class Mathematica extends Tool with QETacticTool with InvGenTool with ODESolverTool with CounterExampleTool with SimulationTool with DerivativeTool with EquationSolverTool with SimplificationTool with AlgebraTool with PDESolverTool with SOSsolveTool with LyapunovSolverTool with ToolOperationManagement
Mathematica/Wolfram Engine tool for quantifier elimination and solving differential equations.
 To do
Code Review: Move noncritical tool implementations into a separate package tactictools
 Alphabetic
 By Inheritance
 Mathematica
 ToolOperationManagement
 LyapunovSolverTool
 SOSsolveTool
 PDESolverTool
 AlgebraTool
 SimplificationTool
 EquationSolverTool
 DerivativeTool
 SimulationTool
 CounterExampleTool
 ODESolverTool
 ToolInterface
 InvGenTool
 QETacticTool
 Tool
 AnyRef
 Any
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Instance Constructors

new
Mathematica(link: MathematicaLink, name: String)
 link
How to connect to the tool, either JLink or WolframScript. Created by smitsch on 4/27/15.
Value Members

final
def
!=(arg0: Any): Boolean
 Definition Classes
 AnyRef → Any

final
def
##(): Int
 Definition Classes
 AnyRef → Any

final
def
==(arg0: Any): Boolean
 Definition Classes
 AnyRef → Any

final
def
asInstanceOf[T0]: T0
 Definition Classes
 Any

def
cancel(): Boolean
Cancels the current tool operation and returns true on success, false otherwise.
Cancels the current tool operation and returns true on success, false otherwise.
 Definition Classes
 Mathematica → Tool

def
clone(): AnyRef
 Attributes
 protected[java.lang]
 Definition Classes
 AnyRef
 Annotations
 @native() @throws( ... )

def
deriveBy(term: Term, v: Variable): Term
Computes the symbolic partial derivative of the given term by
v
.Computes the symbolic partial derivative of the given term by
v
.d(term)  dv
 term
The term whose partial derivative is sought.
 v
The variable to derive by.
 returns
The partial derivative of
term
byv
.
 Definition Classes
 Mathematica → DerivativeTool

final
def
eq(arg0: AnyRef): Boolean
 Definition Classes
 AnyRef

def
equals(arg0: Any): Boolean
 Definition Classes
 AnyRef → Any

def
finalize(): Unit
 Attributes
 protected[java.lang]
 Definition Classes
 AnyRef
 Annotations
 @throws( classOf[java.lang.Throwable] )

def
findCounterExample(formula: Formula): Option[Map[NamedSymbol, Expression]]
Returns a counterexample for the specified formula.
Returns a counterexample for the specified formula.
 formula
The formula.
 returns
A counterexample, if found. None otherwise.
 Definition Classes
 Mathematica → CounterExampleTool

def
genCLF(sys: List[ODESystem]): Option[Term]
Computes a Common Lyapunov Function for the switched system
sys
.Computes a Common Lyapunov Function for the switched system
sys
. Definition Classes
 Mathematica → LyapunovSolverTool

def
genMLF(sys: List[ODESystem], trans: List[(Int, Int, Formula)]): List[Term]
Computes a Lyapunov function for the switched system
sys
.Computes a Lyapunov function for the switched system
sys
. Definition Classes
 Mathematica → LyapunovSolverTool

def
genODECond(ode: ODESystem, assumptions: Seq[Formula], postCond: Formula): (List[Formula], List[Formula])
Returns the sufficient/necessary condition for postCond to be invariant (left of pair) also returns necessary conditions for the safety question to be true at all with those assumptions (right of pair) In either case, all formulas in the returned list must be valid
Returns the sufficient/necessary condition for postCond to be invariant (left of pair) also returns necessary conditions for the safety question to be true at all with those assumptions (right of pair) In either case, all formulas in the returned list must be valid
 Definition Classes
 Mathematica → InvGenTool

def
getAvailableWorkers: Int
<invalid inheritdoc annotation>
<invalid inheritdoc annotation>
 Definition Classes
 Mathematica → ToolOperationManagement

final
def
getClass(): Class[_]
 Definition Classes
 AnyRef → Any
 Annotations
 @native()

def
getOperationTimeout: Int
Returns the timeout duration.
Returns the timeout duration.
 Definition Classes
 Mathematica → ToolOperationManagement

