(* Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Simprocs for nat numerals.
*)
signature NAT_NUMERAL_SIMPROCS =
sig
val combine_numerals: simproc
val cancel_numerals: simproc list
val cancel_factors: simproc list
val cancel_numeral_factors: simproc list
end;
structure Nat_Numeral_Simprocs =
struct
(*Maps n to #n for n = 0, 1, 2*)
val numeral_syms = [@{thm nat_numeral_0_eq_0} RS sym, @{thm nat_numeral_1_eq_1} RS sym, @{thm numeral_2_eq_2} RS sym];
val numeral_sym_ss = HOL_ss addsimps numeral_syms;
fun rename_numerals th =
simplify numeral_sym_ss (Thm.transfer @{theory} th);
(*Utilities*)
fun mk_number n = HOLogic.number_of_const HOLogic.natT $ HOLogic.mk_numeral n;
fun dest_number t = Int.max (0, snd (HOLogic.dest_number t));
fun find_first_numeral past (t::terms) =
((dest_number t, t, rev past @ terms)
handle TERM _ => find_first_numeral (t::past) terms)
| find_first_numeral past [] = raise TERM("find_first_numeral", []);
val zero = mk_number 0;
val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
(*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 [] = HOLogic.zero
| long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} HOLogic.natT;
(** Other simproc items **)
val bin_simps =
[@{thm nat_numeral_0_eq_0} RS sym, @{thm nat_numeral_1_eq_1} RS sym,
@{thm add_nat_number_of}, @{thm nat_number_of_add_left},
@{thm diff_nat_number_of}, @{thm le_number_of_eq_not_less},
@{thm mult_nat_number_of}, @{thm nat_number_of_mult_left},
@{thm less_nat_number_of},
@{thm Let_number_of}, @{thm nat_number_of}] @
@{thms arith_simps} @ @{thms rel_simps} @ @{thms neg_simps};
(*** CancelNumerals simprocs ***)
val one = mk_number 1;
val mk_times = HOLogic.mk_binop @{const_name HOL.times};
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 @{const_name HOL.times} HOLogic.natT;
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 k, t);
(*Express t as a product of (possibly) a numeral with other factors, sorted*)
fun dest_coeff t =
let val ts = sort TermOrd.term_ord (dest_prod t)
val (n, _, ts') = find_first_numeral [] ts
handle TERM _ => (1, one, ts)
in (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 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;
(*Split up a sum into the list of its constituent terms, on the way removing any
Sucs and counting them.*)
fun dest_Suc_sum (Const ("Suc", _) $ t, (k,ts)) = dest_Suc_sum (t, (k+1,ts))
| dest_Suc_sum (t, (k,ts)) =
let val (t1,t2) = dest_plus t
in dest_Suc_sum (t1, dest_Suc_sum (t2, (k,ts))) end
handle TERM _ => (k, t::ts);
(*Code for testing whether numerals are already used in the goal*)
fun is_numeral (Const(@{const_name Int.number_of}, _) $ w) = true
| is_numeral _ = false;
fun prod_has_numeral t = exists is_numeral (dest_prod t);
(*The Sucs found in the term are converted to a binary numeral. If relaxed is false,
an exception is raised unless the original expression contains at least one
numeral in a coefficient position. This prevents nat_combine_numerals from
introducing numerals to goals.*)
fun dest_Sucs_sum relaxed t =
let val (k,ts) = dest_Suc_sum (t,(0,[]))
in
if relaxed orelse exists prod_has_numeral ts then
if k=0 then ts
else mk_number k :: ts
