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Micromega: tactics for solving arithmetic goals over ordered rings

Authors:Frédéric Besson and Evgeny Makarov

Short description of the tactics

The Psatz module (Require Import Psatz.) gives access to several tactics for solving arithmetic goals over \(\mathbb{Z}\), \(\mathbb{Q}\), and \(\mathbb{R}\) [1]. It also possible to get the tactics for integers by a Require Import Lia, rationals Require Import Lqa and reals Require Import Lra.

  • lia is a decision procedure for linear integer arithmetic;
  • nia is an incomplete proof procedure for integer non-linear arithmetic;
  • lra is a decision procedure for linear (real or rational) arithmetic;
  • nra is an incomplete proof procedure for non-linear (real or rational) arithmetic;
  • psatz D n where D is \(\mathbb{Z}\) or \(\mathbb{Q}\) or \(\mathbb{R}\), and n is an optional integer limiting the proof search depth, is an incomplete proof procedure for non-linear arithmetic. It is based on John Harrison’s HOL Light driver to the external prover csdp [2]. Note that the csdp driver is generating a proof cache which makes it possible to rerun scripts even without csdp.

The tactics solve propositional formulas parameterized by atomic arithmetic expressions interpreted over a domain \(D \in \{\mathbb{Z},\mathbb{Q},\mathbb{R}\}\). The syntax of the formulas is the following:

F ::=  A ∣ P ∣ True ∣ False ∣ F ∧ F ∣ F ∨ F ∣ F ↔ F ∣ F → F ∣ ¬ F
A ::=  p = p ∣ p > p ∣ p < p ∣ p ≥ p ∣ p ≤ p
p ::=  c ∣ x ∣ −p ∣ p − p ∣ p + p ∣ p × p ∣ p ^ n

where \(c\) is a numeric constant, \(x \in D\) is a numeric variable, the operators \(−, +, ×\) are respectively subtraction, addition, and product; \(p ^ n\) is exponentiation by a constant \(n\), \(P\) is an arbitrary proposition. For \(\mathbb{Q}\), equality is not Leibniz equality = but the equality of rationals ==.

For \(\mathbb{Z}\) (resp. \(\mathbb{Q}\)), \(c\) ranges over integer constants (resp. rational constants). For \(\mathbb{R}\), the tactic recognizes as real constants the following expressions:

c ::= R0 | R1 | Rmul(c,c) | Rplus(c,c) | Rminus(c,c) | IZR z | IQR q | Rdiv(c,c) | Rinv c

where \(z\) is a constant in \(\mathbb{Z}\) and \(q\) is a constant in \(\mathbb{Q}\). This includes integer constants written using the decimal notation, i.e., c%R.

Positivstellensatz refutations

The name psatz is an abbreviation for positivstellensatz – literally "positivity theorem" – which generalizes Hilbert’s nullstellensatz. It relies on the notion of Cone. Given a (finite) set of polynomials \(S\), \(\mathit{Cone}(S)\) is inductively defined as the smallest set of polynomials closed under the following rules:

\(\begin{array}{l} \dfrac{p \in S}{p \in \mathit{Cone}(S)} \quad \dfrac{}{p^2 \in \mathit{Cone}(S)} \quad \dfrac{p_1 \in \mathit{Cone}(S) \quad p_2 \in \mathit{Cone}(S) \quad \Join \in \{+,*\}} {p_1 \Join p_2 \in \mathit{Cone}(S)}\\ \end{array}\)

The following theorem provides a proof principle for checking that a set of polynomial inequalities does not have solutions [3].

Theorem (Psatz). Let \(S\) be a set of polynomials. If \(-1\) belongs to \(\mathit{Cone}(S)\), then the conjunction \(\bigwedge_{p \in S} p\ge 0\) is unsatisfiable. A proof based on this theorem is called a positivstellensatz refutation. The tactics work as follows. Formulas are normalized into conjunctive normal form \(\bigwedge_i C_i\) where \(C_i\) has the general form \((\bigwedge_{j\in S_i} p_j \Join 0) \to \mathit{False}\) and \(\Join \in \{>,\ge,=\}\) for \(D\in \{\mathbb{Q},\mathbb{R}\}\) and \(\Join \in \{\ge, =\}\) for \(\mathbb{Z}\).

For each conjunct \(C_i\), the tactic calls an oracle which searches for \(-1\) within the cone. Upon success, the oracle returns a cone expression that is normalized by the ring tactic (see The ring and field tactic families) and checked to be \(-1\).

lra: a decision procedure for linear real and rational arithmetic

lra

This tactic is searching for linear refutations using Fourier elimination [4]. As a result, this tactic explores a subset of the Cone defined as

\(\mathit{LinCone}(S) =\left\{ \left. \sum_{p \in S} \alpha_p \times p~\right|~\alpha_p \mbox{ are positive constants} \right\}\)

The deductive power of lra overlaps with the one of field tactic e.g., \(x = 10 * x / 10\) is solved by lra.

lia: a tactic for linear integer arithmetic

lia

This tactic offers an alternative to the omega and romega tactics. Roughly speaking, the deductive power of lia is the combined deductive power of ring_simplify and omega. However, it solves linear goals that omega and romega do not solve, such as the following so-called omega nightmare [Pug92].

Goal forall x y,   27 <= 11 * x + 13 * y <= 45 ->   -10 <= 7 * x - 9 * y <= 4 -> False.
Toplevel input, characters 52-55: > Goal forall x y, 27 <= 11 * x + 13 * y <= 45 -> -10 <= 7 * x - 9 * y <= 4 -> False. > ^^^ Error: Cannot interpret this number as a value of type nat

The estimation of the relative efficiency of lia vs omega and romega is under evaluation.