def
groebnerBasis(polynomials: List[Term]): List[Term]
Computes the Gröbner Basis of the given set of polynomials (with respect to some fixed monomial order).
Computes the Gröbner Basis of the given set of polynomials (with respect to some fixed monomial order). Gröbner Bases can be made unique for the fixed monomial order, when reduced, modulo scaling by constants.
 returns
The Gröbner Basis of
polynomials
. The Gröbner Basis spans the same ideal aspolynomials
but has unique remainders of polynomialReduce.
 Definition Classes
 Mathematica → AlgebraTool
 See also
polynomialReduce()

def
hashCode(): Int
 Definition Classes
 AnyRef → Any
 Annotations
 @native()

def
init(config: Map[String, String]): Unit
Initializes the tool with toolspecific configuration parameters.
Initializes the tool with toolspecific configuration parameters.
 Definition Classes
 Mathematica → Tool

def
invgen(ode: ODESystem, assumptions: Seq[Formula], postCond: Formula): Seq[Either[Seq[(Formula, String)], Seq[(Formula, String)]]]
Returns a continuous invariant for a safety problem sent to the tool.
Returns a continuous invariant for a safety problem sent to the tool.
 ode
The differential equation for which to generate a continuous invariants.
 assumptions
Assumptions on the initial state of the ODE.
 postCond
What to prove from the invariants.
 returns
A sequence of continuous invariants, each to be proved with a diffcut chain (left=invariant, right=candidate).
 Definition Classes
 Mathematica → InvGenTool

def
isInitialized: Boolean
Checks whether this tool has been initialized already.
Checks whether this tool has been initialized already.
 Definition Classes
 Mathematica → Tool

final
def
isInstanceOf[T0]: Boolean
 Definition Classes
 Any

def
lzzCheck(ode: ODESystem, inv: Formula): Boolean
Fast check whether or not
inv
is worth proving to be an invariant ofode
.Fast check whether or not
inv
is worth proving to be an invariant ofode
. Definition Classes
 Mathematica → InvGenTool

val
name: String
Returns the name of the tool.
Returns the name of the tool.
 Definition Classes
 Mathematica → Tool

final
def
ne(arg0: AnyRef): Boolean
 Definition Classes
 AnyRef

final
def
notify(): Unit
 Definition Classes
 AnyRef
 Annotations
 @native()

final
def
notifyAll(): Unit
 Definition Classes
 AnyRef
 Annotations
 @native()

def
odeSolve(diffSys: DifferentialProgram, diffArg: Variable, iv: Map[Variable, Variable]): Option[Formula]
Returns a formula describing the symbolic solution of the specified differential equation system.
Returns a formula describing the symbolic solution of the specified differential equation system.
 diffSys
The differential equation system
 diffArg
The independent variable of the ODE, usually time
 iv
Names of initial values per variable, e.g., x > x_0
 returns
The solution, if found. None otherwise.
 Definition Classes
 Mathematica → ODESolverTool

def
pdeSolve(diffSys: DifferentialProgram): Iterator[Term]
Computes the symbolic solution of the inverse characteristic partial differential equation corresponding to an ordinary differential equation.
Computes the symbolic solution of the inverse characteristic partial differential equation corresponding to an ordinary differential equation.
 diffSys
The system of differential equations of the form x'=theta,y'=eta.
 returns
A list of solutions for
f
of the inverse characteristic PDEtheta*df/dx + eta*df/dy = 0
if found.
 Definition Classes
 Mathematica → PDESolverTool

