.. _micromega: Micromega: solvers for 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 :math:`\mathbb{Q}`, :math:`\mathbb{R}`, and :math:`\mathbb{Z}` but also :g:`nat` and :g:`N`. It also possible to get the tactics for integers by a ``Require Import Lia``, rationals ``Require Import Lqa`` and reals ``Require Import Lra``. + :tacn:`lia` is a decision procedure for linear integer arithmetic; + :tacn:`nia` is an incomplete proof procedure for integer non-linear arithmetic; + :tacn:`lra` is a decision procedure for linear (real or rational) arithmetic; + :tacn:`nra` is an incomplete proof procedure for non-linear (real or rational) arithmetic; + :tacn:`psatz` ``D n`` where ``D`` is :math:`\mathbb{Z}` or :math:`\mathbb{Q}` or :math:`\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` [#csdp]_. Note that the `csdp` driver generates a *proof cache* which makes it possible to rerun scripts even without `csdp`. .. flag:: Simplex .. deprecated:: 8.14 This flag (set by default) instructs the decision procedures to use the Simplex method for solving linear goals instead of the deprecated Fourier elimination. .. opt:: Dump Arith This option (unset by default) may be set to a file path where debug info will be written. .. cmd:: Show Lia Profile This command prints some statistics about the amount of pivoting operations needed by :tacn:`lia` and may be useful to detect inefficiencies (only meaningful if flag :flag:`Simplex` is set). .. flag:: Lia Cache This flag (set by default) instructs :tacn:`lia` to cache its results in the file `.lia.cache` .. flag:: Nia Cache This flag (set by default) instructs :tacn:`nia` to cache its results in the file `.nia.cache` .. flag:: Nra Cache This flag (set by default) instructs :tacn:`nra` to cache its results in the file `.nra.cache` The tactics solve propositional formulas parameterized by atomic arithmetic expressions interpreted over a domain :math:`D \in \{\mathbb{Z},\mathbb{Q},\mathbb{R}\}`. The syntax for formulas over :math:`\mathbb{Z}` is: .. note the following is not an insertprodn .. prodn:: F ::= {| @A | P | True | False | @F /\\ @F | @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 - :token:`F` is interpreted over either `Prop` or `bool` - :n:`P` is an arbitrary proposition - :n:`c` is a numeric constant of :math:`D` - :n:`x` :math:`\in D` is a numeric variable - :n:`−`, :n:`+` and :n:`*` are respectively subtraction, addition and product - :n:`p ^ n` is exponentiation by a constant :math:`n` When :math:`F` is interpreted over `bool`, the boolean operators are `&&`, `||`, `Bool.eqb`, `Bool.implb`, `Bool.negb` and the comparisons in :math:`A` are also interpreted over the booleans (e.g., for :math:`\mathbb{Z}`, we have `Z.eqb`, `Z.gtb`, `Z.ltb`, `Z.geb`, `Z.leb`). For :math:`\mathbb{Q}`, use the equality of rationals ``==`` rather than Leibniz equality ``=``. For :math:`\mathbb{Z}` (resp. :math:`\mathbb{Q}`), :math:`c` ranges over integer constants (resp. rational constants). For :math:`\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 :math:`z` is a constant in :math:`\mathbb{Z}` and :math:`q` is a constant in :math:`\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 :math:`S`, :math:`\mathit{Cone}(S)` is inductively defined as the smallest set of polynomials closed under the following rules: :math:`\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 [#fnpsatz]_. .. _psatz_thm: **Theorem (Psatz)**. Let :math:`S` be a set of polynomials. If :math:`-1` belongs to :math:`\mathit{Cone}(S)`, then the conjunction :math:`\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 :math:`\bigwedge_i C_i` where :math:`C_i` has the general form :math:`(\bigwedge_{j\in S_i} p_j \Join 0) \to \mathit{False}` and :math:`\Join \in \{>,\ge,=\}` for :math:`D\in \{\mathbb{Q},\mathbb{R}\}` and :math:`\Join \in \{\ge, =\}` for :math:`\mathbb{Z}`. For each conjunct :math:`C_i`, the tactic calls an oracle which searches for :math:`-1` within the cone. Upon success, the oracle returns a *cone expression* that is normalized by the :tacn:`ring` tactic (see :ref:`theringandfieldtacticfamilies`) and checked to be :math:`-1`. `lra`: a decision procedure for linear real and rational arithmetic ------------------------------------------------------------------- .. tacn:: lra This tactic is searching for *linear* refutations. As a result, this tactic explores a subset of the *Cone* defined as :math:`\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 :tacn:`lra` overlaps with the one of :tacn:`field` tactic *e.g.*, :math:`x = 10 * x / 10` is solved by :tacn:`lra`. `lia`: a tactic for linear integer arithmetic --------------------------------------------- .. tacn:: lia This tactic solves linear goals over :g:`Z` by searching for *linear* refutations and cutting planes. :tacn:`lia` provides support for :g:`Z`, :g:`nat`, :g:`positive` and :g:`N` by pre-processing via the :tacn:`zify` tactic. High level view of `lia` ~~~~~~~~~~~~~~~~~~~~~~~~ Over :math:`\mathbb{R}`, *positivstellensatz* refutations are a complete proof principle [#mayfail]_. However, this is not the case over :math:`\mathbb{Z}`. Actually, *positivstellensatz* refutations are not even sufficient to decide linear *integer* arithmetic. The canonical example is :math:`2 * x = 1 \to \mathtt{False}` which is a theorem of :math:`\mathbb{Z}` but not a theorem of :math:`{\mathbb{R}}`. To remedy this weakness, the :tacn:`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 :math:`\mathbb{Z}` by rounding up (rational) constants up-to the closest integer. .. _ceil_thm: .. thm:: Bound on the ceiling function Let :math:`p` be an integer and :math:`c` a rational constant. Then :math:`p \ge c \rightarrow p \ge \lceil{c}\rceil`. For instance, from :math:`2 x = 1` we can deduce + :math:`x \ge 1/2` whose cut plane is :math:`x \ge \lceil{1/2}\rceil = 1`; + :math:`x \le 1/2` whose cut plane is :math:`x \le \lfloor{1/2}\rfloor = 0`. By combining these two facts (in normal form) :math:`x − 1 \ge 0` and :math:`-x \ge 0`, we conclude by exhibiting a *positivstellensatz* refutation: :math:`−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. .. _casesplit_thm: **Theorem**. Let :math:`p` be an integer and :math:`c_1` and :math:`c_2` integer constants. Then: :math:`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 :math:`e` with a small range :math:`[c_1,c_2]`. We generate :math:`c_2 − c_1` subgoals which contexts are enriched with an equation :math:`e = i` for :math:`i \in [c_1,c_2]` and recursively search for a proof. `nra`: a proof procedure for non-linear arithmetic -------------------------------------------------- .. tacn:: 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 :tacn:`lra`. This pre-processing does the following: + If the context contains an arithmetic expression of the form :math:`e[x^2]` where :math:`x` is a monomial, the context is enriched with :math:`x^2 \ge 0`; + For all pairs of hypotheses :math:`e_1 \ge 0`, :math:`e_2 \ge 0`, the context is enriched with :math:`e_1 \times e_2 \ge 0`. After this pre-processing, the linear prover of :tacn:`lra` searches for a proof by abstracting monomials by variables. `nia`: a proof procedure for non-linear integer arithmetic ---------------------------------------------------------- .. tacn:: nia This tactic is a proof procedure for non-linear integer arithmetic. It performs a pre-processing similar to :tacn:`nra`. The obtained goal is solved using the linear integer prover :tacn:`lia`. `psatz`: a proof procedure for non-linear arithmetic ---------------------------------------------------- .. tacn:: psatz @one_term {? @nat_or_var } This tactic explores the *Cone* by increasing degrees – hence the depth parameter :token:`nat_or_var`. 