High level view of lia

Over \(\mathbb{R}\), positivstellensatz refutations are a complete proof principle [5]. However, this is not the case over \(\mathbb{Z}\). Actually, positivstellensatz refutations are not even sufficient to decide linear integer arithmetic. The canonical example is \(2 * x = 1 -> \mathtt{False}\) which is a theorem of \(\mathbb{Z}\) but not a theorem of \({\mathbb{R}}\). To remedy this weakness, the lia tactic is using recursively a combination of:

  • linear positivstellensatz refutations;
  • cutting plane proofs;
  • case split.

Cutting plane proofs

are a way to take into account the discreteness of \(\mathbb{Z}\) by rounding up (rational) constants up-to the closest integer.

Theorem Bound on the ceiling function

Let \(p\) be an integer and \(c\) a rational constant. Then \(p \ge c \rightarrow p \ge \lceil{c}\rceil\).

For instance, from 2 x = 1 we can deduce

  • \(x \ge 1/2\) whose cut plane is \(x \ge \lceil{1/2}\rceil = 1\);
  • \(x \le 1/2\) whose cut plane is \(x \le \lfloor{1/2}\rfloor = 0\).

By combining these two facts (in normal form) \(x − 1 \ge 0\) and \(-x \ge 0\), we conclude by exhibiting a positivstellensatz refutation: \(−1 \equiv x−1 + −x \in \mathit{Cone}({x−1,x})\).

Cutting plane proofs and linear positivstellensatz refutations are a complete proof principle for integer linear arithmetic.

Case split

enumerates over the possible values of an expression.

Theorem. Let \(p\) be an integer and \(c_1\) and \(c_2\) integer constants. Then:

\(c_1 \le p \le c_2 \Rightarrow \bigvee_{x \in [c_1,c_2]} p = x\)

Our current oracle tries to find an expression \(e\) with a small range \([c_1,c_2]\). We generate \(c_2 − c_1\) subgoals which contexts are enriched with an equation \(e = i\) for \(i \in [c_1,c_2]\) and recursively search for a proof.

nra: a proof procedure for non-linear arithmetic

nra

This tactic is an experimental proof procedure for non-linear arithmetic. The tactic performs a limited amount of non-linear reasoning before running the linear prover of lra. This pre-processing does the following:

  • If the context contains an arithmetic expression of the form \(e[x^2]\) where \(x\) is a monomial, the context is enriched with \(x^2 \ge 0\);
  • For all pairs of hypotheses \(e_1 \ge 0\), \(e_2 \ge 0\), the context is enriched with \(e_1 \times e_2 \ge 0\).

After this pre-processing, the linear prover of lra searches for a proof by abstracting monomials by variables.

nia: a proof procedure for non-linear integer arithmetic

nia

This tactic is a proof procedure for non-linear integer arithmetic. It performs a pre-processing similar to nra. The obtained goal is solved using the linear integer prover lia.

psatz: a proof procedure for non-linear arithmetic

psatz

This tactic explores the \(\mathit{Cone}\) by increasing degrees – hence the depth parameter \(n\). In theory, such a proof search is complete – if the goal is provable the search eventually stops. Unfortunately, the external oracle is using numeric (approximate) optimization techniques that might miss a refutation.

To illustrate the working of the tactic, consider we wish to prove the following Coq goal:

Require Import ZArith Psatz.
[Loading ML file quote_plugin.cmxs ... done] [Loading ML file newring_plugin.cmxs ... done] [Loading ML file omega_plugin.cmxs ... done] [Loading ML file r_syntax_plugin.cmxs ... done] [Loading ML file micromega_plugin.cmxs ... done]
Open Scope Z_scope.
Goal forall x, -x^2 >= 0 -> x - 1 >= 0 -> False.
1 subgoal ============================ forall x : Z, - x ^ 2 >= 0 -> x - 1 >= 0 -> False
intro x.
1 subgoal x : Z ============================ - x ^ 2 >= 0 -> x - 1 >= 0 -> False
psatz Z 2.
Toplevel input, characters 0-9: > psatz Z 2. > ^^^^^^^^^ Error: In nested Ltac calls to "psatz (constr) (int_or_var)", "xpsatz" and "psatz_Z (int_or_var) (tactic)", last call failed. Tactic failure: Skipping what remains of this tactic: the complexity of the goal requires the use of a specialized external tool called csdp. Unfortunately Coq isn't aware of the presence of any "csdp" executable in the path. Csdp packages are provided by some OS distributions; binaries and source code can be downloaded from https://projects.coin-or.org/Csdp.

As shown, such a goal is solved by intro x. psatz Z 2.. The oracle returns the cone expression \(2 \times (x-1) + (\mathbf{x-1}) \times (\mathbf{x−1}) + -x^2\) (polynomial hypotheses are printed in bold). By construction, this expression belongs to \(\mathit{Cone}({−x^2,x -1})\). Moreover, by running ring we obtain \(-1\). By Theorem Psatz, the goal is valid.

[1]Support for nat and N is obtained by pre-processing the goal with the zify tactic.
[2]Sources and binaries can be found at https://projects.coin-or.org/Csdp
[3]Variants deal with equalities and strict inequalities.
[4]More efficient linear programming techniques could equally be employed.
[5]In practice, the oracle might fail to produce such a refutation.