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z3/src/smt/seq_axioms.cpp
Nikolaj Bjorner 3e9479d01a a lot of seq churn 
Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>
2020-04-17 18:21:40 -07:00

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/*++
Copyright (c) 2011 Microsoft Corporation
Module Name:
seq_axioms.cpp
Abstract:
Axiomatize string operations that can be reduced to
more basic operations. These axioms are kept outside
of a particular solver: they are mainly solver independent.
Author:
Nikolaj Bjorner (nbjorner) 2020-4-16
Revision History:
--*/
#include "ast/ast_pp.h"
#include "ast/ast_ll_pp.h"
#include "smt/seq_axioms.h"
#include "smt/smt_context.h"
using namespace smt;
seq_axioms::seq_axioms(theory& th, th_rewriter& r):
th(th),
m_rewrite(r),
m(r.m()),
a(m),
seq(m),
m_sk(m, r),
m_digits_initialized(false)
{}
literal seq_axioms::mk_eq(expr* a, expr* b) {
return th.mk_eq(a, b, false);
}
expr_ref seq_axioms::mk_sub(expr* x, expr* y) {
expr_ref result(a.mk_sub(x, y), m);
m_rewrite(result);
return result;
}
expr_ref seq_axioms::mk_len(expr* s) {
expr_ref result(seq.str.mk_length(s), m);
m_rewrite(result);
return result;
}
literal seq_axioms::mk_literal(expr* _e) {
expr_ref e(_e, m);
if (a.is_arith_expr(e)) {
m_rewrite(e);
}
th.ensure_enode(e);
return ctx().get_literal(e);
}
/*
let e = extract(s, i, l)
i is start index, l is length of substring starting at index.
i < 0 => e = ""
i >= |s| => e = ""
l <= 0 => e = ""
0 <= i < |s| & l > 0 => s = xey, |x| = i, |e| = min(l, |s|-i)
l <= 0 => e = ""
this translates to:
0 <= i <= |s| -> s = xey
0 <= i <= |s| -> len(x) = i
0 <= i <= |s| & 0 <= l <= |s| - i -> |e| = l
0 <= i <= |s| & |s| < l + i -> |e| = |s| - i
|e| = 0 <=> i < 0 | |s| <= i | l <= 0 | |s| <= 0
It follows that:
|e| = min(l, |s| - i) for 0 <= i < |s| and 0 < |l|
*/
void seq_axioms::add_extract_axiom(expr* e) {
TRACE("seq", tout << mk_pp(e, m) << "\n";);
expr* s = nullptr, *i = nullptr, *l = nullptr;
VERIFY(seq.str.is_extract(e, s, i, l));
if (is_tail(s, i, l)) {
add_tail_axiom(e, s);
return;
}
if (is_drop_last(s, i, l)) {
add_drop_last_axiom(e, s);
return;
}
if (is_extract_prefix0(s, i, l)) {
add_extract_prefix_axiom(e, s, l);
return;
}
if (is_extract_suffix(s, i, l)) {
add_extract_suffix_axiom(e, s, i);
return;
}
expr_ref x = m_sk.mk_pre(s, i);
expr_ref ls = mk_len(s);
expr_ref lx = mk_len(x);
expr_ref le = mk_len(e);
expr_ref ls_minus_i_l(mk_sub(mk_sub(ls, i), l), m);
expr_ref y = m_sk.mk_post(s, a.mk_add(i, l));
expr_ref xe = mk_concat(x, e);
expr_ref xey = mk_concat(x, e, y);
expr_ref zero(a.mk_int(0), m);
literal i_ge_0 = mk_literal(a.mk_ge(i, zero));
literal i_le_ls = mk_literal(a.mk_le(mk_sub(i, ls), zero));
literal ls_le_i = mk_literal(a.mk_le(mk_sub(ls, i), zero));
literal ls_ge_li = mk_literal(a.mk_ge(ls_minus_i_l, zero));
literal l_ge_0 = mk_literal(a.mk_ge(l, zero));
literal l_le_0 = mk_literal(a.mk_le(l, zero));
literal ls_le_0 = mk_literal(a.mk_le(ls, zero));
literal le_is_0 = mk_eq(le, zero);
// 0 <= i & i <= |s| & 0 <= l => xey = s
// 0 <= i & i <= |s| => |x| = i
// 0 <= i & i <= |s| & l >= 0 & |s| >= l + i => |e| = l
// 0 <= i & i <= |s| & |s| < l + i => |e| = |s| - i
// i < 0 => |e| = 0
// |s| <= i => |e| = 0
// |s| <= 0 => |e| = 0
// l <= 0 => |e| = 0
// |e| = 0 & i >= 0 & |s| > i & |s| > 0 => l <= 0
add_axiom(~i_ge_0, ~i_le_ls, ~l_ge_0, mk_seq_eq(xey, s));