def
polynomialReduce(polynomial: Term, GB: List[Term]): (List[Term], Term)
Computes the multivariate polynomial reduction of
polynomial
divided with respect to the set of polynomialsGB
, which is guaranteed to be unique iffGB
is a Gröbner basis.Computes the multivariate polynomial reduction of
polynomial
divided with respect to the set of polynomialsGB
, which is guaranteed to be unique iffGB
is a Gröbner basis. Returns the list of cofactors and the remainder. Repeatedly divides the leading term ofpolynomial
by a corresponding multiple of a polynomial ofGB
while possible. Each individual reduction divides the leading term ofpolynomial
by the required multiple of the leading term of the polynomial ofGB
such that those cancel. Let l(p) be the leading monomial of p and lc(p) its leading coefficient. Then each round of reduction of p:=polynomial with leading terml*X^v
picks a polynomial g in
GBand turns it into
p := p  l/lc(g) * X^v/l(g) * g
alias
p := p  (l/(lc(g) * l(g)))*X^v * g
The former leading monomial
X^v
no longer occurs in the resulting polynomial and
pgot smaller or is now 0. To determine leading terms, polynomial reduction uses the same fixed monomial order that groeberBasis() uses. The remainders will be unique (independent of the order of divisions) iff
GBis a Gröbner Basis.
 polynomial
the multivariate polynomial to divide by the elements of
GB
until saturation. GB
the set of multivariate polynomials that
polynomial
will repeatedly be divided by. The result of this algorithm is particularly insightful (and has unique remainders) ifGB
is a Gröbner Basis. returns
(coeff, rem) where
rem
is the result of multivariate polynomial division ofpolynomial
byGB
andcoeff
are the respective coefficients of the polynomials inGB
that explain the result. That ispolynomial == coeff(1)*GB(1) + coeff(2)*GB(2) + ... + coeff(n)*GB(n) + rem
alias
rem == polynomial  coeff(1)*GB(1)  coeff(2)*GB(2)  ...  coeff(n)*GB(n)
In addition, the remainder
rem
is small in that its leading monomial cannot be divided by leading monomials of any polynomial inGB
. The resultrem
is unique whenGB
is a Gröbner Basis.
 Definition Classes
 Mathematica → AlgebraTool
polynomialReduce("y^3 + 2*x^2*y".asTerm, List("x^2y".asTerm, "y^2+5".asTerm)) = ((2*y :: 2 + y), 5*y10) // because y^3 + 2*x^2*y == (2*y) * (x^2y) + (2+y) * (y^2+5) + (5*y10)
 See also
groebnerBasis()
Example: 
def
qe(goal: Goal, continueOnFalse: Boolean): (Goal, Formula)
Returns the result of the firsttofinish (sub)goal in
g
(result of the firsttofinish subgoal of OneOf, else result of Atom or AllOf).Returns the result of the firsttofinish (sub)goal in
g
(result of the firsttofinish subgoal of OneOf, else result of Atom or AllOf). WhencontinueOnFalse
is set, continues trying other options in case a finished option's result is false. Definition Classes
 Mathematica → QETacticTool

def
qe(formula: Formula): Lemma
Quantifier elimination on the specified formula, returns an equivalent quantifierfree formula plus Mathematica input/output as evidence
Quantifier elimination on the specified formula, returns an equivalent quantifierfree formula plus Mathematica input/output as evidence
 formula
The formula whose quantifierfree equivalent is sought.
 returns
A lemma showing equivalence between
formula
and a quantifierfree formula, with tool evidence.
 Definition Classes
 Mathematica → QETacticTool

def
quotientRemainder(term: Term, div: Term, x: Variable): (Term, Term)
Computes the quotient and remainder of
term
divided bydiv
.Computes the quotient and remainder of
term
divided bydiv
. term
the polynomial term to divide, considered as a univariate polynomial in variable
v
with coefficients that may have other variables. div
the polynomial term to divide
term
by, considered as a univariate polynomial in variablev
with coefficients that may have other variables.
 Definition Classes
 Mathematica → AlgebraTool
quotientRemainder("6*x^2+4*x+8".asTerm, "2*x".asTerm, Variable("x")) == (3*x+2, 8) // because (6*x^2+4*x+8) == (3*x+2) * (2*x) + 8 // so the result of division is 3*x+2 with remainder 8
Example: 
def
refuteODE(ode: ODESystem, assumptions: Seq[Formula], postCond: Formula): Option[Map[NamedSymbol, Expression]]
Finds counterexamples to an ODE safety conjecture.
Finds counterexamples to an ODE safety conjecture.
 Definition Classes
 Mathematica → InvGenTool

def
restart(): Unit
Restarts the MathKernel with the current configuration
Restarts the MathKernel with the current configuration
 Definition Classes
 Mathematica → Tool

def
setOperationTimeout(timeout: Int): Unit
Sets a maximum duration of this tool's operations (e.g., QE).
Sets a maximum duration of this tool's operations (e.g., QE).
 Definition Classes
 Mathematica → ToolOperationManagement

def
shutdown(): Unit
Closes the connection to Mathematica
Closes the connection to Mathematica
 Definition Classes
 Mathematica → Tool

def
simplify(expr: Term, assumptions: List[Formula]): Term
 Definition Classes
 Mathematica → SimplificationTool

def
simplify(expr: Formula, assumptions: List[Formula]): Formula
 Definition Classes
 Mathematica → SimplificationTool

