans

);

structure LessCancelNumerals = CancelNumeralsFun

 (open CancelNumeralsCommon

  val prove_conv = prove_conv "intless_cancel_numerals"

  val mk_bal   = HOLogic.mk_binrel "op <"

  val dest_bal = HOLogic.dest_bin "op <" HOLogic.intT

  val bal_add1 = less_add_iff1 RS trans

  val bal_add2 = less_add_iff2 RS trans

);

structure LeCancelNumerals = CancelNumeralsFun

 (open CancelNumeralsCommon

  val prove_conv = prove_conv "intle_cancel_numerals"

  val mk_bal   = HOLogic.mk_binrel "op <="

  val dest_bal = HOLogic.dest_bin "op <=" HOLogic.intT

  val bal_add1 = le_add_iff1 RS trans

  val bal_add2 = le_add_iff2 RS trans

);

val cancel_numerals = 

  map prep_simproc

   [("inteq_cancel_numerals",

     prep_pats ["(l::int) + m = n", "(l::int) = m + n", 

		"(l::int) - m = n", "(l::int) = m - n", 

		"(l::int) * m = n", "(l::int) = m * n"], 

     EqCancelNumerals.proc),

    ("intless_cancel_numerals", 

     prep_pats ["(l::int) + m < n", "(l::int) < m + n", 

		"(l::int) - m < n", "(l::int) < m - n", 

		"(l::int) * m < n", "(l::int) < m * n"], 

     LessCancelNumerals.proc),

    ("intle_cancel_numerals", 

     prep_pats ["(l::int) + m <= n", "(l::int) <= m + n", 

		"(l::int) - m <= n", "(l::int) <= m - n", 

		"(l::int) * m <= n", "(l::int) <= m * n"], 

     LeCancelNumerals.proc)];

structure CombineNumeralsData =

  struct

  val add		= op + : int*int -> int 

  val mk_sum    	= long_mk_sum    (*to work for e.g. 2*x + 3*x *)

  val dest_sum		= dest_sum

  val mk_coeff		= mk_coeff

  val dest_coeff	= dest_coeff 1

  val left_distrib	= left_zadd_zmult_distrib RS trans

  val prove_conv        = prove_conv_nohyps "int_combine_numerals"

  val trans_tac          = trans_tac

  val norm_tac = 

     ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@

                                         zminus_simps@zadd_ac))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@int_mult_minus_simps))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps int_minus_from_mult_simps@

                                              zadd_ac@zmult_ac))

  val numeral_simp_tac	= ALLGOALS 

                    (simp_tac (HOL_ss addsimps add_0s@bin_simps))

  val simplify_meta_eq  = simplify_meta_eq (add_0s@mult_1s)

  end;

structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);

val combine_numerals = 

    prep_simproc ("int_combine_numerals",

		  prep_pats ["(i::int) + j", "(i::int) - j"],

		  CombineNumerals.proc);

end;

Addsimprocs Int_Numeral_Simprocs.cancel_numerals;

Addsimprocs [Int_Numeral_Simprocs.combine_numerals];

(*The Abel_Cancel simprocs are now obsolete*)

Delsimprocs [Int_Cancel.sum_conv, Int_Cancel.rel_conv];

(*examples:

print_depth 22;

set timing;

set trace_simp;

fun test s = (Goal s; by (Simp_tac 1)); 

test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";

test "2*u = (u::int)";

test "(i + j + 12 + (k::int)) - 15 = y";

test "(i + j + 12 + (k::int)) - 5 = y";

test "y - b < (b::int)";

test "y - (3*b + c) < (b::int) - 2*c";

test "(2*x - (u*v) + y) - v*3*u = (w::int)";

test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";

test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";

test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";

test "(i + j + 12 + (k::int)) = u + 15 + y";

test "(i + j*2 + 12 + (k::int)) = j + 5 + y";

test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::int)";

test "a + -(b+c) + b = (d::int)";

test "a + -(b+c) - b = (d::int)";

(*negative numerals*)

test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";

test "(i + j + -3 + (k::int)) < u + 5 + y";

test "(i + j + 3 + (k::int)) < u + -6 + y";

test "(i + j + -12 + (k::int)) - 15 = y";

test "(i + j + 12 + (k::int)) - -15 = y";

test "(i + j + -12 + (k::int)) - -15 = y";

*)

(** Constant folding for integer plus and times **)

(*We do not need

    structure Nat_Plus_Assoc = Assoc_Fold (Nat_Plus_Assoc_Data);

    structure Int_Plus_Assoc = Assoc_Fold (Int_Plus_Assoc_Data);

  because combine_numerals does the same thing*)

structure Int_Times_Assoc_Data : ASSOC_FOLD_DATA =

struct

  val ss		= HOL_ss

  val eq_reflection	= eq_reflection

  val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))

  val T	     = HOLogic.intT

  val plus   = Const ("op *", [HOLogic.intT,HOLogic.intT] ---> HOLogic.intT);

  val add_ac = zmult_ac

end;

structure Int_Times_Assoc = Assoc_Fold (Int_Times_Assoc_Data);