add_axiom(~i_ge_0, ~i_le_ls, mk_eq(lx, i));
add_axiom(~i_ge_0, ~i_le_ls, ~l_ge_0, ~ls_ge_li, mk_eq(le, l));
add_axiom(~i_ge_0, ~i_le_ls, ~l_ge_0, ls_ge_li, mk_eq(le, mk_sub(ls, i)));
add_axiom(i_ge_0, le_is_0);
add_axiom(~ls_le_i, le_is_0);
add_axiom(~ls_le_0, le_is_0);
add_axiom(~l_le_0, le_is_0);
add_axiom(~le_is_0, ~i_ge_0, ls_le_i, ls_le_0, l_le_0);
}
void seq_axioms::add_tail_axiom(expr* e, expr* s) {
expr_ref head(m), tail(m);
m_sk.decompose(s, head, tail);
TRACE("seq", tout << "tail " << mk_bounded_pp(e, m, 2) << " " << mk_bounded_pp(s, m, 2) << "\n";);
literal emp = mk_eq_empty(s);
add_axiom(emp, mk_seq_eq(s, mk_concat(head, e)));
add_axiom(~emp, mk_eq_empty(e));
}
void seq_axioms::add_drop_last_axiom(expr* e, expr* s) {
TRACE("seq", tout << "drop last " << mk_bounded_pp(e, m, 2) << " " << mk_bounded_pp(s, m, 2) << "\n";);
literal emp = mk_eq_empty(s);
add_axiom(emp, mk_seq_eq(s, mk_concat(e, seq.str.mk_unit(m_sk.mk_last(s)))));
add_axiom(~emp, mk_eq_empty(e));
}
bool seq_axioms::is_drop_last(expr* s, expr* i, expr* l) {
rational i1;
if (!a.is_numeral(i, i1) || !i1.is_zero()) {
return false;
}
expr_ref l2(m), l1(l, m);
l2 = mk_sub(mk_len(s), a.mk_int(1));
m_rewrite(l1);
m_rewrite(l2);
return l1 == l2;
}
bool seq_axioms::is_tail(expr* s, expr* i, expr* l) {
rational i1;
if (!a.is_numeral(i, i1) || !i1.is_one()) {
return false;
}
expr_ref l2(m), l1(l, m);
l2 = mk_sub(mk_len(s), a.mk_int(1));
m_rewrite(l1);
m_rewrite(l2);
return l1 == l2;
}
bool seq_axioms::is_extract_prefix0(expr* s, expr* i, expr* l) {
rational i1;
return a.is_numeral(i, i1) && i1.is_zero();
}
bool seq_axioms::is_extract_suffix(expr* s, expr* i, expr* l) {
expr_ref len(a.mk_add(l, i), m);
m_rewrite(len);
return seq.str.is_length(len, l) && l == s;
}
/*
0 <= l <= len(s) => s = ey & l = len(e)
len(s) < l => s = e
l < 0 => e = empty
*/
void seq_axioms::add_extract_prefix_axiom(expr* e, expr* s, expr* l) {
TRACE("seq", tout << "prefix " << mk_bounded_pp(e, m, 2) << " " << mk_bounded_pp(s, m, 2) << " " << mk_bounded_pp(l, m, 2) << "\n";);
expr_ref le = mk_len(e);
expr_ref ls = mk_len(s);
expr_ref ls_minus_l(mk_sub(ls, l), m);
expr_ref zero(a.mk_int(0), m);
expr_ref y = m_sk.mk_post(s, l);
expr_ref ey = mk_concat(e, y);
literal l_ge_0 = mk_literal(a.mk_ge(l, zero));
literal l_le_s = mk_literal(a.mk_le(mk_sub(l, ls), zero));
add_axiom(~l_ge_0, ~l_le_s, mk_seq_eq(s, ey));
add_axiom(~l_ge_0, ~l_le_s, mk_eq(l, le));
add_axiom(~l_ge_0, ~l_le_s, mk_eq(ls_minus_l, mk_len(y)));
add_axiom(l_le_s, mk_eq(e, s));
add_axiom(l_ge_0, mk_eq_empty(e));
}
/*
0 <= i <= len(s) => s = xe & i = len(x)
i < 0 => e = empty
i > len(s) => e = empty
*/
void seq_axioms::add_extract_suffix_axiom(expr* e, expr* s, expr* i) {
TRACE("seq", tout << "suffix " << mk_bounded_pp(e, m, 2) << " " << mk_bounded_pp(s, m, 2) << "\n";);
expr_ref x = m_sk.mk_pre(s, i);
expr_ref lx = mk_len(x);
expr_ref ls = mk_len(s);
expr_ref zero(a.mk_int(0), m);
expr_ref xe = mk_concat(x, e);
literal le_is_0 = mk_eq_empty(e);
literal i_ge_0 = mk_literal(a.mk_ge(i, zero));
literal i_le_s = mk_literal(a.mk_le(mk_sub(i, ls), zero));
add_axiom(~i_ge_0, ~i_le_s, mk_seq_eq(s, xe));
add_axiom(~i_ge_0, ~i_le_s, mk_eq(i, lx));
add_axiom(i_ge_0, le_is_0);
add_axiom(i_le_s, le_is_0);
}
/*
encode that s is not contained in of xs1
where s1 is all of s, except the last element.
