Packages

package root

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

Definition Classes: root

package cmu

Definition Classes: edu

package cs

Definition Classes: cmu

package ls

Definition Classes: cs

package keymaerax

Definition Classes: ls

package cdgl

Definition Classes: keymaerax

package kaisar

Definition Classes: cdgl

AndEL

AndER

AndI

BAssignE

BAssignI

BChoiceEL

BChoiceER

BChoiceI

BComposeE

BComposeI

BDualE

BDualI

BRandomE

BRandomI

BRepeatEL

BRepeatER

BRepeatI

BSolve

BTestE

BTestI

Compare

ConstantMetric

Context

DAssignE

DAssignI

DChoiceE

DChoiceIL

DChoiceIR

DComposeE

DComposeI

DDualE

DDualI

DGo

DRandomE

DRandomI

DRepeatE

DRepeatI

DSolve

DStop

DTestEL

DTestER

DTestI

EquivEL

EquivER

EquivI

ExistsE

ExistsI

FalseE

ForallE

ForallI

Hyp

ImplyE

ImplyI

Judgement

Metric

Mon

NotE

NotI

OrE

OrIL

OrIR

Proof

ProofChecker

ProofException

TermTactics

Triv

edu.cmu.cs.ls.keymaerax.cdgl

ConstantMetric

case class ConstantMetric(t: Term, theGhost: Variable, child: Proof) extends Metric with Product with Serializable

Constant metric requires real scalar term t to decrease by at least positive constant n each iteration, and stores previous metric value in theGhost. Termination condition isZero is inequational because t is permitted to become negative

Linear Supertypes

Serializable, Serializable, Product, Equals, Metric, AnyRef, Any

Ordering

Alphabetic
By Inheritance

Inherited

ConstantMetric
Serializable
Serializable
Product
Equals
Metric
AnyRef
Any

Hide All
Show All

Visibility

Public
All

Instance Constructors

new ConstantMetric(t: Term, theGhost: Variable, child: Proof)

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
val child: Proof
def clone(): AnyRef

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@native() @throws( ... )
def decreased: Formula
Formula which expresses that the metric has decreased in the given iteration
Formula which expresses that the metric has decreased in the given iteration

Definition Classes
ConstantMetric → Metric
final def eq(arg0: AnyRef): Boolean

Definition Classes
AnyRef
def finalize(): Unit

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( classOf[java.lang.Throwable] )
final def getClass(): Class[_]

Definition Classes
AnyRef → Any
Annotations
@native()
def ghost: Formula
Formula which remembers initial metric value in ghost variable(s)
Formula which remembers initial metric value in ghost variable(s)

Definition Classes
ConstantMetric → Metric
def isAdmissible(boundTaboo: Set[Variable], freeTaboo: Set[Variable]): Boolean
boundTaboo
bound variables of loop body
freeTaboo
Free variables of context, variant, and postcondition
returns
true if metric is admissible

Definition Classes
ConstantMetric → Metric
final def isInstanceOf[T0]: Boolean

Definition Classes
Any
def isZero: Formula
Formula which holds when the metric has converged
Formula which holds when the metric has converged

Definition Classes
ConstantMetric → Metric
var n: Option[Term]
final def ne(arg0: AnyRef): Boolean

Definition Classes
AnyRef
def nonZero: Formula
Formula which holds when the metric has not converged
Formula which holds when the metric has not converged

Definition Classes
ConstantMetric → Metric
final def notify(): Unit

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

Definition Classes
AnyRef
Annotations
@native()
def setFact(fact: Formula): Unit
fact
Result of checking witness.

Definition Classes
ConstantMetric → Metric
final def synchronized[T0](arg0: ⇒ T0): T0

Definition Classes
AnyRef
val t: Term
val theGhost: Variable
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( ... )
def witness: Proof
Witness of admissibility
Witness of admissibility

Definition Classes
ConstantMetric → Metric

Packages

KeYmaera X: An aXiomatic Tactical Theorem Prover

Package Structure

Entry Points

References

ConstantMetric

case class ConstantMetric(t: Term, theGhost: Variable, child: Proof) extends Metric with Product with Serializable

Instance Constructors

Value Members

Inherited from Serializable

Inherited from Serializable

Inherited from Product

Inherited from Equals

Inherited from Metric

Inherited from AnyRef

Inherited from Any

Ungrouped

Packages

KeYmaera X: An aXiomatic Tactical Theorem Prover

Package Structure

Entry Points

References

ConstantMetric 

case class ConstantMetric(t: Term, theGhost: Variable, child: Proof) extends Metric with Product with Serializable

Instance Constructors

Value Members

Inherited from Serializable

Inherited from Serializable

Inherited from Product

Inherited from Equals

Inherited from Metric

Inherited from AnyRef

Inherited from Any

Ungrouped

ConstantMetric