else raise TERM("Nat_Numeral_Simprocs.dest_Sucs_sum", [t])
end;
(*Simplify 1*n and n*1 to n*)
val add_0s = map rename_numerals [@{thm add_0}, @{thm add_0_right}];
val mult_1s = map rename_numerals [@{thm nat_mult_1}, @{thm nat_mult_1_right}];
(*Final simplification: cancel + and *; replace Numeral0 by 0 and Numeral1 by 1*)
(*And these help the simproc return False when appropriate, which helps
the arith prover.*)
val contra_rules = [@{thm add_Suc}, @{thm add_Suc_right}, @{thm Zero_not_Suc},
@{thm Suc_not_Zero}, @{thm le_0_eq}];
val simplify_meta_eq =
Arith_Data.simplify_meta_eq
([@{thm nat_numeral_0_eq_0}, @{thm numeral_1_eq_Suc_0}, @{thm add_0}, @{thm add_0_right},
@{thm mult_0}, @{thm mult_0_right}, @{thm mult_1}, @{thm mult_1_right}] @ contra_rules);
(*** Applying CancelNumeralsFun ***)
structure CancelNumeralsCommon =
struct
val mk_sum = (fn T:typ => mk_sum)
val dest_sum = dest_Sucs_sum true
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff
val find_first_coeff = find_first_coeff []
fun trans_tac _ = Arith_Data.trans_tac
val norm_ss1 = Numeral_Simprocs.num_ss addsimps numeral_syms @ add_0s @ mult_1s @
[@{thm Suc_eq_plus1_left}] @ @{thms add_ac}
val norm_ss2 = Numeral_Simprocs.num_ss addsimps bin_simps @ @{thms add_ac} @ @{thms mult_ac}
fun norm_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
val numeral_simp_ss = HOL_ss addsimps add_0s @ bin_simps;
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss));
val simplify_meta_eq = simplify_meta_eq
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 =" HOLogic.natT
val bal_add1 = @{thm nat_eq_add_iff1} RS trans
val bal_add2 = @{thm nat_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} HOLogic.natT
val bal_add1 = @{thm nat_less_add_iff1} RS trans
val bal_add2 = @{thm nat_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} HOLogic.natT
val bal_add1 = @{thm nat_le_add_iff1} RS trans
val bal_add2 = @{thm nat_le_add_iff2} RS trans
);
structure DiffCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binop @{const_name HOL.minus}
val dest_bal = HOLogic.dest_bin @{const_name HOL.minus} HOLogic.natT
val bal_add1 = @{thm nat_diff_add_eq1} RS trans
val bal_add2 = @{thm nat_diff_add_eq2} RS trans
);
val cancel_numerals =
map (Arith_Data.prep_simproc @{theory})
[("nateq_cancel_numerals",
["(l::nat) + m = n", "(l::nat) = m + n",
"(l::nat) * m = n", "(l::nat) = m * n",
"Suc m = n", "m = Suc n"],
K EqCancelNumerals.proc),
("natless_cancel_numerals",
["(l::nat) + m < n", "(l::nat) < m + n",
"(l::nat) * m < n", "(l::nat) < m * n",
"Suc m < n", "m < Suc n"],
K LessCancelNumerals.proc),
("natle_cancel_numerals",
["(l::nat) + m <= n", "(l::nat) <= m + n",
"(l::nat) * m <= n", "(l::nat) <= m * n",
"Suc m <= n", "m <= Suc n"],
K LeCancelNumerals.proc),
("natdiff_cancel_numerals",
["((l::nat) + m) - n", "(l::nat) - (m + n)",
"(l::nat) * m - n", "(l::nat) - m * n",
"Suc m - n", "m - Suc n"],
K DiffCancelNumerals.proc)];
(*** Applying CombineNumeralsFun ***)
structure CombineNumeralsData =
struct
type coeff = int
val iszero = (fn x => x = 0)
val add = op +
val mk_sum = (fn T:typ => long_mk_sum) (*to work for 2*x + 3*x *)
val dest_sum = dest_Sucs_sum false
