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961 lines
36 KiB
C++
961 lines
36 KiB
C++
/*++
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Copyright (c) 2011 Microsoft Corporation
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Module Name:
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purify_arith_tactic.h
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Abstract:
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Tactic for eliminating arithmetic operators: DIV, IDIV, MOD,
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TO_INT, and optionally (OP_IRRATIONAL_ALGEBRAIC_NUM).
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This tactic uses the simplifier for also eliminating:
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OP_SUB, OP_UMINUS, OP_POWER (optionally), OP_REM, OP_IS_INT.
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Author:
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Leonardo de Moura (leonardo) 2011-12-30.
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Revision History:
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--*/
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#include "tactic/tactical.h"
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#include "ast/rewriter/rewriter_def.h"
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#include "ast/arith_decl_plugin.h"
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#include "math/polynomial/algebraic_numbers.h"
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#include "tactic/core/nnf_tactic.h"
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#include "tactic/core/simplify_tactic.h"
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#include "ast/rewriter/th_rewriter.h"
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#include "ast/converters/generic_model_converter.h"
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#include "ast/ast_smt2_pp.h"
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#include "ast/ast_pp.h"
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#include "ast/rewriter/expr_replacer.h"
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/*
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----
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Some of the rules needed in the conversion are implemented in
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arith_rewriter.cpp. Here is a summary of these rules:
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(^ t (/ p q)) --> (^ (^ t (/ 1 q)) p)
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(^ t n) --> t*...*t
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when integer power expansion is requested
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(is-int t) --> t = (to-real (to-int t))
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(rem t1 t2) --> ite(t2 >= 0, (mod t1 t2), -(mod t1 t2))
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----
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The tactic implements a set of transformation rules. These rules
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create fresh constants or (existential) variables, and add new
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constraints to the context.
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The context is the set of asserted formulas or a quantifier.
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A rule is represented as:
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From --> To | C
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It means, any expression that matches From is replaced by To,
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and the constraints C are added to the context.
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For clarity reasons, I write the constraints using ad-hoc notation.
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Rules
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(^ t 0) --> k | t != 0 implies k = 1, t = 0 implies k = 0^0
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where k is fresh
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0^0 is a constant used to capture the meaning of (^ 0 0).
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(^ t (/ 1 n)) --> k | t = k^n
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when n is odd
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where k is fresh
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(^ t (/ 1 n)) --> k | t >= 0 implies t = k^n, t < 0 implies t = neg-root(t, n)
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when n is even
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where k is fresh
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neg-root is a function symbol used to capture the meaning of a negative root
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(root-obj p(x) i) --> k | p(k) = 0, l < k < u
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when root object elimination is requested
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where k is fresh
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(l, u) is an isolating interval for the i-th root of p.
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(to-int t) --> k | 0 <= to-real(k) - t < 1
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where k is a fresh integer constant/variable
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(/ t1 t2) --> k | t2 != 0 implies k*t2 = t1, t2 = 0 implies k = div-0(t1)
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where k is fresh
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div-0 is a function symbol used to capture the meaning of division by 0.
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Remark: If it can be shown that t2 != 0, then the div-0(t1) function application
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vanishes from the formula.
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(div t1 t2) --> k1 | t2 = 0 \/ t1 = k1 * t2 + k2,