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: .. needs csdp .. coqdoc:: Require Import ZArith Psatz. Open Scope Z_scope. Goal forall x, -x^2 >= 0 -> x - 1 >= 0 -> False. intro x. psatz Z 2. As shown, such a goal is solved by ``intro x. psatz Z 2.``. The oracle returns the cone expression :math:`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 :math:`\mathit{Cone}({−x^2,x -1})`. Moreover, by running :tacn:`ring` we obtain :math:`-1`. By Theorem :ref:`Psatz `, the goal is valid. `zify`: pre-processing of arithmetic goals ------------------------------------------ .. tacn:: zify This tactic is internally called by :tacn:`lia` to support additional types, e.g., :g:`nat`, :g:`positive` and :g:`N`. Additional support is provided by the following modules: + For boolean operators (e.g., :g:`Nat.leb`), require the module :g:`ZifyBool`. + For comparison operators (e.g., :g:`Z.compare`), require the module :g:`ZifyComparison`. + For native 63 bit integers, require the module :g:`ZifyInt63`. :tacn:`zify` can also be extended by rebinding the tactics `Zify.zify_pre_hook` and `Zify.zify_post_hook` that are respectively run in the first and the last steps of :tacn:`zify`. + To support :g:`Z.div` and :g:`Z.modulo`: ``Ltac Zify.zify_post_hook ::= Z.div_mod_to_equations``. + To support :g:`Z.quot` and :g:`Z.rem`: ``Ltac Zify.zify_post_hook ::= Z.quot_rem_to_equations``. + To support :g:`Z.div`, :g:`Z.modulo`, :g:`Z.quot` and :g:`Z.rem`: either ``Ltac Zify.zify_post_hook ::= Z.to_euclidean_division_equations`` or ``Ltac Zify.zify_convert_to_euclidean_division_equations_flag ::= constr:(true)``. The :tacn:`zify` tactic can be extended with new types and operators by declaring and registering new typeclass instances using the following commands. The typeclass declarations can be found in the module ``ZifyClasses`` and the default instances can be found in the module ``ZifyInst``. .. cmd:: Add Zify @add_zify @qualid .. insertprodn add_zify add_zify .. prodn:: add_zify ::= {| InjTyp | BinOp | UnOp | CstOp | BinRel | UnOpSpec | BinOpSpec } | {| PropOp | PropBinOp | PropUOp | Saturate } Registers an instance of the specified typeclass. The typeclass type (e.g. :g:`BinOp Z.mul` or :g:`BinRel (@eq Z)`) has the additional constraint that the non-implicit argument (here, :g:`Z.mul` or :g:`(@eq Z)`) is either a :n:`@reference` (here, :g:`Z.mul`) or the application of a :n:`@reference` (here, :g:`@eq`) to a sequence of :n:`@one_term`. .. cmd:: Show Zify @show_zify .. insertprodn show_zify show_zify .. prodn:: show_zify ::= {| InjTyp | BinOp | UnOp | CstOp | BinRel | UnOpSpec | BinOpSpec | Spec } Prints instances for the specified typeclass. For instance, :cmd:`Show Zify` ``InjTyp`` prints the list of types that supported by :tacn:`zify` i.e., :g:`Z`, :g:`nat`, :g:`positive` and :g:`N`. .. [#csdp] Sources and binaries can be found at https://projects.coin-or.org/Csdp .. [#fnpsatz] Variants deal with equalities and strict inequalities. .. [#mayfail] In practice, the oracle might fail to produce such a refutation. .. comment in original TeX: .. %% \paragraph{The {\tt sos} tactic} -- where {\tt sos} stands for \emph{sum of squares} -- tries to prove that a .. %% single polynomial $p$ is positive by expressing it as a sum of squares \emph{i.e.,} $\sum_{i\in S} p_i^2$. .. %% This amounts to searching for $p$ in the cone without generators \emph{i.e.}, $Cone(\{\})$.