s = "" or s = s1*(unit c)
s = "" or !contains(x*s1, s)
*/
void seq_axioms::tightest_prefix(expr* s, expr* x) {
expr_ref s1 = m_sk.mk_first(s);
expr_ref c = m_sk.mk_last(s);
expr_ref s1c = mk_concat(s1, seq.str.mk_unit(c));
literal s_eq_emp = mk_eq_empty(s);
add_axiom(s_eq_emp, mk_seq_eq(s, s1c));
add_axiom(s_eq_emp, ~mk_literal(seq.str.mk_contains(mk_concat(x, s1), s)));
}
/*
[[str.indexof]](w, w2, i) is the smallest n such that for some some w1, w3
- w = w1w2w3
- i <= n = |w1|
if [[str.contains]](w, w2) = true, |w2| > 0 and i >= 0.
[[str.indexof]](w,w2,i) = -1 otherwise.
let i = Index(t, s, offset):
// index of s in t starting at offset.
|t| = 0 => |s| = 0 or indexof(t,s,offset) = -1
|t| = 0 & |s| = 0 => indexof(t,s,offset) = 0
offset >= len(t) => |s| = 0 or i = -1
len(t) != 0 & !contains(t, s) => i = -1
offset = 0 & len(t) != 0 & contains(t, s) => t = xsy & i = len(x)
tightest_prefix(x, s)
0 <= offset < len(t) => xy = t &
len(x) = offset &
(-1 = indexof(y, s, 0) => -1 = i) &
(indexof(y, s, 0) >= 0 => indexof(t, s, 0) + offset = i)
offset < 0 => i = -1
optional lemmas:
(len(s) > len(t) -> i = -1)
(len(s) <= len(t) -> i <= len(t)-len(s))
*/
void seq_axioms::add_indexof_axiom(expr* i) {
expr* s = nullptr, *t = nullptr, *offset = nullptr;
rational r;
VERIFY(seq.str.is_index(i, t, s) ||
seq.str.is_index(i, t, s, offset));
expr_ref minus_one(a.mk_int(-1), m);
expr_ref zero(a.mk_int(0), m);
expr_ref xsy(m);
literal cnt = mk_literal(seq.str.mk_contains(t, s));
literal i_eq_m1 = mk_eq(i, minus_one);
literal i_eq_0 = mk_eq(i, zero);
literal s_eq_empty = mk_eq_empty(s);
literal t_eq_empty = mk_eq_empty(t);
// |t| = 0 => |s| = 0 or indexof(t,s,offset) = -1
// ~contains(t,s) <=> indexof(t,s,offset) = -1
add_axiom(cnt, i_eq_m1);
add_axiom(~t_eq_empty, s_eq_empty, i_eq_m1);
if (!offset || (a.is_numeral(offset, r) && r.is_zero())) {
expr_ref x = m_sk.mk_indexof_left(t, s);
expr_ref y = m_sk.mk_indexof_right(t, s);
xsy = mk_concat(x, s, y);
expr_ref lenx = mk_len(x);
// |s| = 0 => indexof(t,s,0) = 0
// contains(t,s) & |s| != 0 => t = xsy & indexof(t,s,0) = |x|
add_axiom(~s_eq_empty, i_eq_0);
add_axiom(~cnt, s_eq_empty, mk_seq_eq(t, xsy));
add_axiom(~cnt, s_eq_empty, mk_eq(i, lenx));
add_axiom(~cnt, mk_literal(a.mk_ge(i, zero)));
tightest_prefix(s, x);
}
else {
// offset >= len(t) => |s| = 0 or indexof(t, s, offset) = -1
// offset > len(t) => indexof(t, s, offset) = -1
// offset = len(t) & |s| = 0 => indexof(t, s, offset) = offset
expr_ref len_t = mk_len(t);
literal offset_ge_len = mk_literal(a.mk_ge(mk_sub(offset, len_t), zero));
literal offset_le_len = mk_literal(a.mk_le(mk_sub(offset, len_t), zero));
literal i_eq_offset = mk_eq(i, offset);
add_axiom(~offset_ge_len, s_eq_empty, i_eq_m1);