Addsimprocs [Int_Times_Assoc.conv];

(** The same for the naturals **)

structure Nat_Times_Assoc_Data : ASSOC_FOLD_DATA =

struct

  val ss		= HOL_ss

  val eq_reflection	= eq_reflection

  val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))

  val T	     = HOLogic.natT

  val plus   = Const ("op *", [HOLogic.natT,HOLogic.natT] ---> HOLogic.natT);

  val add_ac = mult_ac

end;

structure Nat_Times_Assoc = Assoc_Fold (Nat_Times_Assoc_Data);

Addsimprocs [Nat_Times_Assoc.conv];

(*** decision procedure for linear arithmetic ***)

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

(* Linear arithmetic                                                         *)

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

(*

Instantiation of the generic linear arithmetic package for int.

*)

(* Update parameters of arithmetic prover *)

local

(* reduce contradictory <= to False *)

val add_rules = simp_thms @ bin_arith_simps @ bin_rel_simps @

                [zadd_0, zadd_0_right, zdiff_def,

		 zadd_zminus_inverse, zadd_zminus_inverse2, 

		 zmult_0, zmult_0_right, 

		 zmult_1, zmult_1_right, 

		 zmult_minus1, zmult_minus1_right,

		 zminus_zadd_distrib, zminus_zminus, zmult_assoc,

                 Zero_int_def, int_0, zadd_int RS sym, int_Suc];

val simprocs = [Int_Times_Assoc.conv, Int_Numeral_Simprocs.combine_numerals]@

               Int_Numeral_Simprocs.cancel_numerals;

val add_mono_thms_int =

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

                 (fn prems => [cut_facts_tac prems 1,

                      asm_simp_tac (simpset() addsimps [zadd_zle_mono]) 1]))

    ["(i <= j) & (k <= l) ==> i + k <= j + (l::int)",

     "(i  = j) & (k <= l) ==> i + k <= j + (l::int)",

     "(i <= j) & (k  = l) ==> i + k <= j + (l::int)",

     "(i  = j) & (k  = l) ==> i + k  = j + (l::int)"

    ];

in

val int_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_int,

    mult_mono_thms = mult_mono_thms,

    inj_thms = [zle_int RS iffD2,int_int_eq RS iffD2] @ inj_thms,

    lessD = lessD @ [add1_zle_eq RS iffD2],

    simpset = simpset addsimps add_rules

                      addsimprocs simprocs

                      addcongs [if_weak_cong]}),

  arith_inj_const ("IntDef.int", HOLogic.natT --> HOLogic.intT),

  arith_discrete ("IntDef.int", true)];

end;

let

val int_arith_simproc_pats =

  map (fn s => Thm.read_cterm (Theory.sign_of (the_context())) (s, HOLogic.boolT))

      ["(m::int) < n","(m::int) <= n", "(m::int) = n"];

val fast_int_arith_simproc = mk_simproc

  "fast_int_arith" int_arith_simproc_pats Fast_Arith.lin_arith_prover;

in

Addsimprocs [fast_int_arith_simproc]

end;

(* Some test data

Goal "!!a::int. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d";

by (fast_arith_tac 1);

Goal "!!a::int. [| a < b; c < d |] ==> a-d+ 2 <= b+(-c)";

by (fast_arith_tac 1);

Goal "!!a::int. [| a < b; c < d |] ==> a+c+ Numeral1 < b+d";

by (fast_arith_tac 1);

Goal "!!a::int. [| a <= b; b+b <= c |] ==> a+a <= c";

by (fast_arith_tac 1);

Goal "!!a::int. [| a+b <= i+j; a<=b; i<=j |] \

\     ==> a+a <= j+j";

by (fast_arith_tac 1);

Goal "!!a::int. [| a+b < i+j; a<b; i<j |] \

\     ==> a+a - - -1 < j+j - 3";

by (fast_arith_tac 1);

Goal "!!a::int. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k";

by (arith_tac 1);

Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \

\     ==> a <= l";

by (fast_arith_tac 1);

Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \

\     ==> a+a+a+a <= l+l+l+l";

by (fast_arith_tac 1);

Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \

\     ==> a+a+a+a+a <= l+l+l+l+i";

by (fast_arith_tac 1);

Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \

\     ==> a+a+a+a+a+a <= l+l+l+l+i+l";

by (fast_arith_tac 1);

Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \

\     ==> 6*a <= 5*l+i";

by (fast_arith_tac 1);

*)