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t2 = 0 \/ 0 <= k2,
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t2 = 0 \/ k2 < |t2|,
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t2 != 0 \/ k1 = idiv-0(t1),
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t2 != 0 \/ k2 = mod-0(t1)
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k1 is a fresh name for (div t1 t2)
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k2 is a fresh name for (mod t1 t2)
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(mod t1 t2) --> k2 | same constraints as above
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*/
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struct purify_arith_proc {
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arith_util & m_util;
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goal & m_goal;
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bool m_produce_proofs;
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bool m_elim_root_objs;
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bool m_elim_inverses;
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bool m_complete;
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ast_mark m_unsafe_exprs;
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bool m_unsafe_found;
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obj_map<app, std::pair<expr*, expr*> > m_sin_cos;
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expr_ref_vector m_pinned;
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purify_arith_proc(goal & g, arith_util & u, bool produce_proofs, bool elim_root_objs, bool elim_inverses, bool complete):
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m_util(u),
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m_goal(g),
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m_produce_proofs(produce_proofs),
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m_elim_root_objs(elim_root_objs),
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m_elim_inverses(elim_inverses),
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m_complete(complete),
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m_unsafe_found(false),
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m_pinned(m()) {
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}
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arith_util & u() {
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return m_util;
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}
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ast_manager & m() {
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return u().get_manager();
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}
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struct find_unsafe_proc {
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purify_arith_proc& m_owner;
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find_unsafe_proc(purify_arith_proc& o) : m_owner(o) {}
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void operator()(app* a) {
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if (!m_owner.u().is_sin(a) &&
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!m_owner.u().is_cos(a)) {
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for (unsigned i = 0; i < a->get_num_args(); ++i) {
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m_owner.m_unsafe_exprs.mark(a->get_arg(i), true);
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}
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}
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}
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void operator()(quantifier *q) {}
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void operator()(var* v) {}
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};
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void find_unsafe() {
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if (m_unsafe_found) return;
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find_unsafe_proc proc(*this);
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expr_fast_mark1 visited;
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unsigned sz = m_goal.size();
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for (unsigned i = 0; i < sz; i++) {
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expr * curr = m_goal.form(i);
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for_each_expr_core<find_unsafe_proc, expr_fast_mark1, true, true>(proc, visited, curr);
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}
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m_unsafe_found = true;
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}
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bool convert_basis(expr* theta, expr*& x, expr*& y) {
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if (!is_uninterp_const(theta)) {
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return false;
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}
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find_unsafe();
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if (m_unsafe_exprs.is_marked(theta)) {
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return false;
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}
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std::pair<expr*, expr*> pair;
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if (!m_sin_cos.find(to_app(theta), pair)) {
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pair.first = m().mk_fresh_const(nullptr, u().mk_real());
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pair.second = m().mk_fresh_const(nullptr, u().mk_real());
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m_sin_cos.insert(to_app(theta), pair);
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m_pinned.push_back(pair.first);
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m_pinned.push_back(pair.second);
