(* Title: ZF/int_arith.ML
Author: Larry Paulson
Simprocs for linear arithmetic.
*)
signature INT_NUMERAL_SIMPROCS =
sig
val cancel_numerals: simproc list
val combine_numerals: simproc
val combine_numerals_prod: simproc
end
structure Int_Numeral_Simprocs: INT_NUMERAL_SIMPROCS =
struct
(* abstract syntax operations *)
fun mk_bit 0 = @{term "0"}
| mk_bit 1 = @{term "succ(0)"}
| mk_bit _ = raise TERM ("mk_bit", []);
fun dest_bit @{term "0"} = 0
| dest_bit @{term "succ(0)"} = 1
| dest_bit t = raise TERM ("dest_bit", [t]);
fun mk_bin i =
let
fun term_of [] = @{term Pls}
| term_of [~1] = @{term Min}
| term_of (b :: bs) = @{term Bit} $ term_of bs $ mk_bit b;
in term_of (Numeral_Syntax.make_binary i) end;
fun dest_bin tm =
let
fun bin_of @{term Pls} = []
| bin_of @{term Min} = [~1]
| bin_of (@{term Bit} $ bs $ b) = dest_bit b :: bin_of bs
| bin_of _ = raise TERM ("dest_bin", [tm]);
in Numeral_Syntax.dest_binary (bin_of tm) end;
(*Utilities*)
fun mk_numeral i = @{const integ_of} $ mk_bin i;
fun dest_numeral (Const(@{const_name integ_of}, _) $ w) = dest_bin w
| dest_numeral t = raise TERM ("dest_numeral", [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 = FOLogic.mk_binop @{const_name "zadd"};
(*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);
(*decompose additions AND subtractions as a sum*)
fun dest_summing (pos, Const (@{const_name "zadd"}, _) $ t $ u, ts) =
dest_summing (pos, t, dest_summing (pos, u, ts))
| dest_summing (pos, Const (@{const_name "zdiff"}, _) $ t $ u, ts) =
dest_summing (pos, t, dest_summing (not pos, u, ts))
| dest_summing (pos, t, ts) =
if pos then t::ts else @{const zminus} $ t :: ts;
fun dest_sum t = dest_summing (true, t, []);
val one = mk_numeral 1;
val mk_times = FOLogic.mk_binop @{const_name "zmult"};
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 = FOLogic.dest_bin @{const_name "zmult"} @{typ i};
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_numeral k, t);
(*Express t as a product of (possibly) a numeral with other sorted terms*)
fun dest_coeff sign (Const (@{const_name "zminus"}, _) $ t) = dest_coeff (~sign) t
| dest_coeff sign t =
let val ts = sort Term_Ord.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 = [@{thm zadd_0_intify}, @{thm zadd_0_right_intify}];
val mult_1s = [@{thm zmult_1_intify}, @{thm zmult_1_right_intify},
@{thm zmult_minus1}, @{thm zmult_minus1_right}];
val tc_rules = [@{thm integ_of_type}, @{thm intify_in_int},
@{thm int_of_type}, @{thm zadd_type}, @{thm zdiff_type}, @{thm zmult_type}] @
@{thms bin.intros};
val intifys = [@{thm intify_ident}, @{thm zadd_intify1}, @{thm zadd_intify2},
@{thm zdiff_intify1}, @{thm zdiff_intify2}, @{thm zmult_intify1}, @{thm zmult_intify2},
@{thm zless_intify1}, @{thm zless_intify2}, @{thm zle_intify1}, @{thm zle_intify2}];
(*To perform binary arithmetic*)
val bin_simps = [@{thm add_integ_of_left}] @ @{thms bin_arith_simps} @ @{thms bin_rel_simps};
(*To evaluate binary negations of coefficients*)
val zminus_simps = @{thms NCons_simps} @
[@{thm integ_of_minus} RS @{thm sym},
@{thm bin_minus_1}, @{thm bin_minus_0}, @{thm bin_minus_Pls}, @{thm bin_minus_Min},
@{thm bin_pred_1}, @{thm bin_pred_0}, @{thm bin_pred_Pls}, @{thm bin_pred_Min}];
(*To let us treat subtraction as addition*)
val diff_simps = [@{thm zdiff_def}, @{thm zminus_zadd_distrib}, @{thm zminus_zminus}];
(*push the unary minus down*)
val int_minus_mult_eq_1_to_2 = @{lemma "$- w $* z = w $* $- z" by simp};
(*to extract again any uncancelled minuses*)
val int_minus_from_mult_simps =
[@{thm zminus_zminus}, @{thm zmult_zminus}, @{thm zmult_zminus_right}];
(*combine unary minus with numeric literals, however nested within a product*)
