(* Authors: Larry Paulson and Tobias Nipkow
Simprocs and decision procedure for numerals and linear arithmetic.
*)
structure Int_Numeral_Simprocs =
struct
(*reorientation simprules using ==, for the following simproc*)
val meta_zero_reorient = @{thm zero_reorient} RS eq_reflection
val meta_one_reorient = @{thm one_reorient} RS eq_reflection
val meta_number_of_reorient = @{thm number_of_reorient} RS eq_reflection
(*reorientation simplification procedure: reorients (polymorphic)
0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a Int.*)
fun reorient_proc sg _ (_ $ t $ u) =
case u of
Const(@{const_name HOL.zero}, _) => NONE
| Const(@{const_name HOL.one}, _) => NONE
| Const(@{const_name Int.number_of}, _) $ _ => NONE
| _ => SOME (case t of
Const(@{const_name HOL.zero}, _) => meta_zero_reorient
| Const(@{const_name HOL.one}, _) => meta_one_reorient
| Const(@{const_name Int.number_of}, _) $ _ => meta_number_of_reorient)
val reorient_simproc =
Arith_Data.prep_simproc ("reorient_simproc", ["0=x", "1=x", "number_of w = x"], reorient_proc);
(** Utilities **)
fun mk_number T n = HOLogic.number_of_const T $ HOLogic.mk_numeral n;
fun find_first_numeral past (t::terms) =
((snd (HOLogic.dest_number t), rev past @ terms)
handle TERM _ => find_first_numeral (t::past) terms)
| find_first_numeral past [] = raise TERM("find_first_numeral", []);
val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
fun mk_minus t =
let val T = Term.fastype_of t
in Const (@{const_name HOL.uminus}, T --> T) $ t end;
(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
fun mk_sum T [] = mk_number T 0
| mk_sum T [t,u] = mk_plus (t, u)
| mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
(*this version ALWAYS includes a trailing zero*)
fun long_mk_sum T [] = mk_number T 0
| long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT;
(*decompose additions AND subtractions as a sum*)
fun dest_summing (pos, Const (@{const_name HOL.plus}, _) $ t $ u, ts) =
dest_summing (pos, t, dest_summing (pos, u, ts))
| dest_summing (pos, Const (@{const_name HOL.minus}, _) $ t $ u, ts) =
dest_summing (pos, t, dest_summing (not pos, u, ts))
| dest_summing (pos, t, ts) =
if pos then t::ts else mk_minus t :: ts;
fun dest_sum t = dest_summing (true, t, []);
val mk_diff = HOLogic.mk_binop @{const_name HOL.minus};
val dest_diff = HOLogic.dest_bin @{const_name HOL.minus} Term.dummyT;
val mk_times = HOLogic.mk_binop @{const_name HOL.times};
fun one_of T = Const(@{const_name HOL.one},T);
(* build product with trailing 1 rather than Numeral 1 in order to avoid the
unnecessary restriction to type class number_ring
which is not required for cancellation of common factors in divisions.
*)
fun mk_prod T =
let val one = one_of T
fun mk [] = one
| mk [t] = t
| mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
in mk end;
(*This version ALWAYS includes a trailing one*)
fun long_mk_prod T [] = one_of T
| long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
val dest_times = HOLogic.dest_bin @{const_name HOL.times} Term.dummyT;
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, t) = mk_times (mk_number (Term.fastype_of t) k, t);
(*Express t as a product of (possibly) a numeral with other sorted terms*)
fun dest_coeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_coeff (~sign) t
| dest_coeff sign t =
let val ts = sort TermOrd.term_ord (dest_prod t)
val (n, ts') = find_first_numeral [] ts
handle TERM _ => (1, ts)
in (sign*n, mk_prod (Term.fastype_of t) 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;
(*Fractions as pairs of ints. Can't use Rat.rat because the representation
needs to preserve negative values in the denominator.*)
fun mk_frac (p, q) = if q = 0 then raise Div else (p, q);
(*Don't reduce fractions; sums must be proved by rule add_frac_eq.