add_axiom(offset_le_len, i_eq_m1);
add_axiom(~offset_ge_len, ~offset_le_len, ~s_eq_empty, i_eq_offset);
expr_ref x = m_sk.mk_indexof_left(t, s, offset);
expr_ref y = m_sk.mk_indexof_right(t, s, offset);
expr_ref indexof0(seq.str.mk_index(y, s, zero), m);
expr_ref offset_p_indexof0(a.mk_add(offset, indexof0), m);
literal offset_ge_0 = mk_literal(a.mk_ge(offset, zero));
// 0 <= offset & offset < len(t) => t = xy
// 0 <= offset & offset < len(t) => len(x) = offset
// 0 <= offset & offset < len(t) & indexof(y,s,0) = -1 => -1 = i
// 0 <= offset & offset < len(t) & indexof(y,s,0) >= 0 =>
// -1 = indexof(y,s,0) + offset = indexof(t, s, offset)
add_axiom(~offset_ge_0, offset_ge_len, mk_seq_eq(t, mk_concat(x, y)));
add_axiom(~offset_ge_0, offset_ge_len, mk_eq(mk_len(x), offset));
add_axiom(~offset_ge_0, offset_ge_len,
~mk_eq(indexof0, minus_one), i_eq_m1);
add_axiom(~offset_ge_0, offset_ge_len,
~mk_literal(a.mk_ge(indexof0, zero)),
mk_eq(offset_p_indexof0, i));
// offset < 0 => -1 = i
add_axiom(offset_ge_0, i_eq_m1);
}
}
/**
!contains(t, s) => i = -1
|t| = 0 => |s| = 0 or i = -1
|t| = 0 & |s| = 0 => i = 0
|t| != 0 & contains(t, s) => t = xsy & i = len(x)
|s| = 0 or s = s_head*s_tail
|s| = 0 or !contains(s_tail*y, s)
*/
void seq_axioms::add_last_indexof_axiom(expr* i) {
expr* s = nullptr, *t = nullptr;
VERIFY(seq.str.is_last_index(i, t, s));
expr_ref minus_one(a.mk_int(-1), m);
expr_ref zero(a.mk_int(0), m);
expr_ref s_head(m), s_tail(m);
expr_ref x = m_sk.mk_last_indexof_left(t, s);
expr_ref y = m_sk.mk_last_indexof_right(t, s);
m_sk.decompose(s, s_head, s_tail);
literal cnt = mk_literal(seq.str.mk_contains(t, s));
literal cnt2 = mk_literal(seq.str.mk_contains(mk_concat(s_tail, y), s));
literal i_eq_m1 = mk_eq(i, minus_one);
literal i_eq_0 = mk_eq(i, zero);
literal s_eq_empty = mk_eq_empty(s);
literal t_eq_empty = mk_eq_empty(t);
expr_ref xsy = mk_concat(x, s, y);
add_axiom(cnt, i_eq_m1);
add_axiom(~t_eq_empty, s_eq_empty, i_eq_m1);
add_axiom(~t_eq_empty, ~s_eq_empty, i_eq_0);
add_axiom(t_eq_empty, ~cnt, mk_seq_eq(t, xsy));
add_axiom(t_eq_empty, ~cnt, mk_eq(i, mk_len(x)));
add_axiom(s_eq_empty, mk_eq(s, mk_concat(s_head, s_tail)));
add_axiom(s_eq_empty, ~cnt2);
}
/*
let r = replace(a, s, t)
a = "" => s = "" or r = a
contains(a, s) or r = a
s = "" => r = t+a
tightest_prefix(s, x)
(contains(a, s) -> r = xty & a = xsy) &
(!contains(a, s) -> r = a)
*/
void seq_axioms::add_replace_axiom(expr* r) {
expr* u = nullptr, *s = nullptr, *t = nullptr;
VERIFY(seq.str.is_replace(r, u, s, t));
expr_ref x = m_sk.mk_indexof_left(u, s);
expr_ref y = m_sk.mk_indexof_right(u, s);
expr_ref xty = mk_concat(x, t, y);
expr_ref xsy = mk_concat(x, s, y);
literal a_emp = mk_eq_empty(u, true);