def
simplify(expr: Expression, assumptions: List[Formula]): Expression
Simplifies the given expression
expr
, under the list of assumptions.Simplifies the given expression
expr
, under the list of assumptions. expr
The formula or term to simplify.
 assumptions
The list of logical formulas whose conjunction is assumed to hold during the simplification. The assumptions are allowed to contain additional conjunctions.
 returns
A simplified version of
expr
.
 Definition Classes
 Mathematica → SimplificationTool
simplify("a*x^2+b^2 > a*x^3+b*abs(b)".asFormula, "x>1".asFormula :: "b>0".asFormula::Nil) == "a<0".asFormula simplify("a*x^2+b^2 > a*x^3+b*abs(b)".asFormula, "x>1 && b>0".asFormula::Nil) == "a<0".asFormula
Example: 
def
simulate(initial: Formula, stateRelation: Formula, steps: Int = 10, n: Int = 1): Simulation
Returns a list of simulated states, where consecutive states in the list satisfy 'stateRelation'.
Returns a list of simulated states, where consecutive states in the list satisfy 'stateRelation'. The state relation is a modalityfree firstorder formula. The simulation starts in a state where initial holds (firstorder formula).
 initial
A firstorder formula describing the initial state.
 stateRelation
A firstorder formula describing the relation between consecutive states. The relationship is by name convention: postfix 'pre': prior state; no postfix: posterior state.
 steps
The length of the simulation run (i.e., number of states).
 n
The number of simulations (different initial states) to create.
 returns
'n' lists (length 'steps') of simulated states.
 Definition Classes
 Mathematica → SimulationTool

def
simulateRun(initial: SimState, stateRelation: Formula, steps: Int = 10): SimRun
Returns a list of simulated states, where consecutive states in the list satisfy 'stateRelation'.
Returns a list of simulated states, where consecutive states in the list satisfy 'stateRelation'. The state relation is a modalityfree firstorder formula. The simulation starts in the specified initial state.
 initial
The initial state: concrete values .
 stateRelation
A firstorder formula describing the relation between consecutive states. The relationship is by name convention: postfix 'pre': prior state; no postfix: posterior state.
 steps
The length of the simulation run (i.e., number of states).
 returns
A list (length 'steps') of simulated states.
 Definition Classes
 Mathematica → SimulationTool

def
solve(equations: Formula, vars: List[Expression]): Option[Formula]
Computes the symbolic solution of an equation system written as a conjunction of equations.
Computes the symbolic solution of an equation system written as a conjunction of equations.
 equations
The system of equations as a conjunction of equations.
 vars
The variables or symbols to solve for. Within reason, it may also be possible to solve for compound expressions like solve for j(z).
 returns
The solution if found; None otherwise The solution should be a conjunction of explicit equations for the vars. Or a disjunction of a conjunction of explicit equations for the vars.
 Definition Classes
 Mathematica → EquationSolverTool
solve("z+1=3&x+5=z1".asFormula, Variable("z")::Variable("x")::Nil) == Some("z=2&x=4")
Example: 
def
sosSolve(polynomials: List[Term], variables: List[Term], degree: Int, timeout: Option[Int]): Result
Returns a continuous invariant for a safety problem sent to the tool.
Returns a continuous invariant for a safety problem sent to the tool.
 returns
(1 + sos, cofactors) such that (cofactors, polynomials).zipped.map(Times) = 1 + sos.
 Definition Classes
 Mathematica → SOSsolveTool

final
def
synchronized[T0](arg0: ⇒ T0): T0
 Definition Classes
 AnyRef

def
toString(): String
 Definition Classes
 AnyRef → Any

final
def
wait(): Unit
 Definition Classes
 AnyRef
 Annotations
 @throws( ... )

final
def
wait(arg0: Long, arg1: Int): Unit
 Definition Classes
 AnyRef
 Annotations
 @throws( ... )

