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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.

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 structures
- Expression - Differential dynamic logic expressions: Term, Formula, Program
- Sequent - Sequents of formulas
- Provable - Proof certificates transformed by rules/axioms
- Rule - Proof rules as well as USubstOne for (one-pass) uniform substitutions and renaming.
- StaticSemantics - Static semantics with free and bound variable analysis
- Instead of constructing dL expressions from constructors, it is easier to use the KeYmaeraXParser.
edu.cmu.cs.ls.keymaerax.parser - Parser and pretty printer with concrete syntax and notation for differential dynamic logic.
- Concrete syntax for input language Differential Dynamic Logic
- KeYmaeraXPrettyPrinter - Pretty printer producing concrete KeYmaera X syntax
- KeYmaeraXParser - Parser reading concrete KeYmaera X syntax
- KeYmaeraXArchiveParser - Parser reading KeYmaera X model and proof archive .kyx files
- DLParser - Combinator parser reading concrete KeYmaera X syntax
- DLArchiveParser - Combinator parser reading KeYmaera X model and proof archive .kyx files
edu.cmu.cs.ls.keymaerax.infrastruct - Prover infrastructure outside the kernel
- UnificationMatch - Unification algorithm
- RenUSubst - 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 functionality
- ExpressionTraversal - Generic traversal functionality for expressions
edu.cmu.cs.ls.keymaerax.bellerophon - Bellerophon tactic language and tactic interpreter
- BelleExpr - Tactic language expressions
- SequentialInterpreter - Sequential tactic interpreter for Bellerophon tactics
edu.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 logic
- SequentCalculus - Sequent Calculus for propositional and first-order logic
- HybridProgramCalculus - Hybrid Program Calculus for differential dynamic logic
- DifferentialEquationCalculus - Differential Equation Calculus for differential dynamic logic
- UnifyUSCalculus - Unification-based 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 conclusions
edu.cmu.cs.ls.keymaerax.lemma - Lemma mechanism
- Lemma - 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 back-end solvers
- MathematicaQETool - Mathematica interface for real arithmetic.
- Z3QETool - Z3 interface for real arithmetic.
edu.cmu.cs.ls.keymaerax.tools.ext - Extended back-ends 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 - Command-line launcher for KeYmaera X supports command-line argument -help to obtain usage information
edu.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 - Meta-information on all derivation steps (axioms, derived axioms, proof rules, tactics) with user-interface 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. 219-265, 2017.

2. Nathan Fulton, Stefan Mitsch, Jan-David 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. 527-538. Springer, 2015.

3. André Platzer. Logical Foundations of Cyber-Physical Systems. Springer, 2018. Videos

Definition Classes: root

package edu

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package cmu

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package cs

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package ls

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package keymaerax

Definition Classes: ls

package cdgl

Definition Classes: keymaerax

package kaisar

Definition Classes: cdgl

AndEL

AndER

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BAssignE

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BChoiceEL

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BTestE

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DAssignE

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edu.cmu.cs.ls.keymaerax.cdgl

ProofChecker

object ProofChecker

Checks proof terms.

See also: ProofChecker.apply

Linear Supertypes

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ProofChecker
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type Indices = Map[Either[String, String], Option[Int]]

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final def !=(arg0: Any): Boolean

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final def ##(): Int

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final def ==(arg0: Any): Boolean

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def apply(G: Context, M: Proof): Formula
Checks whether G |- M : P for some P
Checks whether G |- M : P for some P
G
Context in which to check term
M
Term to check
returns
Unique P such that G |- M : P, if any

Exceptions thrown
[[ProofException]] if M does not check in context G
final def asInstanceOf[T0]: T0

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def clone(): AnyRef

Attributes
protected[java.lang]
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@native() @throws( ... )
def deriveFormula(p: Formula, ode: DifferentialProgram): Formula
returns
compute differential of formula p
def deriveFormula(p: Formula, map: Map[Variable, Term]): Formula
returns
compute differential of formula p

See also
kaisar.ProofChecker
def deriveTerm(e: Term, ode: DifferentialProgram): Term
returns
compute differential of term f
def deriveTerm(f: Term, map: Map[Variable, Term]): Term
returns
compute differential of term f
final def eq(arg0: AnyRef): Boolean

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def equals(arg0: Any): Boolean

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def finalize(): Unit

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protected[java.lang]
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@throws( classOf[java.lang.Throwable] )
def formulaDerivative(p: Formula, map: Map[Variable, Term]): Formula
final def getClass(): Class[_]

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Annotations
@native()
def ghostVar(explicit: Option[Variable], base: Variable, taboos: List[Expression]): Variable
explicit
Optional, explicit variable annotation
base
Base variable which can be reindexed to find ghost
taboos
Expressions in which ghost must be fresh
returns
ghost variable fresh in taboos, either explicit or renaming of base.

Exceptions thrown
[[ProofException]] if explicit ghost is not fresh
def ghostVars(explicit: Option[List[Variable]], base: List[Variable], taboos: List[Expression]): List[Variable]
explicit
Optional, explicit variable annotations
base
Base variables which can be reindexed to find ghost
taboos
Expressions in which ghosts must be fresh
returns
ghost variables fresh in taboos, either explicit or renamings of base.

Exceptions thrown
[[ProofException]] if explicit ghosts are not fresh
def hashCode(): Int

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@native()
final def isInstanceOf[T0]: Boolean

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final def ne(arg0: AnyRef): Boolean

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final def notify(): Unit

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@native()
final def notifyAll(): Unit

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@native()
def qeValid(p: Formula): Boolean
p
formula of first-order constructive arithmetic
returns
whether formula is constructively valid
def solve(tvar: Variable, xys: List[(Variable, Variable)], ode: ODESystem): List[(Variable, Term)]
Solves an ODE
Solves an ODE
tvar
Variable for time
xys
Renamings for bound variables of ODE
ode
ODE to solve
returns
solution terms for each bound variable of ODE

Exceptions thrown
[[ProofException]] if ode is not integrable
final def synchronized[T0](arg0: ⇒ T0): T0

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def toString(): String

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final def wait(): Unit

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@throws( ... )
final def wait(arg0: Long, arg1: Int): Unit

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@throws( ... )
final def wait(arg0: Long): Unit

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@native() @throws( ... )

Packages

KeYmaera X: An aXiomatic Tactical Theorem Prover

Package Structure

Entry Points

References

ProofChecker

object ProofChecker

Type Members

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Packages

KeYmaera X: An aXiomatic Tactical Theorem Prover

Package Structure

Entry Points

References

ProofChecker 

object ProofChecker

Type Members

Value Members

Inherited from AnyRef

Inherited from Any

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ProofChecker