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m_pinned.push_back(theta);
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// TBD for model conversion
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}
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x = pair.first;
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y = pair.second;
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return true;
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}
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struct bin_def {
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expr* x, *y, *d;
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bin_def(expr* x, expr* y, expr* d): x(x), y(y), d(d) {}
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};
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struct rw_cfg : public default_rewriter_cfg {
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purify_arith_proc & m_owner;
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obj_hashtable<func_decl> m_cannot_purify;
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obj_map<app, expr*> m_app2fresh;
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obj_map<app, proof*> m_app2pr;
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expr_ref_vector m_pinned;
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expr_ref_vector m_new_cnstrs;
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proof_ref_vector m_new_cnstr_prs;
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svector<bin_def> m_divs, m_idivs, m_mods;
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expr_ref m_ipower0, m_rpower0;
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expr_ref m_subst;
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proof_ref m_subst_pr;
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expr_ref_vector m_new_vars;
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rw_cfg(purify_arith_proc & o):
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m_owner(o),
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m_pinned(o.m()),
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m_new_cnstrs(o.m()),
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m_new_cnstr_prs(o.m()),
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m_ipower0(o.m()),
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m_rpower0(o.m()),
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m_subst(o.m()),
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m_subst_pr(o.m()),
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m_new_vars(o.m()) {
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init_cannot_purify();
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}
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ast_manager & m() { return m_owner.m(); }
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arith_util & u() { return m_owner.u(); }
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bool produce_proofs() const { return m_owner.m_produce_proofs; }
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bool complete() const { return m_owner.m_complete; }
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bool elim_root_objs() const { return m_owner.m_elim_root_objs; }
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bool elim_inverses() const { return m_owner.m_elim_inverses; }
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void init_cannot_purify() {
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struct proc {
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rw_cfg& o;
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proc(rw_cfg& o):o(o) {}
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void operator()(app* a) {
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for (expr* arg : *a) {
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if (!is_ground(arg)) {
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auto* f = a->get_decl();
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o.m_cannot_purify.insert(f);
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break;
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}
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}
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}
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void operator()(expr* ) {}
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};
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expr_fast_mark1 visited;
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proc p(*this);
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unsigned sz = m_owner.m_goal.size();
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for (unsigned i = 0; i < sz; i++) {
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expr* f = m_owner.m_goal.form(i);
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for_each_expr_core<proc, expr_fast_mark1, true, true>(p, visited, f);
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}
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}
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expr * mk_fresh_var(bool is_int) {
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expr * r = m().mk_fresh_const(nullptr, is_int ? u().mk_int() : u().mk_real());
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m_new_vars.push_back(r);
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return r;
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}
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expr * mk_fresh_real_var() { return mk_fresh_var(false); }
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expr * mk_fresh_int_var() { return mk_fresh_var(true); }
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expr * mk_int_zero() { return u().mk_numeral(rational(0), true); }
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expr * mk_real_zero() { return u().mk_numeral(rational(0), false); }
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expr * mk_real_one() { return u().mk_numeral(rational(1), false); }