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff
val left_distrib = @{thm left_add_mult_distrib} RS trans
val prove_conv = Arith_Data.prove_conv_nohyps
fun trans_tac _ = Arith_Data.trans_tac
val norm_ss1 = Numeral_Simprocs.num_ss addsimps numeral_syms @ add_0s @ mult_1s @ [@{thm Suc_eq_plus1}] @ @{thms add_ac}
val norm_ss2 = Numeral_Simprocs.num_ss addsimps bin_simps @ @{thms add_ac} @ @{thms mult_ac}
fun norm_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
val numeral_simp_ss = HOL_ss addsimps add_0s @ bin_simps;
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
val simplify_meta_eq = simplify_meta_eq
end;
structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
val combine_numerals =
Arith_Data.prep_simproc @{theory}
("nat_combine_numerals", ["(i::nat) + j", "Suc (i + j)"], K CombineNumerals.proc);
(*** Applying CancelNumeralFactorFun ***)
structure CancelNumeralFactorCommon =
struct
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff
fun trans_tac _ = Arith_Data.trans_tac
val norm_ss1 = Numeral_Simprocs.num_ss addsimps
numeral_syms @ add_0s @ mult_1s @ [@{thm Suc_eq_plus1_left}] @ @{thms add_ac}
val norm_ss2 = Numeral_Simprocs.num_ss addsimps bin_simps @ @{thms add_ac} @ @{thms mult_ac}
fun norm_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
val numeral_simp_ss = HOL_ss addsimps bin_simps
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
val simplify_meta_eq = simplify_meta_eq
end
structure DivCancelNumeralFactor = CancelNumeralFactorFun
(open CancelNumeralFactorCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binop @{const_name Divides.div}
val dest_bal = HOLogic.dest_bin @{const_name Divides.div} HOLogic.natT
val cancel = @{thm nat_mult_div_cancel1} RS trans
val neg_exchanges = false
)
structure DvdCancelNumeralFactor = CancelNumeralFactorFun
(open CancelNumeralFactorCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binrel @{const_name Ring_and_Field.dvd}
val dest_bal = HOLogic.dest_bin @{const_name Ring_and_Field.dvd} HOLogic.natT
val cancel = @{thm nat_mult_dvd_cancel1} RS trans
val neg_exchanges = false
)
structure EqCancelNumeralFactor = CancelNumeralFactorFun
(open CancelNumeralFactorCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT
val cancel = @{thm nat_mult_eq_cancel1} RS trans
val neg_exchanges = false
)
structure LessCancelNumeralFactor = CancelNumeralFactorFun
(open CancelNumeralFactorCommon
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} HOLogic.natT
val cancel = @{thm nat_mult_less_cancel1} RS trans
val neg_exchanges = true
)
structure LeCancelNumeralFactor = CancelNumeralFactorFun
(open CancelNumeralFactorCommon
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} HOLogic.natT
val cancel = @{thm nat_mult_le_cancel1} RS trans
val neg_exchanges = true
)
val cancel_numeral_factors =
map (Arith_Data.prep_simproc @{theory})
[("nateq_cancel_numeral_factors",
["(l::nat) * m = n", "(l::nat) = m * n"],
K EqCancelNumeralFactor.proc),
("natless_cancel_numeral_factors",
["(l::nat) * m < n", "(l::nat) < m * n"],
K LessCancelNumeralFactor.proc),
("natle_cancel_numeral_factors",
["(l::nat) * m <= n", "(l::nat) <= m * n"],
K LeCancelNumeralFactor.proc),