Fractions are reduced later by the cancel_numeral_factor simproc.*)
fun add_frac ((p1, q1), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2);
val mk_divide = HOLogic.mk_binop @{const_name HOL.divide};
(*Build term (p / q) * t*)
fun mk_fcoeff ((p, q), t) =
let val T = Term.fastype_of t
in mk_times (mk_divide (mk_number T p, mk_number T q), t) end;
(*Express t as a product of a fraction with other sorted terms*)
fun dest_fcoeff sign (Const (@{const_name HOL.uminus}, _) $ t) = dest_fcoeff (~sign) t
| dest_fcoeff sign (Const (@{const_name HOL.divide}, _) $ t $ u) =
let val (p, t') = dest_coeff sign t
val (q, u') = dest_coeff 1 u
in (mk_frac (p, q), mk_divide (t', u')) end
| dest_fcoeff sign t =
let val (p, t') = dest_coeff sign t
val T = Term.fastype_of t
in (mk_frac (p, 1), mk_divide (t', one_of T)) end;
(** New term ordering so that AC-rewriting brings numerals to the front **)
(*Order integers by absolute value and then by sign. The standard integer
ordering is not well-founded.*)
fun num_ord (i,j) =
(case int_ord (abs i, abs j) of
EQUAL => int_ord (Int.sign i, Int.sign j)
| ord => ord);
(*This resembles TermOrd.term_ord, but it puts binary numerals before other
non-atomic terms.*)
local open Term
in
fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) =
(case numterm_ord (t, u) of EQUAL => TermOrd.typ_ord (T, U) | ord => ord)
| numterm_ord
(Const(@{const_name Int.number_of}, _) $ v, Const(@{const_name Int.number_of}, _) $ w) =
num_ord (HOLogic.dest_numeral v, HOLogic.dest_numeral w)
| numterm_ord (Const(@{const_name Int.number_of}, _) $ _, _) = LESS
| numterm_ord (_, Const(@{const_name Int.number_of}, _) $ _) = GREATER
| numterm_ord (t, u) =
(case int_ord (size_of_term t, size_of_term u) of
EQUAL =>
let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
(case TermOrd.hd_ord (f, g) of EQUAL => numterms_ord (ts, us) | ord => ord)
end
| ord => ord)
and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)
end;
fun numtermless tu = (numterm_ord tu = LESS);
val num_ss = HOL_ss settermless numtermless;
(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic isn't complicated by the abstract 0 and 1.*)
val numeral_syms = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym];
(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
val add_0s = @{thms add_0s};
val mult_1s = @{thms mult_1s mult_1_left mult_1_right divide_1};
(*Simplify inverse Numeral1, a/Numeral1*)
val inverse_1s = [@{thm inverse_numeral_1}];
val divide_1s = [@{thm divide_numeral_1}];
(*To perform binary arithmetic. The "left" rewriting handles patterns
created by the Int_Numeral_Simprocs, such as 3 * (5 * x). *)
val simps = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym,
@{thm add_number_of_left}, @{thm mult_number_of_left}] @
@{thms arith_simps} @ @{thms rel_simps};
(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
during re-arrangement*)
val non_add_simps =
subtract Thm.eq_thm [@{thm add_number_of_left}, @{thm number_of_add} RS sym] simps;
(*To evaluate binary negations of coefficients*)
val minus_simps = [@{thm numeral_m1_eq_minus_1} RS sym, @{thm number_of_minus} RS sym] @
@{thms minus_bin_simps} @ @{thms pred_bin_simps};
(*To let us treat subtraction as addition*)
val diff_simps = [@{thm diff_minus}, @{thm minus_add_distrib}, @{thm minus_minus}];
(*To let us treat division as multiplication*)
val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];
(*push the unary minus down: - x * y = x * - y *)
val minus_mult_eq_1_to_2 =
[@{thm mult_minus_left}, @{thm minus_mult_right}] MRS trans |> standard;
(*to extract again any uncancelled minuses*)
val minus_from_mult_simps =
[@{thm minus_minus}, @{thm mult_minus_left}, @{thm mult_minus_right}];
(*combine unary minus with numeric literals, however nested within a product*)
val mult_minus_simps =
[@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2];
val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
diff_simps @ minus_simps @ @{thms add_ac}
val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
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 = K Arith_Data.trans_tac
fun norm_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
end;
structure EqCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
val bal_add1 = @{thm eq_add_iff1} RS trans