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bool already_processed(app * t, expr_ref & result, proof_ref & result_pr) {
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expr * r;
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if (m_app2fresh.find(t, r)) {
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result = r;
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if (produce_proofs())
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result_pr = m_app2pr.find(t);
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return true;
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}
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return false;
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}
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void mk_def_proof(expr * k, expr * def, proof_ref & result_pr) {
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result_pr = nullptr;
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if (produce_proofs()) {
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expr * eq = m().mk_eq(def, k);
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proof * pr1 = m().mk_def_intro(eq);
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result_pr = m().mk_apply_def(def, k, pr1);
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}
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}
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void push_cnstr_pr(proof * def_pr) {
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if (produce_proofs())
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m_new_cnstr_prs.push_back(m().mk_th_lemma(u().get_family_id(), m_new_cnstrs.back(), 1, &def_pr));
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}
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void push_cnstr_pr(proof * def_pr1, proof * def_pr2) {
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if (produce_proofs()) {
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proof * prs[2] = { def_pr1, def_pr2 };
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m_new_cnstr_prs.push_back(m().mk_th_lemma(u().get_family_id(), m_new_cnstrs.back(), 2, prs));
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}
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}
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void push_cnstr(expr * cnstr) {
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m_new_cnstrs.push_back(cnstr);
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TRACE("purify_arith", tout << mk_pp(cnstr, m()) << "\n";);
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}
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void cache_result(app * t, expr * r, proof * pr) {
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m_app2fresh.insert(t, r);
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m_pinned.push_back(t);
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m_pinned.push_back(r);
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if (produce_proofs()) {
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m_app2pr.insert(t, pr);
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m_pinned.push_back(pr);
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}
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}
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expr * OR(expr * arg1, expr * arg2) { return m().mk_or(arg1, arg2); }
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expr * AND(expr * arg1, expr * arg2) { return m().mk_and(arg1, arg2); }
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expr * EQ(expr * lhs, expr * rhs) { return m().mk_eq(lhs, rhs); }
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expr * NOT(expr * arg) { return m().mk_not(arg); }
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void process_div(func_decl * f, unsigned num, expr * const * args, expr_ref & result, proof_ref & result_pr) {
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app_ref t(m());
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t = m().mk_app(f, num, args);
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if (already_processed(t, result, result_pr))
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return;
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expr * k = mk_fresh_real_var();
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result = k;
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mk_def_proof(k, t, result_pr);
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cache_result(t, result, result_pr);
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expr * x = args[0];
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expr * y = args[1];
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// y = 0 \/ y*k = x
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push_cnstr(OR(EQ(y, mk_real_zero()),
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EQ(u().mk_mul(y, k), x)));
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push_cnstr_pr(result_pr);
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rational r;
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if (complete()) {
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// y != 0 \/ k = div-0(x)
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push_cnstr(OR(NOT(EQ(y, mk_real_zero())),
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EQ(k, u().mk_div(x, mk_real_zero()))));
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push_cnstr_pr(result_pr);
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}
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m_divs.push_back(bin_def(x, y, k));
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}
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void process_idiv(func_decl * f, unsigned num, expr * const * args, expr_ref & result, proof_ref & result_pr) {
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app_ref div_app(m());
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div_app = m().mk_app(f, num, args);
