(* Title: ZF/Integ/int_arith.ML
ID: $Id$
Author: Larry Paulson
Copyright 2000 University of Cambridge
Simprocs for linear arithmetic.
*)
(** To simplify inequalities involving integer negation and literals,
such as -x = #3
**)
Addsimps [inst "y" "integ_of(?w)" zminus_equation,
inst "x" "integ_of(?w)" equation_zminus];
AddIffs [inst "y" "integ_of(?w)" zminus_zless,
inst "x" "integ_of(?w)" zless_zminus];
AddIffs [inst "y" "integ_of(?w)" zminus_zle,
inst "x" "integ_of(?w)" zle_zminus];
Addsimps [inst "s" "integ_of(?w)" (thm "Let_def")];
(*** Simprocs for numeric literals ***)
(** Combining of literal coefficients in sums of products **)
Goal "(x $< y) <-> (x$-y $< #0)";
by (simp_tac (simpset() addsimps zcompare_rls) 1);
qed "zless_iff_zdiff_zless_0";
Goal "[| x: int; y: int |] ==> (x = y) <-> (x$-y = #0)";
by (asm_simp_tac (simpset() addsimps zcompare_rls) 1);
qed "eq_iff_zdiff_eq_0";
Goal "(x $<= y) <-> (x$-y $<= #0)";
by (asm_simp_tac (simpset() addsimps zcompare_rls) 1);
qed "zle_iff_zdiff_zle_0";
(** For combine_numerals **)
Goal "i$*u $+ (j$*u $+ k) = (i$+j)$*u $+ k";
by (simp_tac (simpset() addsimps [zadd_zmult_distrib]@zadd_ac) 1);
qed "left_zadd_zmult_distrib";
(** For cancel_numerals **)
val rel_iff_rel_0_rls = map (inst "y" "?u$+?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$*u $+ m = j$*u $+ n) <-> ((i$-j)$*u $+ m = intify(n))";
by (simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]) 1);
by (simp_tac (simpset() addsimps zcompare_rls) 1);
by (simp_tac (simpset() addsimps zadd_ac) 1);
qed "eq_add_iff1";
Goal "(i$*u $+ m = j$*u $+ n) <-> (intify(m) = (j$-i)$*u $+ n)";
by (simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]) 1);
by (simp_tac (simpset() addsimps zcompare_rls) 1);
by (simp_tac (simpset() addsimps zadd_ac) 1);
qed "eq_add_iff2";
Goal "(i$*u $+ m $< j$*u $+ n) <-> ((i$-j)$*u $+ m $< n)";
by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@
zadd_ac@rel_iff_rel_0_rls) 1);
qed "less_add_iff1";
Goal "(i$*u $+ m $< j$*u $+ n) <-> (m $< (j$-i)$*u $+ n)";
by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@
zadd_ac@rel_iff_rel_0_rls) 1);
qed "less_add_iff2";
Goal "(i$*u $+ m $<= j$*u $+ n) <-> ((i$-j)$*u $+ m $<= n)";
by (simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]) 1);
by (simp_tac (simpset() addsimps zcompare_rls) 1);
by (simp_tac (simpset() addsimps zadd_ac) 1);
qed "le_add_iff1";
Goal "(i$*u $+ m $<= j$*u $+ n) <-> (m $<= (j$-i)$*u $+ n)";
by (simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]) 1);
by (simp_tac (simpset() addsimps zcompare_rls) 1);
by (simp_tac (simpset() addsimps zadd_ac) 1);
qed "le_add_iff2";
structure Int_Numeral_Simprocs =
struct
(*Utilities*)
val integ_of_const = Const ("Bin.integ_of", iT --> iT);
fun mk_numeral n = integ_of_const $ NumeralSyntax.mk_bin n;
(*Decodes a binary INTEGER*)
fun dest_numeral (Const("Bin.integ_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 = FOLogic.mk_binop "Int.zadd";
val iT = Ind_Syntax.iT;
val zminus_const = Const ("Int.zminus", iT --> iT);
(*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 = FOLogic.dest_bin "Int.zadd" iT;
(*decompose additions AND subtractions as a sum*)
fun dest_summing (pos, Const ("Int.zadd", _) $ t $ u, ts) =
dest_summing (pos, t, dest_summing (pos, u, ts))
| dest_summing (pos, Const ("Int.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 zminus_const$t :: ts;
fun dest_sum t = dest_summing (true, t, []);
val mk_diff = FOLogic.mk_binop "Int.zdiff";
val dest_diff = FOLogic.dest_bin "Int.zdiff" iT;
val one = mk_numeral 1;
val mk_times = FOLogic.mk_binop "Int.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 "Int.zmult" iT;
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 ("Int.zminus", _) $ 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_intify, zadd_0_right_intify];
val mult_1s = [zmult_1_intify, zmult_1_right_intify,
zmult_minus1, zmult_minus1_right];
val tc_rules = [integ_of_type, intify_in_int,
int_of_type, zadd_type, zdiff_type, zmult_type] @
thms "bin.intros";
val intifys = [intify_ident, zadd_intify1, zadd_intify2,
zdiff_intify1, zdiff_intify2, zmult_intify1, zmult_intify2,
zless_intify1, zless_intify2, zle_intify1, zle_intify2];
(*To perform binary arithmetic*)
val bin_simps = [add_integ_of_left] @ bin_arith_simps @ bin_rel_simps;
(*To evaluate binary negations of coefficients*)
val zminus_simps = NCons_simps @
[integ_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];
fun prep_simproc (name, pats, proc) =
Simplifier.simproc (the_context ()) 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 []
fun trans_tac _ = ArithData.gen_trans_tac iff_trans
val norm_ss1 = ZF_ss addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ 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 @ zadd_ac @ 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 (SIMPSET' (fn simpset => asm_simp_tac (Simplifier.inherit_context ss simpset)))
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 = eq_add_iff1 RS iff_trans
val bal_add2 = eq_add_iff2 RS iff_trans
);
structure LessCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = ArithData.prove_conv "intless_cancel_numerals"
val mk_bal = FOLogic.mk_binrel "Int.zless"
val dest_bal = FOLogic.dest_bin "Int.zless" iT
val bal_add1 = less_add_iff1 RS iff_trans
val bal_add2 = less_add_iff2 RS iff_trans
);
structure LeCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = ArithData.prove_conv "intle_cancel_numerals"
val mk_bal = FOLogic.mk_binrel "Int.zle"
val dest_bal = FOLogic.dest_bin "Int.zle" iT
val bal_add1 = le_add_iff1 RS iff_trans
val bal_add2 = le_add_iff2 RS iff_trans
);
val cancel_numerals =
map prep_simproc
[("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 = IntInf.int
val iszero = (fn x => x = 0)
val add = IntInf.+
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 = left_zadd_zmult_distrib RS trans
val prove_conv = prove_conv_nohyps "int_combine_numerals"
fun trans_tac _ = ArithData.gen_trans_tac trans
val norm_ss1 = ZF_ss addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ 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 @ zadd_ac @ 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 ("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 = IntInf.int
val iszero = (fn x => x = 0)
val add = IntInf.*
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 = zmult_assoc RS sym RS trans
val prove_conv = prove_conv_nohyps "int_combine_numerals_prod"
fun trans_tac _ = ArithData.gen_trans_tac trans
val norm_ss1 = ZF_ss addsimps mult_1s @ diff_simps @ zminus_simps
val norm_ss2 = ZF_ss addsimps [zmult_zminus_right RS sym] @
bin_simps @ 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 ("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 trace_simp;
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";
*)