("natdiv_cancel_numeral_factors",
["((l::nat) * m) div n", "(l::nat) div (m * n)"],
K DivCancelNumeralFactor.proc),
("natdvd_cancel_numeral_factors",
["((l::nat) * m) dvd n", "(l::nat) dvd (m * n)"],
K DvdCancelNumeralFactor.proc)];
(*** Applying ExtractCommonTermFun ***)
(*this version ALWAYS includes a trailing one*)
fun long_mk_prod [] = one
| long_mk_prod (t :: ts) = mk_times (t, mk_prod ts);
(*Find first term that matches u*)
fun find_first_t past u [] = raise TERM("find_first_t", [])
| find_first_t past u (t::terms) =
if u aconv t then (rev past @ terms)
else find_first_t (t::past) u terms
handle TERM _ => find_first_t (t::past) u terms;
(** Final simplification for the CancelFactor simprocs **)
val simplify_one = Arith_Data.simplify_meta_eq
[@{thm mult_1_left}, @{thm mult_1_right}, @{thm div_1}, @{thm numeral_1_eq_Suc_0}];
fun cancel_simplify_meta_eq ss cancel_th th =
simplify_one ss (([th, cancel_th]) MRS trans);
structure CancelFactorCommon =
struct
val mk_sum = (fn T:typ => long_mk_prod)
val dest_sum = dest_prod
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff
val find_first = find_first_t []
fun trans_tac _ = Arith_Data.trans_tac
val norm_ss = HOL_ss addsimps mult_1s @ @{thms mult_ac}
fun norm_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss))
val simplify_meta_eq = cancel_simplify_meta_eq
end;
structure EqCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT
fun simp_conv _ _ = SOME @{thm nat_mult_eq_cancel_disj}
);
structure LessCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
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} HOLogic.natT
fun simp_conv _ _ = SOME @{thm nat_mult_less_cancel_disj}
);
structure LeCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
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} HOLogic.natT
fun simp_conv _ _ = SOME @{thm nat_mult_le_cancel_disj}
);
structure DivideCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binop @{const_name Divides.div}
val dest_bal = HOLogic.dest_bin @{const_name Divides.div} HOLogic.natT
fun simp_conv _ _ = SOME @{thm nat_mult_div_cancel_disj}
);
structure DvdCancelFactor = ExtractCommonTermFun
(open CancelFactorCommon
val prove_conv = Arith_Data.prove_conv
val mk_bal = HOLogic.mk_binrel @{const_name Ring_and_Field.dvd}
val dest_bal = HOLogic.dest_bin @{const_name Ring_and_Field.dvd} HOLogic.natT
fun simp_conv _ _ = SOME @{thm nat_mult_dvd_cancel_disj}
);
val cancel_factor =
map (Arith_Data.prep_simproc @{theory})
[("nat_eq_cancel_factor",
["(l::nat) * m = n", "(l::nat) = m * n"],
K EqCancelFactor.proc),
("nat_less_cancel_factor",
["(l::nat) * m < n", "(l::nat) < m * n"],
K LessCancelFactor.proc),
("nat_le_cancel_factor",
["(l::nat) * m <= n", "(l::nat) <= m * n"],
K LeCancelFactor.proc),
("nat_divide_cancel_factor",
["((l::nat) * m) div n", "(l::nat) div (m * n)"],
K DivideCancelFactor.proc),
("nat_dvd_cancel_factor",
["((l::nat) * m) dvd n", "(l::nat) dvd (m * n)"],
K DvdCancelFactor.proc)];
end;
Addsimprocs Nat_Numeral_Simprocs.cancel_numerals;
Addsimprocs [Nat_Numeral_Simprocs.combine_numerals];
Addsimprocs Nat_Numeral_Simprocs.cancel_numeral_factors;
Addsimprocs Nat_Numeral_Simprocs.cancel_factor;