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if (already_processed(div_app, result, result_pr))
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return;
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expr * k1 = mk_fresh_int_var();
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result = k1;
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mk_def_proof(k1, div_app, result_pr);
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cache_result(div_app, result, result_pr);
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expr * k2 = mk_fresh_int_var();
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app_ref mod_app(m());
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proof_ref mod_pr(m());
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expr * x = args[0];
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expr * y = args[1];
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mod_app = u().mk_mod(x, y);
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mk_def_proof(k2, mod_app, mod_pr);
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cache_result(mod_app, k2, mod_pr);
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m_mods.push_back(bin_def(x, y, k2));
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// (div x y) --> k1 | y = 0 \/ x = k1 * y + k2,
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// y = 0 \/ 0 <= k2,
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// y = 0 \/ k2 < |y|,
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// y != 0 \/ k1 = idiv-0(x),
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// y != 0 \/ k2 = mod-0(x)
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// We can write y = 0 \/ k2 < |y| as:
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// y > 0 implies k2 < y ---> y <= 0 \/ k2 < y
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// y < 0 implies k2 < -y ---> y >= 0 \/ k2 < -y
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//
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expr * zero = mk_int_zero();
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push_cnstr(OR(EQ(y, zero), EQ(x, u().mk_add(u().mk_mul(k1, y), k2))));
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push_cnstr_pr(result_pr, mod_pr);
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push_cnstr(OR(EQ(y, zero), u().mk_le(zero, k2)));
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push_cnstr_pr(mod_pr);
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push_cnstr(OR(u().mk_le(y, zero), u().mk_lt(k2, y)));
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push_cnstr_pr(mod_pr);
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push_cnstr(OR(u().mk_ge(y, zero), u().mk_lt(k2, u().mk_mul(u().mk_numeral(rational(-1), true), y))));
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push_cnstr_pr(mod_pr);
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rational r;
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if (complete() && (!u().is_numeral(y, r) || r.is_zero())) {
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push_cnstr(OR(NOT(EQ(y, zero)), EQ(k1, u().mk_idiv(x, zero))));
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push_cnstr_pr(result_pr);
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push_cnstr(OR(NOT(EQ(y, zero)), EQ(k2, u().mk_mod(x, zero))));
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push_cnstr_pr(mod_pr);
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}
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m_idivs.push_back(bin_def(x, y, k1));
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}
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void process_mod(func_decl * f, unsigned num, expr * const * args, expr_ref & result, proof_ref & result_pr) {
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app_ref t(m());
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t = m().mk_app(f, num, args);
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if (already_processed(t, result, result_pr))
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return;
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process_idiv(f, num, args, result, result_pr); // it will create mod
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VERIFY(already_processed(t, result, result_pr));
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}
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void process_to_int(func_decl * f, unsigned num, expr * const * args, expr_ref & result, proof_ref & result_pr) {
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app_ref t(m());
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t = m().mk_app(f, num, args);
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if (already_processed(t, result, result_pr))
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return;
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expr * k = mk_fresh_int_var();
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result = k;
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mk_def_proof(k, t, result_pr);
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cache_result(t, result, result_pr);
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expr * x = args[0];
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// x - to-real(k) >= 0
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expr * diff = u().mk_add(x, u().mk_mul(u().mk_numeral(rational(-1), false), u().mk_to_real(k)));
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push_cnstr(u().mk_ge(diff, mk_real_zero()));
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push_cnstr_pr(result_pr);
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// not(x - to-real(k) >= 1)
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push_cnstr(NOT(u().mk_ge(diff, u().mk_numeral(rational(1), false))));
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push_cnstr_pr(result_pr);
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}
