(* 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 < (#0::int))";
by (simp_tac (simpset() addsimps zcompare_rls) 1);
qed "zless_iff_zdiff_zless_0";
Goal "(x = y) = (x-y = (#0::int))";
by (simp_tac (simpset() addsimps zcompare_rls) 1);
qed "eq_iff_zdiff_eq_0";
Goal "(x <= y) = (x-y <= (#0::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 $
NumeralSyntax.mk_bin n;
(*Decodes a binary INTEGER*)
fun dest_numeral (Const("Numeral.number_of", _) $ w) =
(NumeralSyntax.dest_bin w
handle Match => 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+#0; 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 #1*n and n*#1 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];
(*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 [zmult_zminus_right RS sym]@
bin_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 [zmult_zminus_right RS sym]@
bin_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 @
[int_0, 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];
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, lessD, simpset} =>
{add_mono_thms = add_mono_thms @ add_mono_thms_int,
lessD = lessD @ [add1_zle_eq RS iffD2],
simpset = simpset addsimps add_rules
addsimprocs simprocs
addcongs [if_weak_cong]}),
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+ #1 < 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);
*)