literal s_emp = mk_eq_empty(u, true);
literal cnt = mk_literal(seq.str.mk_contains(u, s));
add_axiom(~a_emp, s_emp, mk_seq_eq(r, u));
add_axiom(cnt, mk_seq_eq(r, u));
add_axiom(~s_emp, mk_seq_eq(r, mk_concat(t, u)));
add_axiom(~cnt, a_emp, s_emp, mk_seq_eq(u, xsy));
add_axiom(~cnt, a_emp, s_emp, mk_seq_eq(r, xty));
ctx().force_phase(cnt);
tightest_prefix(s, x);
}
/*
let e = at(s, i)
0 <= i < len(s) -> s = xey & len(x) = i & len(e) = 1
i < 0 \/ i >= len(s) -> e = empty
*/
void seq_axioms::add_at_axiom(expr* e) {
TRACE("seq", tout << "at-axiom: " << ctx().get_scope_level() << " " << mk_bounded_pp(e, m) << "\n";);
expr* s = nullptr, *i = nullptr;
VERIFY(seq.str.is_at(e, s, i));
expr_ref zero(a.mk_int(0), m);
expr_ref one(a.mk_int(1), m);
expr_ref emp(seq.str.mk_empty(m.get_sort(e)), m);
expr_ref len_s = mk_len(s);
literal i_ge_0 = mk_literal(a.mk_ge(i, zero));
literal i_ge_len_s = mk_literal(a.mk_ge(mk_sub(i, mk_len(s)), zero));
expr_ref len_e = mk_len(e);
rational iv;
if (a.is_numeral(i, iv) && iv.is_unsigned()) {
expr_ref_vector es(m);
expr_ref nth(m);
unsigned k = iv.get_unsigned();
for (unsigned j = 0; j <= k; ++j) {
es.push_back(seq.str.mk_unit(mk_nth(s, a.mk_int(j))));
}
nth = es.back();
es.push_back(m_sk.mk_tail(s, i));
add_axiom(~i_ge_0, i_ge_len_s, mk_seq_eq(s, seq.str.mk_concat(es)));
add_axiom(~i_ge_0, i_ge_len_s, mk_seq_eq(nth, e));
}
else {
expr_ref x = m_sk.mk_pre(s, i);
expr_ref y = m_sk.mk_tail(s, i);
expr_ref xey = mk_concat(x, e, y);
expr_ref len_x = mk_len(x);
add_axiom(~i_ge_0, i_ge_len_s, mk_seq_eq(s, xey));
add_axiom(~i_ge_0, i_ge_len_s, mk_eq(i, len_x));
}
add_axiom(i_ge_0, mk_eq(e, emp));
add_axiom(~i_ge_len_s, mk_eq(e, emp));
add_axiom(~i_ge_0, i_ge_len_s, mk_eq(one, len_e));
add_axiom(mk_literal(a.mk_le(len_e, one)));
}
/**
i >= 0 i < len(s) => unit(nth_i(s, i)) = at(s, i)
nth_i(unit(nth_i(s, i)), 0) = nth_i(s, i)
*/
void seq_axioms::add_nth_axiom(expr* e) {
expr* s = nullptr, *i = nullptr;
rational n;
zstring str;
VERIFY(seq.str.is_nth_i(e, s, i));
if (seq.str.is_string(s, str) && a.is_numeral(i, n) &&
n.is_unsigned() && n.get_unsigned() < str.length()) {
app_ref ch(seq.str.mk_char(str[n.get_unsigned()]), m);
add_axiom(mk_eq(ch, e));
}
else {
expr_ref zero(a.mk_int(0), m);
literal i_ge_0 = mk_literal(a.mk_ge(i, zero));
literal i_ge_len_s = mk_literal(a.mk_ge(mk_sub(i, mk_len(s)), zero));
// at(s,i) = [nth(s,i)]
expr_ref rhs(s, m);
expr_ref lhs(seq.str.mk_unit(e), m);
if (!seq.str.is_at(s) || zero != i) rhs = seq.str.mk_at(s, i);
m_rewrite(rhs);
add_axiom(~i_ge_0, i_ge_len_s, mk_eq(lhs, rhs));
}
}
void seq_axioms::add_itos_axiom(expr* e) {
expr* n = nullptr;
TRACE("seq", tout << mk_pp(e, m) << "\n";);
VERIFY(seq.str.is_itos(e, n));
// itos(n) = "" <=> n < 0