val int_mult_minus_simps =
[@{thm zmult_assoc}, @{thm zmult_zminus} RS @{thm sym}, int_minus_mult_eq_1_to_2];
fun prep_simproc thy (name, pats, proc) =
Simplifier.simproc_global thy name pats proc;
structure CancelNumeralsCommon =
struct
val mk_sum = (fn T:typ => 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 = ArithData.gen_trans_tac @{thm iff_trans}
val norm_ss1 = ZF_ss addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ @{thms zadd_ac}
val norm_ss2 = ZF_ss addsimps bin_simps @ int_mult_minus_simps @ intifys
val norm_ss3 = ZF_ss addsimps int_minus_from_mult_simps @ @{thms zadd_ac} @ @{thms zmult_ac} @ tc_rules @ intifys
fun norm_tac ss =
ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2))
THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss3))
val numeral_simp_ss = ZF_ss addsimps add_0s @ bin_simps @ tc_rules @ intifys
fun numeral_simp_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
THEN ALLGOALS (asm_simp_tac (simpset_of (Simplifier.the_context ss)))
val simplify_meta_eq = ArithData.simplify_meta_eq (add_0s @ mult_1s)
end;
structure EqCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = ArithData.prove_conv "inteq_cancel_numerals"
val mk_bal = FOLogic.mk_eq
val dest_bal = FOLogic.dest_eq
val bal_add1 = @{thm eq_add_iff1} RS @{thm iff_trans}
val bal_add2 = @{thm eq_add_iff2} RS @{thm iff_trans}
);
structure LessCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = ArithData.prove_conv "intless_cancel_numerals"
val mk_bal = FOLogic.mk_binrel @{const_name "zless"}
val dest_bal = FOLogic.dest_bin @{const_name "zless"} @{typ i}
val bal_add1 = @{thm less_add_iff1} RS @{thm iff_trans}
val bal_add2 = @{thm less_add_iff2} RS @{thm iff_trans}
);
structure LeCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = ArithData.prove_conv "intle_cancel_numerals"
val mk_bal = FOLogic.mk_binrel @{const_name "zle"}
val dest_bal = FOLogic.dest_bin @{const_name "zle"} @{typ i}
val bal_add1 = @{thm le_add_iff1} RS @{thm iff_trans}
val bal_add2 = @{thm le_add_iff2} RS @{thm iff_trans}
);
val cancel_numerals =
map (prep_simproc @{theory})
[("inteq_cancel_numerals",
["l $+ m = n", "l = m $+ n",
"l $- m = n", "l = m $- n",
"l $* m = n", "l = m $* n"],
K EqCancelNumerals.proc),
("intless_cancel_numerals",
["l $+ m $< n", "l $< m $+ n",
"l $- m $< n", "l $< m $- n",
"l $* m $< n", "l $< m $* n"],
K LessCancelNumerals.proc),
("intle_cancel_numerals",
["l $+ m $<= n", "l $<= m $+ n",
"l $- m $<= n", "l $<= m $- n",
"l $* m $<= n", "l $<= m $* n"],
K LeCancelNumerals.proc)];
(*version without the hyps argument*)
fun prove_conv_nohyps name tacs sg = ArithData.prove_conv name tacs sg [];
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_sum
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff 1
val left_distrib = @{thm left_zadd_zmult_distrib} RS @{thm trans}
val prove_conv = prove_conv_nohyps "int_combine_numerals"
val trans_tac = ArithData.gen_trans_tac @{thm trans}
val norm_ss1 = ZF_ss addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ @{thms zadd_ac} @ intifys
val norm_ss2 = ZF_ss addsimps bin_simps @ int_mult_minus_simps @ intifys
val norm_ss3 = ZF_ss addsimps int_minus_from_mult_simps @ @{thms zadd_ac} @ @{thms zmult_ac} @ tc_rules @ intifys
fun norm_tac ss =
ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2))
THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss3))
val numeral_simp_ss = ZF_ss addsimps add_0s @ bin_simps @ tc_rules @ intifys
fun numeral_simp_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
val simplify_meta_eq = ArithData.simplify_meta_eq (add_0s @ mult_1s)
end;
structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
val combine_numerals =
prep_simproc @{theory}
("int_combine_numerals", ["i $+ j", "i $- j"], K CombineNumerals.proc);
(** Constant folding for integer multiplication **)
(*The trick is to regard products as sums, e.g. #3 $* x $* #4 as
the "sum" of #3, x, #4; the literals are then multiplied*)
structure CombineNumeralsProdData =
struct
type coeff = int
val iszero = (fn x => x = 0)
val add = op *
val mk_sum = (fn T:typ => mk_prod)
val dest_sum = dest_prod
fun mk_coeff(k,t) = if t=one then mk_numeral k
else raise TERM("mk_coeff", [])
fun dest_coeff t = (dest_numeral t, one) (*We ONLY want pure numerals.*)
val left_distrib = @{thm zmult_assoc} RS @{thm sym} RS @{thm trans}
val prove_conv = prove_conv_nohyps "int_combine_numerals_prod"
val trans_tac = ArithData.gen_trans_tac @{thm trans}
val norm_ss1 = ZF_ss addsimps mult_1s @ diff_simps @ zminus_simps
val norm_ss2 = ZF_ss addsimps [@{thm zmult_zminus_right} RS @{thm sym}] @
bin_simps @ @{thms zmult_ac} @ tc_rules @ intifys
fun norm_tac ss =
ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2))
val numeral_simp_ss = ZF_ss addsimps bin_simps @ tc_rules @ intifys
fun numeral_simp_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
val simplify_meta_eq = ArithData.simplify_meta_eq (mult_1s);
end;
structure CombineNumeralsProd = CombineNumeralsFun(CombineNumeralsProdData);
val combine_numerals_prod =
prep_simproc @{theory}
("int_combine_numerals_prod", ["i $* j"], K CombineNumeralsProd.proc);
end;
Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
Addsimprocs [Int_Numeral_Simprocs.combine_numerals,
Int_Numeral_Simprocs.combine_numerals_prod];
(*examples:*)
(*
print_depth 22;
set timing;
set simp_trace;
fun test s = (Goal s; by (Asm_simp_tac 1));
val sg = #sign (rep_thm (topthm()));
val t = FOLogic.dest_Trueprop (Logic.strip_assums_concl(getgoal 1));
val (t,_) = FOLogic.dest_eq t;
(*combine_numerals_prod (products of separate literals) *)
test "#5 $* x $* #3 = y";
test "y2 $+ ?x42 = y $+ y2";
test "oo : int ==> l $+ (l $+ #2) $+ oo = oo";
test "#9$*x $+ y = x$*#23 $+ z";
test "y $+ x = x $+ z";
test "x : int ==> x $+ y $+ z = x $+ z";
test "x : int ==> y $+ (z $+ x) = z $+ x";
test "z : int ==> x $+ y $+ z = (z $+ y) $+ (x $+ w)";
test "z : int ==> x$*y $+ z = (z $+ y) $+ (y$*x $+ w)";
test "#-3 $* x $+ y $<= x $* #2 $+ z";
test "y $+ x $<= x $+ z";
test "x $+ y $+ z $<= x $+ z";
test "y $+ (z $+ x) $< z $+ x";
test "x $+ y $+ z $< (z $+ y) $+ (x $+ w)";
test "x$*y $+ z $< (z $+ y) $+ (y$*x $+ w)";
test "l $+ #2 $+ #2 $+ #2 $+ (l $+ #2) $+ (oo $+ #2) = uu";
test "u : int ==> #2 $* u = u";
test "(i $+ j $+ #12 $+ k) $- #15 = y";
test "(i $+ j $+ #12 $+ k) $- #5 = y";
test "y $- b $< b";
test "y $- (#3 $* b $+ c) $< b $- #2 $* c";
test "(#2 $* x $- (u $* v) $+ y) $- v $* #3 $* u = w";
test "(#2 $* x $* u $* v $+ (u $* v) $* #4 $+ y) $- v $* u $* #4 = w";
test "(#2 $* x $* u $* v $+ (u $* v) $* #4 $+ y) $- v $* u = w";
test "u $* v $- (x $* u $* v $+ (u $* v) $* #4 $+ y) = w";
test "(i $+ j $+ #12 $+ k) = u $+ #15 $+ y";
test "(i $+ j $* #2 $+ #12 $+ k) = 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";
test "a $+ $-(b$+c) $+ b = d";
test "a $+ $-(b$+c) $- b = d";
(*negative numerals*)
test "(i $+ j $+ #-2 $+ k) $- (u $+ #5 $+ y) = zz";
test "(i $+ j $+ #-3 $+ k) $< u $+ #5 $+ y";
test "(i $+ j $+ #3 $+ k) $< u $+ #-6 $+ y";
test "(i $+ j $+ #-12 $+ k) $- #15 = y";
test "(i $+ j $+ #12 $+ k) $- #-15 = y";
test "(i $+ j $+ #-12 $+ k) $- #-15 = y";
(*Multiplying separated numerals*)
Goal "#6 $* ($# x $* #2) = uu";
Goal "#4 $* ($# x $* $# x) $* (#2 $* $# x) = uu";
*)