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br_status process_power(func_decl * f, unsigned num, expr * const * args, expr_ref & result, proof_ref & result_pr) {
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rational y;
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if (!u().is_numeral(args[1], y))
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return BR_FAILED;
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if (y.is_int() && !y.is_zero())
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return BR_FAILED;
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app_ref t(m());
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t = m().mk_app(f, num, args);
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if (already_processed(t, result, result_pr))
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return BR_DONE;
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expr * x = args[0];
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bool is_int = u().is_int(x);
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expr * k = mk_fresh_var(is_int);
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result = k;
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mk_def_proof(k, t, result_pr);
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cache_result(t, result, result_pr);
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expr_ref zero(u().mk_numeral(rational(0), is_int), m());
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expr_ref one(u().mk_numeral(rational(1), is_int), m());
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if (y.is_zero()) {
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|
expr* p0;
|
|
if (is_int) {
|
|
if (!m_ipower0) m_ipower0 = mk_fresh_var(true);
|
|
p0 = m_ipower0;
|
|
}
|
|
else {
|
|
if (!m_rpower0) m_rpower0 = mk_fresh_var(false);
|
|
p0 = m_rpower0;
|
|
}
|
|
|
|
// (^ x 0) --> k | x != 0 implies k = 1, x = 0 implies k = 0^0
|
|
push_cnstr(OR(EQ(x, zero), EQ(k, one)));
|
|
push_cnstr_pr(result_pr);
|
|
push_cnstr(OR(NOT(EQ(x, zero)), EQ(k, p0)));
|
|
push_cnstr_pr(result_pr);
|
|
}
|
|
else if (!is_int) {
|
|
SASSERT(!y.is_int());
|
|
SASSERT(numerator(y).is_one());
|
|
rational n = denominator(y);
|
|
if (!n.is_even()) {
|
|
// (^ x (/ 1 n)) --> k | x = k^n
|
|
// when n is odd
|
|
push_cnstr(EQ(x, u().mk_power(k, u().mk_numeral(n, false))));
|
|
push_cnstr_pr(result_pr);
|
|
}
|
|
else {
|
|
SASSERT(n.is_even());
|
|
// (^ x (/ 1 n)) --> k | x >= 0 implies (x = k^n and k >= 0), x < 0 implies k = neg-root(x, n)
|
|
// when n is even
|
|
push_cnstr(OR(NOT(u().mk_ge(x, zero)),
|
|
AND(EQ(x, u().mk_power(k, u().mk_numeral(n, false))),
|
|
u().mk_ge(k, zero))));
|
|
push_cnstr_pr(result_pr);
|
|
|
|
push_cnstr(OR(u().mk_ge(x, zero),
|
|
EQ(k, u().mk_neg_root(x, u().mk_numeral(n, false)))));
|
|
push_cnstr_pr(result_pr);
|
|
}
|
|
// else {
|
|
// return BR_FAILED;
|
|
// }
|
|
}
|
|
else {
|
|
// root not supported for integers.
|
|
SASSERT(is_int);
|
|
SASSERT(!y.is_int());
|
|
return BR_FAILED;
|
|
}
|
|
return BR_DONE;
|
|
}
|
|
|
|
void process_irrat(app * s, expr_ref & result, proof_ref & result_pr) {
|
|
if (already_processed(s, result, result_pr))
|
|
return;
|
|
|
|
expr * k = mk_fresh_real_var();
|
|
result = k;
|
|
mk_def_proof(k, s, result_pr);
|
|
cache_result(s, result, result_pr);
|
|
|
|
anum_manager & am = u().am();
|
|
anum const & a = u().to_irrational_algebraic_numeral(s);
|
|
scoped_mpz_vector p(am.qm());
|
|
am.get_polynomial(a, p);
|
|
rational lower, upper;
|
|
am.get_lower(a, lower);
|
|
am.get_upper(a, upper);
|
|
unsigned sz = p.size();
|
|
SASSERT(sz > 2);
|
|
ptr_buffer<expr> args;
|
|
for (unsigned i = 0; i < sz; i++) {
|
|
if (am.qm().is_zero(p[i]))
|
|
continue;
|
|
rational coeff = rational(p[i]);
|
|
if (i == 0) {
|
|
args.push_back(u().mk_numeral(coeff, false));
|
|
}
|
|
else {
|
|
expr * m;
|
|
if (i == 1)
|
|
m = k;
|
|
else
|
|
m = u().mk_power(k, u().mk_numeral(rational(i), false));
|
|
args.push_back(u().mk_mul(u().mk_numeral(coeff, false), m));
|
|
}
|
|
}
|
|
SASSERT(args.size() >= 2);
|
|
push_cnstr(EQ(u().mk_add(args.size(), args.data()), mk_real_zero()));
|
|
push_cnstr_pr(result_pr);
|
|
push_cnstr(u().mk_lt(u().mk_numeral(lower, false), k));
|
|
push_cnstr_pr(result_pr);
|
|
push_cnstr(u().mk_lt(k, u().mk_numeral(upper, false)));
|
|
push_cnstr_pr(result_pr);
|
|
}
|
|
|
|
br_status process_sin_cos(bool first, func_decl *f, expr * theta, expr_ref & result, proof_ref & result_pr) {
|
|
expr* x, *y;
|
|
if (m_owner.convert_basis(theta, x, y)) {
|
|
result = first ? x : y;
|
|
app_ref t(m().mk_app(f, theta), m());
|
|
mk_def_proof(result, t, result_pr);
|
|
cache_result(t, result, result_pr);
|
|
push_cnstr(EQ(mk_real_one(), u().mk_add(u().mk_mul(x, x), u().mk_mul(y, y))));
|
|
push_cnstr_pr(result_pr);
|
|
return BR_DONE;
|
|
}
|
|
else {
|
|
expr_ref s(u().mk_sin(theta), m());
|
|
expr_ref c(u().mk_cos(theta), m());
|
|
expr_ref axm(EQ(mk_real_one(), u().mk_add(u().mk_mul(s, s), u().mk_mul(c, c))), m());
|
|
push_cnstr(axm);
|
|
push_cnstr_pr(m().mk_asserted(axm));
|
|
return BR_FAILED;
|
|
}
|
|
}
|
|
|
|
br_status process_sin(func_decl *f, expr * theta, expr_ref & result, proof_ref & result_pr) {
|
|
return process_sin_cos(true, f, theta, result, result_pr);
|
|
}
|
|
|
|
br_status process_cos(func_decl *f, expr * theta, expr_ref & result, proof_ref & result_pr) {
|
|
return process_sin_cos(false, f, theta, result, result_pr);
|
|
}
|
|
|
|
br_status process_asin(func_decl * f, expr * x, expr_ref & result, proof_ref & result_pr) {
|
|
if (!elim_inverses())
|
|
return BR_FAILED;
|
|
app_ref t(m());
|
|
t = m().mk_app(f, x);
|
|
if (already_processed(t, result, result_pr))
|
|
return BR_DONE;
|
|
|
|
expr * k = mk_fresh_var(false);
|
|
result = k;
|
|
mk_def_proof(k, t, result_pr);
|
|
cache_result(t, result, result_pr);
|
|
|
|
// Constraints:
|
|
// -1 <= x <= 1 implies sin(k) = x, -pi/2 <= k <= pi/2
|
|
// If complete()
|
|
// x < -1 implies k = asin_u(x)
|
|
// x > 1 implies k = asin_u(x)
|
|