expr_ref zero(a.mk_int(0), m);
literal eq1 = mk_literal(seq.str.mk_is_empty(e));
literal ge0 = mk_literal(a.mk_ge(n, zero));
// n >= 0 => itos(n) != ""
// itos(n) = "" or n >= 0
add_axiom(~eq1, ~ge0);
add_axiom(eq1, ge0);
add_axiom(mk_literal(a.mk_ge(mk_len(e), zero)));
// n >= 0 => stoi(itos(n)) = n
app_ref stoi(seq.str.mk_stoi(e), m);
add_axiom(~ge0, th.mk_preferred_eq(stoi, n));
// itos(n) does not start with "0" when n > 0
// n = 0 or at(itos(n),0) != "0"
// alternative: n >= 0 => itos(stoi(itos(n))) = itos(n)
expr_ref zs(seq.str.mk_string(symbol("0")), m);
m_rewrite(zs);
literal eq0 = mk_eq(n, zero);
literal at0 = mk_eq(seq.str.mk_at(e, zero), zs);
add_axiom(eq0, ~at0);
add_axiom(~eq0, mk_eq(e, zs));
}
/**
stoi(s) >= -1
stoi("") = -1
*/
void seq_axioms::add_stoi_axiom(expr* e) {
TRACE("seq", tout << mk_pp(e, m) << "\n";);
expr* s = nullptr;
VERIFY (seq.str.is_stoi(e, s));
add_axiom(mk_literal(a.mk_ge(e, a.mk_int(-1))));
add_axiom(~mk_literal(seq.str.mk_is_empty(s)), mk_eq(seq.str.mk_stoi(s), a.mk_int(-1)));
}
/**
stoi(s) >= 0 =>
s != empty
s = unit(head) + tail
stoi(s) = 10*digit(head) + stoi(tail) or tail = empty
stoi(s) = digit(head) or tail != empty
is_digit(head)
(tail = empty or stoi(tail) >= 0)
*/
void seq_axioms::add_stoi_non_empty_axiom(expr* e) {
expr* s = nullptr;
VERIFY (seq.str.is_stoi(e, s));
expr_ref head(m), tail(m);
m_sk.decompose(s, head, tail);
expr_ref first_char = mk_nth(s, a.mk_int(0));
literal ge0 = mk_literal(a.mk_ge(e, a.mk_int(0)));
literal tail_empty = mk_eq_empty(tail);
expr_ref first_digit = m_sk.mk_digit2int(first_char);
expr_ref stoi_tail(seq.str.mk_stoi(tail), m);
add_axiom(~ge0, ~mk_literal(seq.str.mk_is_empty(s)));
add_axiom(~ge0, mk_seq_eq(s, mk_concat(head, tail)));
add_axiom(~ge0, tail_empty, mk_eq(a.mk_add(a.mk_mul(a.mk_int(10), first_digit), stoi_tail), e));
add_axiom(~ge0, ~tail_empty, mk_eq(first_digit, e));
add_axiom(~ge0, is_digit(first_char));
add_axiom(~ge0, tail_empty, mk_literal(a.mk_ge(stoi_tail, a.mk_int(0))));
}
/**
e1 < e2 => prefix(e1, e2) or e1 = xcy e1 < e2 => prefix(e1, e2) or
c < d e1 < e2 => prefix(e1, e2) or e2 = xdz e1 < e2 => e1 != e2
!(e1 < e2) => prefix(e2, e1) or e2 = xdz !(e1 < e2) => prefix(e2,
e1) or d < c !(e1 < e2) => prefix(e2, e1) or e1 = xcy !(e1 = e2) or
!(e1 < e2) optional: e1 < e2 or e1 = e2 or e2 < e1 !(e1 = e2) or
!(e2 < e1) !(e1 < e2) or !(e2 < e1)
*/
void seq_axioms::add_lt_axiom(expr* n) {
expr* e1 = nullptr, *e2 = nullptr;
VERIFY(seq.str.is_lt(n, e1, e2));
sort* s = m.get_sort(e1);
sort* char_sort = nullptr;
VERIFY(seq.is_seq(s, char_sort));
literal lt = mk_literal(n);
expr_ref x = m_sk.mk(symbol("str.lt.x"), e1, e2);
expr_ref y = m_sk.mk(symbol("str.lt.y"), e1, e2);