(*examples:
print_depth 22;
set timing;
set trace_simp;
fun test s = (Goal s; by (Simp_tac 1));
(*cancel_numerals*)
test "l +( 2) + (2) + 2 + (l + 2) + (oo + 2) = (uu::nat)";
test "(2*length xs < 2*length xs + j)";
test "(2*length xs < length xs * 2 + j)";
test "2*u = (u::nat)";
test "2*u = Suc (u)";
test "(i + j + 12 + (k::nat)) - 15 = y";
test "(i + j + 12 + (k::nat)) - 5 = y";
test "Suc u - 2 = y";
test "Suc (Suc (Suc u)) - 2 = y";
test "(i + j + 2 + (k::nat)) - 1 = y";
test "(i + j + 1 + (k::nat)) - 2 = y";
test "(2*x + (u*v) + y) - v*3*u = (w::nat)";
test "(2*x*u*v + 5 + (u*v)*4 + y) - v*u*4 = (w::nat)";
test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::nat)";
test "Suc (Suc (2*x*u*v + u*4 + y)) - u = w";
test "Suc ((u*v)*4) - v*3*u = w";
test "Suc (Suc ((u*v)*3)) - v*3*u = w";
test "(i + j + 12 + (k::nat)) = u + 15 + y";
test "(i + j + 32 + (k::nat)) - (u + 15 + y) = zz";
test "(i + j + 12 + (k::nat)) = u + 5 + y";
(*Suc*)
test "(i + j + 12 + k) = Suc (u + y)";
test "Suc (Suc (Suc (Suc (Suc (u + y))))) <= ((i + j) + 41 + k)";
test "(i + j + 5 + k) < Suc (Suc (Suc (Suc (Suc (u + y)))))";
test "Suc (Suc (Suc (Suc (Suc (u + y))))) - 5 = v";
test "(i + j + 5 + k) = Suc (Suc (Suc (Suc (Suc (Suc (Suc (u + y)))))))";
test "2*y + 3*z + 2*u = Suc (u)";
test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = Suc (u)";
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::nat)";
test "6 + 2*y + 3*z + 4*u = Suc (vv + 2*u + z)";
test "(2*n*m) < (3*(m*n)) + (u::nat)";
test "(Suc (Suc (Suc (Suc (Suc (Suc (case length (f c) of 0 => 0 | Suc k => k)))))) <= Suc 0)";
test "Suc (Suc (Suc (Suc (Suc (Suc (length l1 + length l2)))))) <= length l1";
test "( (Suc (Suc (Suc (Suc (Suc (length (compT P E A ST mxr e) + length l3)))))) <= length (compT P E A ST mxr e))";
test "( (Suc (Suc (Suc (Suc (Suc (length (compT P E A ST mxr e) + length (compT P E (A Un \<A> e) ST mxr c))))))) <= length (compT P E A ST mxr e))";
(*negative numerals: FAIL*)
test "(i + j + -23 + (k::nat)) < u + 15 + y";
test "(i + j + 3 + (k::nat)) < u + -15 + y";
test "(i + j + -12 + (k::nat)) - 15 = y";
test "(i + j + 12 + (k::nat)) - -15 = y";
test "(i + j + -12 + (k::nat)) - -15 = y";
(*combine_numerals*)
test "k + 3*k = (u::nat)";
test "Suc (i + 3) = u";
test "Suc (i + j + 3 + k) = u";
test "k + j + 3*k + j = (u::nat)";
test "Suc (j*i + i + k + 5 + 3*k + i*j*4) = (u::nat)";
test "(2*n*m) + (3*(m*n)) = (u::nat)";
(*negative numerals: FAIL*)
test "Suc (i + j + -3 + k) = u";
(*cancel_numeral_factors*)
test "9*x = 12 * (y::nat)";
test "(9*x) div (12 * (y::nat)) = z";
test "9*x < 12 * (y::nat)";
test "9*x <= 12 * (y::nat)";
(*cancel_factor*)
test "x*k = k*(y::nat)";
test "k = k*(y::nat)";
test "a*(b*c) = (b::nat)";
test "a*(b*c) = d*(b::nat)*(x*a)";
test "x*k < k*(y::nat)";
test "k < k*(y::nat)";
test "a*(b*c) < (b::nat)";
test "a*(b*c) < d*(b::nat)*(x*a)";
test "x*k <= k*(y::nat)";
test "k <= k*(y::nat)";
test "a*(b*c) <= (b::nat)";
test "a*(b*c) <= d*(b::nat)*(x*a)";
test "(x*k) div (k*(y::nat)) = (uu::nat)";
test "(k) div (k*(y::nat)) = (uu::nat)";
test "(a*(b*c)) div ((b::nat)) = (uu::nat)";
test "(a*(b*c)) div (d*(b::nat)*(x*a)) = (uu::nat)";
*)