expr * one = u().mk_numeral(rational(1), false);
|
|
expr * mone = u().mk_numeral(rational(-1), false);
|
|
expr * pi2 = u().mk_mul(u().mk_numeral(rational(1,2), false), u().mk_pi());
|
|
expr * mpi2 = u().mk_mul(u().mk_numeral(rational(-1,2), false), u().mk_pi());
|
|
// -1 <= x <= 1 implies sin(k) = x, -pi/2 <= k <= pi/2
|
|
push_cnstr(OR(OR(NOT(u().mk_ge(x, mone)),
|
|
NOT(u().mk_le(x, one))),
|
|
AND(EQ(x, u().mk_sin(k)),
|
|
AND(u().mk_ge(k, mpi2),
|
|
u().mk_le(k, pi2)))));
|
|
push_cnstr_pr(result_pr);
|
|
if (complete()) {
|
|
// x < -1 implies k = asin_u(x)
|
|
// x > 1 implies k = asin_u(x)
|
|
push_cnstr(OR(u().mk_ge(x, mone),
|
|
EQ(k, u().mk_u_asin(x))));
|
|
push_cnstr_pr(result_pr);
|
|
push_cnstr(OR(u().mk_le(x, one),
|
|
EQ(k, u().mk_u_asin(x))));
|
|
push_cnstr_pr(result_pr);
|
|
}
|
|
return BR_DONE;
|
|
}
|
|
|
|
br_status process_acos(func_decl * f, expr * x, expr_ref & result, proof_ref & result_pr) {
|
|
if (!elim_inverses())
|
|
return BR_FAILED;
|
|
app_ref t(m());
|
|
t = m().mk_app(f, x);
|
|
if (already_processed(t, result, result_pr))
|
|
return BR_DONE;
|
|
|
|
expr * k = mk_fresh_var(false);
|
|
result = k;
|
|
mk_def_proof(k, t, result_pr);
|
|
cache_result(t, result, result_pr);
|
|
|
|
// Constraints:
|
|
// -1 <= x <= 1 implies cos(k) = x, 0 <= k <= pi
|
|
// If complete()
|
|
// x < -1 implies k = acos_u(x)
|
|
// x > 1 implies k = acos_u(x)
|
|
expr * one = u().mk_numeral(rational(1), false);
|
|
expr * mone = u().mk_numeral(rational(-1), false);
|
|
expr * pi = u().mk_pi();
|
|
expr * zero = u().mk_numeral(rational(0), false);
|
|
// -1 <= x <= 1 implies cos(k) = x, 0 <= k <= pi
|
|
push_cnstr(OR(OR(NOT(u().mk_ge(x, mone)),
|
|
NOT(u().mk_le(x, one))),
|
|
AND(EQ(x, u().mk_cos(k)),
|
|
AND(u().mk_ge(k, zero),
|
|
u().mk_le(k, pi)))));
|
|
push_cnstr_pr(result_pr);
|
|
if (complete()) {
|
|
// x < -1 implies k = acos_u(x)
|
|
// x > 1 implies k = acos_u(x)
|
|
push_cnstr(OR(u().mk_ge(x, mone),
|
|
EQ(k, u().mk_u_acos(x))));
|
|
push_cnstr_pr(result_pr);
|
|
push_cnstr(OR(u().mk_le(x, one),
|
|
EQ(k, u().mk_u_acos(x))));
|
|
push_cnstr_pr(result_pr);
|
|
}
|
|
return BR_DONE;
|
|
}
|
|
|
|
br_status process_atan(func_decl * f, expr * x, expr_ref & result, proof_ref & result_pr) {
|
|
if (!elim_inverses())
|
|
return BR_FAILED;
|
|
app_ref t(m());
|
|
t = m().mk_app(f, x);
|
|
if (already_processed(t, result, result_pr))
|
|
return BR_DONE;
|
|
|
|
expr * k = mk_fresh_var(false);
|
|
result = k;
|
|
mk_def_proof(k, t, result_pr);
|
|
cache_result(t, result, result_pr);
|
|
|
|
// Constraints:
|
|
// tan(k) = x, -pi/2 < k < pi/2
|
|
expr * pi2 = u().mk_mul(u().mk_numeral(rational(1,2), false), u().mk_pi());
|
|
expr * mpi2 = u().mk_mul(u().mk_numeral(rational(-1,2), false), u().mk_pi());
|
|
push_cnstr(AND(EQ(x, u().mk_tan(k)),
|
|
AND(u().mk_gt(k, mpi2),
|
|
u().mk_lt(k, pi2))));
|
|
push_cnstr_pr(result_pr);
|
|
return BR_DONE;
|
|
}
|
|
|
|
br_status reduce_app(func_decl * f, unsigned num, expr * const * args, expr_ref & result, proof_ref & result_pr) {
|
|
if (f->get_family_id() != u().get_family_id())
|
|
return BR_FAILED;
|
|
if (m_cannot_purify.contains(f))
|
|
return BR_FAILED;
|
|
switch (f->get_decl_kind()) {
|
|
case OP_DIV:
|
|
process_div(f, num, args, result, result_pr);
|
|
return BR_DONE;
|
|
case OP_IDIV:
|
|
if (!m_cannot_purify.empty())
|
|
return BR_FAILED;
|
|
process_idiv(f, num, args, result, result_pr);
|
|
return BR_DONE;
|
|
case OP_MOD:
|
|
if (!m_cannot_purify.empty())
|
|
return BR_FAILED;
|
|
process_mod(f, num, args, result, result_pr);
|
|
return BR_DONE;
|
|
case OP_TO_INT:
|
|
process_to_int(f, num, args, result, result_pr);
|
|
return BR_DONE;
|
|
case OP_POWER:
|
|
return process_power(f, num, args, result, result_pr);
|
|
case OP_ASIN:
|
|
return process_asin(f, args[0], result, result_pr);
|
|
case OP_ACOS:
|
|
return process_acos(f, args[0], result, result_pr);
|
|
case OP_SIN:
|
|
return process_sin(f, args[0], result, result_pr);
|
|
case OP_COS:
|
|
return process_cos(f, args[0], result, result_pr);
|
|
case OP_ATAN:
|
|
return process_atan(f, args[0], result, result_pr);
|
|
default:
|
|
return BR_FAILED;
|
|
}
|
|
}
|
|
|
|
bool get_subst(expr * s, expr * & t, proof * & t_pr) {
|
|
if (is_quantifier(s)) {
|
|
m_owner.process_quantifier(*this, to_quantifier(s), m_subst, m_subst_pr);
|
|
t = m_subst.get();
|
|
t_pr = m_subst_pr.get();
|
|
return true;
|
|
}
|
|
else if (u().is_irrational_algebraic_numeral(s) && elim_root_objs()) {
|
|
process_irrat(to_app(s), m_subst, m_subst_pr);
|
|
t = m_subst.get();
|
|
t_pr = m_subst_pr.get();
|
|
return true;
|
|
}
|
|
return false;
|
|
}
|
|
};
|
|
|
|
expr * mk_real_zero() { return u().mk_numeral(rational(0), false); }
|
|
|
|
struct rw : public rewriter_tpl<rw_cfg> {
|
|
rw_cfg m_cfg;
|
|
rw(purify_arith_proc & o):
|
|
rewriter_tpl<rw_cfg>(o.m(), o.m_produce_proofs, m_cfg),
|
|
m_cfg(o) {
|
|
}
|
|
};
|
|
|
|
struct rw_rec : public rewriter_tpl<rw_cfg> {
|
|
rw_cfg& m_cfg;
|
|
rw_rec(rw_cfg& cfg):
|
|
rewriter_tpl<rw_cfg>(cfg.m(), cfg.produce_proofs(), cfg),
|
|
m_cfg(cfg) {
|
|
}
|
|
};
|
|
|
|
void process_quantifier(rw_cfg& cfg, quantifier * q, expr_ref & result, proof_ref & result_pr) {
|
|
result_pr = nullptr;
|
|
rw_rec r(cfg);
|
|
expr_ref new_body(m());
|
|
proof_ref new_body_pr(m());
|
|
r(q->get_expr(), new_body, new_body_pr);
|
|
TRACE("purify_arith",
|
|
tout << "body: " << mk_ismt2_pp(q->get_expr(), m()) << "\nnew_body: " << new_body << "\n";);
|
|
result = m().update_quantifier(q, new_body);
|
|
if (m_produce_proofs) {
|
|
result_pr = m().mk_rewrite(q->get_expr(), new_body);
|
|
result_pr = m().mk_quant_intro(q, to_quantifier(result.get()), result_pr);
|
|
}
|
|
}
|
|
|
|
void operator()(model_converter_ref & mc, bool produce_models) {
|
|
rw r(*this);
|
|
// purify
|
|
expr_ref new_curr(m());
|
|
proof_ref new_pr(m());
|
|
unsigned sz = m_goal.size();
|
|
for (unsigned i = 0; !m_goal.inconsistent() && i < sz; i++) {
|
|
expr * curr = m_goal.form(i);
|
|
r(curr, new_curr, new_pr);
|
|
if (m_produce_proofs) {
|
|