expr_ref z = m_sk.mk(symbol("str.lt.z"), e1, e2);
expr_ref c = m_sk.mk(symbol("str.lt.c"), e1, e2, char_sort);
expr_ref d = m_sk.mk(symbol("str.lt.d"), e1, e2, char_sort);
expr_ref xcy = mk_concat(x, seq.str.mk_unit(c), y);
expr_ref xdz = mk_concat(x, seq.str.mk_unit(d), z);
literal eq = mk_eq(e1, e2);
literal pref21 = mk_literal(seq.str.mk_prefix(e2, e1));
literal pref12 = mk_literal(seq.str.mk_prefix(e1, e2));
literal e1xcy = mk_eq(e1, xcy);
literal e2xdz = mk_eq(e2, xdz);
literal ltcd = mk_literal(seq.mk_lt(c, d));
literal ltdc = mk_literal(seq.mk_lt(d, c));
add_axiom(~lt, pref12, e2xdz);
add_axiom(~lt, pref12, e1xcy);
add_axiom(~lt, pref12, ltcd);
add_axiom(lt, pref21, e1xcy);
add_axiom(lt, pref21, ltdc);
add_axiom(lt, pref21, e2xdz);
add_axiom(~eq, ~lt);
}
/**
e1 <= e2 <=> e1 < e2 or e1 = e2
*/
void seq_axioms::add_le_axiom(expr* n) {
expr* e1 = nullptr, *e2 = nullptr;
VERIFY(seq.str.is_le(n, e1, e2));
literal lt = mk_literal(seq.str.mk_lex_lt(e1, e2));
literal le = mk_literal(n);
literal eq = mk_eq(e1, e2);
add_axiom(~le, lt, eq);
add_axiom(~eq, le);
add_axiom(~lt, le);
}
void seq_axioms::add_unit_axiom(expr* n) {
expr* u = nullptr;
VERIFY(seq.str.is_unit(n, u));
add_axiom(mk_eq(u, m_sk.mk_unit_inv(n)));
}
void seq_axioms::add_suffix_axiom(expr* e) {
expr* e1 = nullptr, *e2 = nullptr;
VERIFY(seq.str.is_suffix(e, e1, e2));
literal lit = mk_literal(e);
literal e1_gt_e2 = mk_literal(a.mk_ge(mk_sub(mk_len(e1), mk_len(e2)), a.mk_int(1)));
sort* char_sort = nullptr;
VERIFY(seq.is_seq(m.get_sort(e1), char_sort));
expr_ref x = m_sk.mk(symbol("seq.suffix.x"), e1, e2);
expr_ref y = m_sk.mk(symbol("seq.suffix.y"), e1, e2);
expr_ref z = m_sk.mk(symbol("seq.suffix.z"), e1, e2);
expr_ref c = m_sk.mk(symbol("seq.suffix.c"), e1, e2, nullptr, nullptr, char_sort);
expr_ref d = m_sk.mk(symbol("seq.suffix.d"), e1, e2, nullptr, nullptr, char_sort);
add_axiom(lit, e1_gt_e2, mk_seq_eq(e1, mk_concat(y, seq.str.mk_unit(c), x)));
add_axiom(lit, e1_gt_e2, mk_seq_eq(e2, mk_concat(z, seq.str.mk_unit(d), x)));
add_axiom(lit, e1_gt_e2, ~mk_eq(c, d));
}
void seq_axioms::add_prefix_axiom(expr* e) {
expr* e1 = nullptr, *e2 = nullptr;
VERIFY(seq.str.is_prefix(e, e1, e2));
literal lit = mk_literal(e);
literal e1_gt_e2 = mk_literal(a.mk_ge(mk_sub(mk_len(e1), mk_len(e2)), a.mk_int(1)));
sort* char_sort = nullptr;
VERIFY(seq.is_seq(m.get_sort(e1), char_sort));
expr_ref x = m_sk.mk(symbol("seq.prefix.x"), e1, e2);
expr_ref y = m_sk.mk(symbol("seq.prefix.y"), e1, e2);
expr_ref z = m_sk.mk(symbol("seq.prefix.z"), e1, e2);
expr_ref c = m_sk.mk(symbol("seq.prefix.c"), e1, e2, nullptr, nullptr, char_sort);
expr_ref d = m_sk.mk(symbol("seq.prefix.d"), e1, e2, nullptr, nullptr, char_sort);