val bal_add2 = @{thm eq_add_iff2} RS trans
);
structure LessCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binrel @{const_name HOL.less}
val dest_bal = HOLogic.dest_bin @{const_name HOL.less} Term.dummyT
val bal_add1 = @{thm less_add_iff1} RS trans
val bal_add2 = @{thm less_add_iff2} RS trans
);
structure LeCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binrel @{const_name HOL.less_eq}
val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} Term.dummyT
val bal_add1 = @{thm le_add_iff1} RS trans
val bal_add2 = @{thm le_add_iff2} RS trans
);
val cancel_numerals =
map Arith_Data.prep_simproc
[("inteq_cancel_numerals",
["(l::'a::number_ring) + m = n",
"(l::'a::number_ring) = m + n",
"(l::'a::number_ring) - m = n",
"(l::'a::number_ring) = m - n",
"(l::'a::number_ring) * m = n",
"(l::'a::number_ring) = m * n"],
K EqCancelNumerals.proc),
("intless_cancel_numerals",
["(l::'a::{ordered_idom,number_ring}) + m < n",
"(l::'a::{ordered_idom,number_ring}) < m + n",
"(l::'a::{ordered_idom,number_ring}) - m < n",
"(l::'a::{ordered_idom,number_ring}) < m - n",
"(l::'a::{ordered_idom,number_ring}) * m < n",
"(l::'a::{ordered_idom,number_ring}) < m * n"],
K LessCancelNumerals.proc),
("intle_cancel_numerals",
["(l::'a::{ordered_idom,number_ring}) + m <= n",
"(l::'a::{ordered_idom,number_ring}) <= m + n",
"(l::'a::{ordered_idom,number_ring}) - m <= n",
"(l::'a::{ordered_idom,number_ring}) <= m - n",
"(l::'a::{ordered_idom,number_ring}) * m <= n",
"(l::'a::{ordered_idom,number_ring}) <= m * n"],
K LeCancelNumerals.proc)];
structure CombineNumeralsData =
struct
type coeff = int
val iszero = (fn x => x = 0)
val add = op +
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 = @{thm combine_common_factor} RS trans
val prove_conv = Arith_Data.prove_conv_nohyps
val trans_tac = K Arith_Data.trans_tac
fun norm_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
end;
structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
(*Version for fields, where coefficients can be fractions*)
structure FieldCombineNumeralsData =
struct
type coeff = int * int
val iszero = (fn (p, q) => p = 0)
val add = add_frac
val mk_sum = long_mk_sum
val dest_sum = dest_sum
val mk_coeff = mk_fcoeff
val dest_coeff = dest_fcoeff 1
val left_distrib = @{thm combine_common_factor} RS trans
val prove_conv = Arith_Data.prove_conv_nohyps
val trans_tac = K Arith_Data.trans_tac
val norm_ss1a = norm_ss1 addsimps inverse_1s @ divide_simps
fun norm_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1a))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
val numeral_simp_ss = HOL_ss addsimps add_0s @ simps @ [@{thm add_frac_eq}]
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s @ divide_1s)
end;
structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);
val combine_numerals =
Arith_Data.prep_simproc
("int_combine_numerals",
["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"],
K CombineNumerals.proc);
val field_combine_numerals =
Arith_Data.prep_simproc
("field_combine_numerals",
["(i::'a::{number_ring,field,division_by_zero}) + j",
"(i::'a::{number_ring,field,division_by_zero}) - j"],
K FieldCombineNumerals.proc);
(** Constant folding for multiplication in semirings **)
(*We do not need folding for addition: combine_numerals does the same thing*)
structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
struct
val assoc_ss = HOL_ss addsimps @{thms mult_ac}
val eq_reflection = eq_reflection
fun is_numeral (Const(@{const_name Int.number_of}, _) $ _) = true
| is_numeral _ = false;
end;
structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
val assoc_fold_simproc =
Arith_Data.prep_simproc
("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],
K Semiring_Times_Assoc.proc);
end;
Addsimprocs [Int_Numeral_Simprocs.reorient_simproc];
Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
Addsimprocs [Int_Numeral_Simprocs.field_combine_numerals];
Addsimprocs [Int_Numeral_Simprocs.assoc_fold_simproc];
(*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";
*)
(*** decision procedure for linear arithmetic ***)
(*---------------------------------------------------------------------------*)
(* Linear arithmetic *)
(*---------------------------------------------------------------------------*)
(*
Instantiation of the generic linear arithmetic package for int.