proof * pr = m_goal.pr(i);
|
|
new_pr = m().mk_modus_ponens(pr, new_pr);
|
|
}
|
|
m_goal.update(i, new_curr, new_pr, m_goal.dep(i));
|
|
}
|
|
|
|
// add constraints
|
|
sz = r.cfg().m_new_cnstrs.size();
|
|
TRACE("purify_arith", tout << r.cfg().m_new_cnstrs << "\n";);
|
|
TRACE("purify_arith", tout << r.cfg().m_new_cnstr_prs << "\n";);
|
|
for (unsigned i = 0; i < sz; i++) {
|
|
m_goal.assert_expr(r.cfg().m_new_cnstrs.get(i), m_produce_proofs ? r.cfg().m_new_cnstr_prs.get(i) : nullptr, nullptr);
|
|
}
|
|
auto const& divs = r.cfg().m_divs;
|
|
auto const& idivs = r.cfg().m_idivs;
|
|
auto const& mods = r.cfg().m_mods;
|
|
for (unsigned i = 0; i < divs.size(); ++i) {
|
|
auto const& p1 = divs[i];
|
|
for (unsigned j = i + 1; j < divs.size(); ++j) {
|
|
auto const& p2 = divs[j];
|
|
m_goal.assert_expr(m().mk_implies(
|
|
m().mk_and(m().mk_eq(p1.x, p2.x), m().mk_eq(p1.y, p2.y)),
|
|
m().mk_eq(p1.d, p2.d)));
|
|
}
|
|
}
|
|
for (unsigned i = 0; i < mods.size(); ++i) {
|
|
auto const& p1 = mods[i];
|
|
for (unsigned j = i + 1; j < mods.size(); ++j) {
|
|
auto const& p2 = mods[j];
|
|
m_goal.assert_expr(m().mk_implies(
|
|
m().mk_and(m().mk_eq(p1.x, p2.x), m().mk_eq(p1.y, p2.y)),
|
|
m().mk_eq(p1.d, p2.d)));
|
|
}
|
|
}
|
|
for (unsigned i = 0; i < idivs.size(); ++i) {
|
|
auto const& p1 = idivs[i];
|
|
for (unsigned j = i + 1; j < idivs.size(); ++j) {
|
|
auto const& p2 = idivs[j];
|
|
m_goal.assert_expr(m().mk_implies(
|
|
m().mk_and(m().mk_eq(p1.x, p2.x), m().mk_eq(p1.y, p2.y)),
|
|
m().mk_eq(p1.d, p2.d)));
|
|
}
|
|
}
|
|
|
|
// add generic_model_converter to eliminate auxiliary variables from model
|
|
if (produce_models) {
|
|
generic_model_converter * fmc = alloc(generic_model_converter, m(), "purify");
|
|
mc = fmc;
|
|
obj_map<app, expr*> & f2v = r.cfg().m_app2fresh;
|
|
for (auto const& kv : f2v) {
|
|
app * v = to_app(kv.m_value);
|
|
SASSERT(is_uninterp_const(v));
|
|
fmc->hide(v->get_decl());
|
|
}
|
|
if (!divs.empty()) {
|
|
expr_ref body(u().mk_real(0), m());
|
|
expr_ref v0(m().mk_var(0, u().mk_real()), m());
|
|
expr_ref v1(m().mk_var(1, u().mk_real()), m());
|
|
for (auto const& p : divs) {
|
|
body = m().mk_ite(m().mk_and(m().mk_eq(v0, p.x), m().mk_eq(v1, p.y)), p.d, body);
|
|
}
|
|
fmc->add(u().mk_div0(), body);
|
|
}
|
|
if (!mods.empty()) {
|
|
expr_ref body(u().mk_int(0), m());
|
|
expr_ref v0(m().mk_var(0, u().mk_int()), m());
|
|
expr_ref v1(m().mk_var(1, u().mk_int()), m());
|
|
for (auto const& p : mods) {
|
|
body = m().mk_ite(m().mk_and(m().mk_eq(v0, p.x), m().mk_eq(v1, p.y)), p.d, body);
|
|
}
|
|
|
|
fmc->add(u().mk_mod0(), body);
|
|
body = m().mk_ite(u().mk_ge(v1, u().mk_int(0)), body, u().mk_uminus(body));
|
|
fmc->add(u().mk_rem0(), body);
|
|
}
|
|
if (!idivs.empty()) {
|
|
expr_ref body(u().mk_int(0), m());
|
|
expr_ref v0(m().mk_var(0, u().mk_int()), m());
|
|
expr_ref v1(m().mk_var(1, u().mk_int()), m());
|
|
for (auto const& p : idivs) {
|
|
body = m().mk_ite(m().mk_and(m().mk_eq(v0, p.x), m().mk_eq(v1, p.y)), p.d, body);
|
|
}
|
|
fmc->add(u().mk_idiv0(), body);
|
|
}
|
|
if (r.cfg().m_ipower0) {
|
|
fmc->add(u().mk_ipower0(), r.cfg().m_ipower0);
|
|
}
|
|
if (r.cfg().m_rpower0) {
|
|
fmc->add(u().mk_rpower0(), r.cfg().m_rpower0);
|
|
}
|
|
}
|
|
if (produce_models && !m_sin_cos.empty()) {
|
|
generic_model_converter* emc = alloc(generic_model_converter, m(), "purify_sin_cos");
|
|
mc = concat(mc.get(), emc);
|
|
for (auto const& kv : m_sin_cos) {
|
|
emc->add(kv.m_key->get_decl(),
|
|
m().mk_ite(u().mk_ge(kv.m_value.first, mk_real_zero()), u().mk_acos(kv.m_value.second),
|
|
u().mk_add(u().mk_acos(u().mk_uminus(kv.m_value.second)), u().mk_pi())));
|
|
}
|
|
|
|
}
|
|
}
|
|
|
|
|
|
};
|
|
|
|
class purify_arith_tactic : public tactic {
|
|
arith_util m_util;
|
|
params_ref m_params;
|
|
public:
|
|
purify_arith_tactic(ast_manager & m, params_ref const & p):
|
|
m_util(m),
|
|
m_params(p) {
|
|
}
|
|
|
|
tactic * translate(ast_manager & m) override {
|
|
return alloc(purify_arith_tactic, m, m_params);
|
|
}
|
|
|
|
char const* name() const override { return "purify_arith"; }
|
|
|
|
void updt_params(params_ref const & p) override {
|
|
m_params.append(p);
|
|
}
|
|
|
|
void collect_param_descrs(param_descrs & r) override {
|
|
r.insert("complete", CPK_BOOL,
|
|
"add constraints to make sure that any interpretation of a underspecified arithmetic operators is a function. The result will include additional uninterpreted functions/constants: /0, div0, mod0, 0^0, neg-root", "true");
|
|
r.insert("elim_root_objects", CPK_BOOL,
|
|
"eliminate root objects.", "true");
|
|
r.insert("elim_inverses", CPK_BOOL,
|
|
"eliminate inverse trigonometric functions (asin, acos, atan).", "true");
|
|
th_rewriter::get_param_descrs(r);
|
|
}
|
|
|
|
void operator()(goal_ref const & g,
|
|
goal_ref_buffer & result) override {
|
|
try {
|
|
tactic_report report("purify-arith", *g);
|
|
TRACE("goal", g->display(tout););
|
|
bool produce_proofs = g->proofs_enabled();
|
|
bool produce_models = g->models_enabled();
|
|
bool elim_root_objs = m_params.get_bool("elim_root_objects", true);
|
|
bool elim_inverses = m_params.get_bool("elim_inverses", true);
|
|
bool complete = m_params.get_bool("complete", true);
|
|
purify_arith_proc proc(*(g.get()), m_util, produce_proofs, elim_root_objs, elim_inverses, complete);
|
|
model_converter_ref mc;
|
|
proc(mc, produce_models);
|
|
g->add(mc.get());
|
|
g->inc_depth();
|
|
result.push_back(g.get());
|
|
}
|
|
catch (rewriter_exception & ex) {
|
|
throw tactic_exception(ex.msg());
|
|
}
|
|
}
|
|
|
|
void cleanup() override {
|
|
}
|
|
|
|
};
|
|
|
|
tactic * mk_purify_arith_tactic(ast_manager & m, params_ref const & p) {
|
|
params_ref elim_rem_p = p;
|
|
elim_rem_p.set_bool("elim_rem", true);
|
|
|
|
params_ref skolemize_p;
|
|
skolemize_p.set_bool("skolemize", false);
|
|
|
|
return and_then(using_params(mk_snf_tactic(m, skolemize_p), skolemize_p),
|
|
using_params(mk_simplify_tactic(m, elim_rem_p), elim_rem_p),
|
|
alloc(purify_arith_tactic, m, p),
|
|
mk_simplify_tactic(m, p));
|
|
}
|