add_axiom(lit, e1_gt_e2, mk_seq_eq(e1, mk_concat(x, seq.str.mk_unit(c), y)));
add_axiom(lit, e1_gt_e2, mk_seq_eq(e2, mk_concat(x, seq.str.mk_unit(d), z)), mk_seq_eq(e2, x));
add_axiom(lit, e1_gt_e2, ~mk_eq(c, d));
}
literal seq_axioms::is_digit(expr* ch) {
literal isd = mk_literal(m_sk.mk_is_digit(ch));
expr_ref d2i = m_sk.mk_digit2int(ch);
expr_ref _lo(seq.mk_le(seq.mk_char('0'), ch), m);
expr_ref _hi(seq.mk_le(ch, seq.mk_char('9')), m);
literal lo = mk_literal(_lo);
literal hi = mk_literal(_hi);
add_axiom(~lo, ~hi, isd);
add_axiom(~isd, lo);
add_axiom(~isd, hi);
return isd;
}
// n >= 0 & len(e) >= i + 1 => is_digit(e_i) for i = 0..k-1
// n >= 0 & len(e) = k => n = sum 10^i*digit(e_i)
// n < 0 & len(e) = k => \/_i ~is_digit(e_i) for i = 0..k-1
// 10^k <= n < 10^{k+1}-1 => len(e) => k
void seq_axioms::add_si_axiom(expr* e, expr* n, unsigned k) {
zstring s;
expr_ref ith_char(m), num(m), coeff(m);
expr_ref_vector nums(m), chars(m);
expr_ref len = mk_len(e);
literal len_eq_k = th.mk_preferred_eq(len, a.mk_int(k));
literal ge0 = mk_literal(a.mk_ge(n, a.mk_int(0)));
literal_vector digits;
digits.push_back(~len_eq_k);
digits.push_back(ge0);
ensure_digit_axiom();
for (unsigned i = 0; i < k; ++i) {
ith_char = mk_nth(e, a.mk_int(i));
literal isd = is_digit(ith_char);
literal len_ge_i1 = mk_literal(a.mk_ge(len, a.mk_int(i+1)));
add_axiom(~len_ge_i1, ~ge0, isd);
digits.push_back(~isd);
chars.push_back(seq.str.mk_unit(ith_char));
nums.push_back(m_sk.mk_digit2int(ith_char));
}
ctx().mk_th_axiom(th.get_id(), digits.size(), digits.c_ptr());
rational c(1);
for (unsigned i = k; i-- > 0; c *= rational(10)) {
coeff = a.mk_int(c);
nums[i] = a.mk_mul(coeff, nums.get(i));
}
num = a.mk_add(nums.size(), nums.c_ptr());
m_rewrite(num);
add_axiom(~len_eq_k, ~ge0, th.mk_preferred_eq(n, num));
add_axiom(~len_eq_k, ~ge0, th.mk_preferred_eq(e, seq.str.mk_concat(chars)));
SASSERT(k > 0);
rational lb = power(rational(10), k - 1);
rational ub = power(rational(10), k) - 1;
literal lbl = mk_literal(a.mk_ge(n, a.mk_int(lb)));
literal ge_k = mk_literal(a.mk_ge(len, a.mk_int(k)));
// n >= lb => len(s) >= k
add_axiom(~lbl, ge_k);
}
void seq_axioms::ensure_digit_axiom() {
if (!m_digits_initialized) {
for (unsigned i = 0; i < 10; ++i) {
expr_ref cnst(seq.mk_char('0'+i), m);
add_axiom(mk_eq(m_sk.mk_digit2int(cnst), a.mk_int(i)));
}
ctx().push_trail(value_trail<context, bool>(m_digits_initialized));
m_digits_initialized = true;
}
}
expr_ref seq_axioms::add_length_limit(expr* s, unsigned k) {
expr_ref bound_tracker = m_sk.mk_length_limit(s, k);
literal bound_predicate = mk_literal(a.mk_le(mk_len(s), a.mk_int(k)));
add_axiom(~mk_literal(bound_tracker), bound_predicate);
return bound_tracker;
}
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