*)
structure Int_Arith =
struct
(* Update parameters of arithmetic prover *)
(* reduce contradictory =/</<= to False *)
(* Evaluation of terms of the form "m R n" where R is one of "=", "<=" or "<",
and m and n are ground terms over rings (roughly speaking).
That is, m and n consist only of 1s combined with "+", "-" and "*".
*)
val zeroth = (symmetric o mk_meta_eq) @{thm of_int_0};
val lhss0 = [@{cpat "0::?'a::ring"}];
fun proc0 phi ss ct =
let val T = ctyp_of_term ct
in if typ_of T = @{typ int} then NONE else
SOME (instantiate' [SOME T] [] zeroth)
end;
val zero_to_of_int_zero_simproc =
make_simproc {lhss = lhss0, name = "zero_to_of_int_zero_simproc",
proc = proc0, identifier = []};
val oneth = (symmetric o mk_meta_eq) @{thm of_int_1};
val lhss1 = [@{cpat "1::?'a::ring_1"}];
fun proc1 phi ss ct =
let val T = ctyp_of_term ct
in if typ_of T = @{typ int} then NONE else
SOME (instantiate' [SOME T] [] oneth)
end;
val one_to_of_int_one_simproc =
make_simproc {lhss = lhss1, name = "one_to_of_int_one_simproc",
proc = proc1, identifier = []};
val allowed_consts =
[@{const_name "op ="}, @{const_name "HOL.times"}, @{const_name "HOL.uminus"},
@{const_name "HOL.minus"}, @{const_name "HOL.plus"},
@{const_name "HOL.zero"}, @{const_name "HOL.one"}, @{const_name "HOL.less"},
@{const_name "HOL.less_eq"}];
fun check t = case t of
Const(s,t) => if s = @{const_name "HOL.one"} then not (t = @{typ int})
else s mem_string allowed_consts
| a$b => check a andalso check b
| _ => false;
val conv =
Simplifier.rewrite
(HOL_basic_ss addsimps
((map (fn th => th RS sym) [@{thm of_int_add}, @{thm of_int_mult},
@{thm of_int_diff}, @{thm of_int_minus}])@
[@{thm of_int_less_iff}, @{thm of_int_le_iff}, @{thm of_int_eq_iff}])
addsimprocs [zero_to_of_int_zero_simproc,one_to_of_int_one_simproc]);
fun sproc phi ss ct = if check (term_of ct) then SOME (conv ct) else NONE
val lhss' =
[@{cpat "(?x::?'a::ring_char_0) = (?y::?'a)"},
@{cpat "(?x::?'a::ordered_idom) < (?y::?'a)"},
@{cpat "(?x::?'a::ordered_idom) <= (?y::?'a)"}]
val zero_one_idom_simproc =
make_simproc {lhss = lhss' , name = "zero_one_idom_simproc",
proc = sproc, identifier = []}
val add_rules =
simp_thms @ @{thms arith_simps} @ @{thms rel_simps} @ @{thms arith_special} @
@{thms int_arith_rules}
val nat_inj_thms = [@{thm zle_int} RS iffD2, @{thm int_int_eq} RS iffD2]
val int_numeral_base_simprocs = Int_Numeral_Simprocs.assoc_fold_simproc :: zero_one_idom_simproc
:: Int_Numeral_Simprocs.combine_numerals
:: Int_Numeral_Simprocs.cancel_numerals;
val setup =
Lin_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
{add_mono_thms = add_mono_thms,
mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} :: mult_mono_thms,
inj_thms = nat_inj_thms @ inj_thms,
lessD = lessD @ [@{thm zless_imp_add1_zle}],
neqE = neqE,
simpset = simpset addsimps add_rules
addsimprocs int_numeral_base_simprocs
addcongs [if_weak_cong]}) #>
arith_inj_const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) #>
arith_discrete @{type_name Int.int}
val fast_int_arith_simproc =
Simplifier.simproc (the_context ())
"fast_int_arith"
["(m::'a::{ordered_idom,number_ring}) < n",
"(m::'a::{ordered_idom,number_ring}) <= n",
"(m::'a::{ordered_idom,number_ring}) = n"] (K Lin_Arith.lin_arith_simproc);
end;
Addsimprocs [Int_Arith.fast_int_arith_simproc];