final
def
wait(arg0: Long): Unit
 Definition Classes
 AnyRef
 Annotations
 @native() @throws( ... )
KeYmaera X: An aXiomatic Tactical Theorem Prover
KeYmaera X is a theorem prover for differential dynamic logic (dL), a logic for specifying and verifying properties of hybrid systems with mixed discrete and continuous dynamics. Reasoning about complicated hybrid systems requires support for sophisticated proof techniques, efficient computation, and a user interface that crystallizes salient properties of the system. KeYmaera X allows users to specify custom proof search techniques as tactics, execute tactics in parallel, and interface with partial proofs via an extensible user interface.
http://keymaeraX.org/
Concrete syntax for input language Differential Dynamic Logic
Package Structure
Main documentation entry points for KeYmaera X API:
edu.cmu.cs.ls.keymaerax.core
 KeYmaera X kernel, proof certificates, main data structuresExpression
 Differential dynamic logic expressions:Term
,Formula
,Program
Sequent
 Sequents of formulasProvable
 Proof certificates transformed by rules/axiomsRule
 Proof rules as well asUSubstOne
for (onepass) uniform substitutions and renaming.StaticSemantics
 Static semantics with free and bound variable analysisKeYmaeraXParser
.edu.cmu.cs.ls.keymaerax.parser
 Parser and pretty printer with concrete syntax and notation for differential dynamic logic.KeYmaeraXPrettyPrinter
 Pretty printer producing concrete KeYmaera X syntaxKeYmaeraXParser
 Parser reading concrete KeYmaera X syntaxKeYmaeraXArchiveParser
 Parser reading KeYmaera X model and proof archive.kyx
filesDLParser
 Combinator parser reading concrete KeYmaera X syntaxDLArchiveParser
 Combinator parser reading KeYmaera X model and proof archive.kyx
filesedu.cmu.cs.ls.keymaerax.infrastruct
 Prover infrastructure outside the kernelUnificationMatch
 Unification algorithmRenUSubst
 Renaming Uniform Substitution quickly combining kernel's renaming and substitution.Context
 Representation for contexts of formulas in which they occur.Augmentors
 Augmenting formula and expression data structures with additional functionalityExpressionTraversal
 Generic traversal functionality for expressionsedu.cmu.cs.ls.keymaerax.bellerophon
 Bellerophon tactic language and tactic interpreterBelleExpr
 Tactic language expressionsSequentialInterpreter
 Sequential tactic interpreter for Bellerophon tacticsedu.cmu.cs.ls.keymaerax.btactics
 Bellerophon tactic library for conducting proofs.TactixLibrary
 Main KeYmaera X tactic library including many proof tactics.HilbertCalculus
 Hilbert Calculus for differential dynamic logicSequentCalculus
 Sequent Calculus for propositional and firstorder logicHybridProgramCalculus
 Hybrid Program Calculus for differential dynamic logicDifferentialEquationCalculus
 Differential Equation Calculus for differential dynamic logicUnifyUSCalculus
 Unificationbased uniform substitution calculus underlying the other calculi[edu.cmu.cs.ls.keymaerax.btactics.UnifyUSCalculus.ForwardTactic ForwardTactic]
 Forward tactic framework for conducting proofs from premises to conclusionsedu.cmu.cs.ls.keymaerax.lemma
 Lemma mechanismLemma
 Lemmas are Provables stored under a name, e.g., in files.LemmaDB
 Lemma database stored in files or database etc.edu.cmu.cs.ls.keymaerax.tools.qe
 Real arithmetic backend solversMathematicaQETool
 Mathematica interface for real arithmetic.Z3QETool
 Z3 interface for real arithmetic.edu.cmu.cs.ls.keymaerax.tools.ext
 Extended backends for noncritical ODE solving, counterexamples, algebra, simplifiers, etc.Mathematica
 Mathematica interface for ODE solving, algebra, simplification, invariant generation, etc.Z3
 Z3 interface for real arithmetic including simplifiers.Entry Points
Additional entry points and usage points for KeYmaera X API:
edu.cmu.cs.ls.keymaerax.launcher.KeYmaeraX
 Commandline launcher for KeYmaera X supports commandline argumenthelp
to obtain usage informationedu.cmu.cs.ls.keymaerax.btactics.AxIndex
 Axiom indexing data structures with keys and recursors for canonical proof strategies.edu.cmu.cs.ls.keymaerax.btactics.DerivationInfo
 Metainformation on all derivation steps (axioms, derived axioms, proof rules, tactics) with userinterface info.edu.cmu.cs.ls.keymaerax.bellerophon.UIIndex
 Index determining which canonical reasoning steps to display on the KeYmaera X User Interface.edu.cmu.cs.ls.keymaerax.btactics.Ax
 Registry for derived axioms and axiomatic proof rules that are proved from the core.References
Full references on KeYmaera X are provided at http://keymaeraX.org/. The main references are the following:
1. André Platzer. A complete uniform substitution calculus for differential dynamic logic. Journal of Automated Reasoning, 59(2), pp. 219265, 2017.
2. Nathan Fulton, Stefan Mitsch, JanDavid Quesel, Marcus Völp and André Platzer. KeYmaera X: An axiomatic tactical theorem prover for hybrid systems. In Amy P. Felty and Aart Middeldorp, editors, International Conference on Automated Deduction, CADE'15, Berlin, Germany, Proceedings, volume 9195 of LNCS, pp. 527538. Springer, 2015.
3. André Platzer. Logical Foundations of CyberPhysical Systems. Springer, 2018. Videos