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 parser

Parser & Pretty-Printer for Differential Dynamic Logic

Defines the concrete syntax of differential dynamic logic as used in KeYmaera X. Provides a parser to read string and file inputs with differential dynamic logic. Conversely provides a pretty-printer to format differential dynamic logic expression data structure as human readable concrete syntax.

PrettyPrinter: Expression => String
Parser: String => Expression

Usage Overview

The default pretty-printer for the concrete syntax in KeYmaera X is in edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXPrettyPrinter. The corresponding parser for the concrete syntax in edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXParser. Also see Grammar of KeYmaera X Syntax

val parser = KeYmaeraXParser
val pp = KeYmaeraXPrettyPrinter
val input = "x^2>=0 & x<44 -> [x:=2;{x'=1&x<=10}]x>=1"
val parse = parser(input)
println("Parsed:   " + parse)
val print = pp(parse)
println("Printed:  " + print)

Printing Differential Dynamic Logic

edu.cmu.cs.ls.keymaerax.parser.PrettyPrinter defines the interface for all differential dynamic logic pretty printers in KeYmaera X.

PrettyPrinter: Expression => String

edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXPrettyPrinter implements the pretty-printer for the concrete syntax of differential dynamic logic used in KeYmaera X. A pretty-printer is an injective function from differential dynamic logic expressions to strings.

Printing formulas to strings is straightforward using edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXPrettyPrinter.apply:

val pp = KeYmaeraXPrettyPrinter
// "x < -y"
val fml0 = Less(Variable("x"),Neg(Variable("y")))
val fml0str = pp(fml0)
// "true -> [x:=1;]x>=0"
val fml1 = Imply(True, Box(Assign(Variable("x"), Number(1)), GreaterEqual(Variable("x"), Number(0))))
val fml1str = pp(fml1)

Printing Fully Parenthesized Forms

Fully parenthesized strings are obtained using the edu.cmu.cs.ls.keymaerax.parser.FullPrettyPrinter printer:

val pp = FullPrettyPrinter
// "x < -(y)"
val fml0 = Less(Variable("x"),Neg(Variable("y")))
val fml0str = pp(fml0)
// "true -> ([x:=1;](x>=0))"
val fml1 = Imply(True, Box(Assign(Variable("x"), Number(1)), GreaterEqual(Variable("x"), Number(0))))
val fml1str = pp(fml1)

The fully-parenthesized pretty printer corresponding to a pretty printer can also be obtained using edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXPrinter.fullPrinter

Parsing Differential Dynamic Logic

edu.cmu.cs.ls.keymaerax.parser.Parser defines the interface for all differential dynamic logic parsers in KeYmaera X.

Parser: String => Expression

edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXParser implements the parser for the concrete syntax of differential dynamic logic used in KeYmaera X. A parser is a function from input strings to differential dynamic logic expressions.

Parsing formulas from strings is straightforward using edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXParser.apply:

val parser = KeYmaeraXParser
val fml0 = parser("x!=5")
val fml1 = parser("x>0 -> [x:=x+1;]x>1")
val fml2 = parser("x>=0 -> [{x'=2}]x>=0")

The parser parses any dL expression, so will also accept terms or programs from strings, which will lead to appropriate types:

val parser = KeYmaeraXParser
// formulas
val fml0 = parser("x!=5")
val fml1 = parser("x>0 -> [x:=x+1;]x>1")
val fml2 = parser("x>=0 -> [{x'=2}]x>=0")
// terms
val term0 = parser("x+5")
val term1 = parser("x^2-2*x+7")
val term2 = parser("x*y/3+(x-1)^2+5*z")
// programs
val prog0 = parser("x:=1;")
val prog1 = parser("x:=1;x:=5;x:=-1;")
val prog2 = parser("x:=1;{{x'=5}++x:=0;}")

Parsing Only Formulas of Differential Dynamic Logic

The parser that only parses formulas is obtained via edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXParser.formulaParser

// the formula parser only accepts formulas
val parser = KeYmaeraXParser.formulaParser
// formulas
val fml0 = parser("x!=5")
val fml1 = parser("x>0 -> [x:=x+1;]x>1")
val fml2 = parser("x>=0 -> [{x'=2}]x>=0")
// terms will cause exceptions
try { parser("x+5") } catch {case e: ParseException => println("Rejected")}
// programs will cause exceptions
try { parser("x:=1;") } catch {case e: ParseException => println("Rejected")}

Similarly, a parser that only parses terms is obtained via edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXParser.termParser and a parser that only parses programs is obtained via edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXParser.programParser

Parsing Pretty-Printed Strings

Corresponding parsers and pretty-printers match with one another. Parsing a pretty-printed expression results in the original expression again:

parse(print(e)) == e

edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXParser and are inverses in this sense. The converse print(parse(s)) == s is not quite the case, because there can be minor spacing and superfluous parentheses differences. The following slightly weaker form still holds:

parse(print(parse(s))) == parse(s)

Parsing the pretty-print of an expression with compatible printers and parsers always gives the original expression back:

val parser = KeYmaeraXParser
val pp = KeYmaeraXPrettyPrinter
val fml = Imply(True, Box(Assign(Variable("x"), Number(1)), GreaterEqual(Variable("x"), Number(0))))
// something like "true -> [x:=1;]x>=0" modulo spacing
val print = pp(fml)
val reparse = parser(print)
if (fml == reparse) println("Print and reparse successful") else println("Discrepancy")

Pretty-Printing Parsed Strings

It can be quite helpful to print an expression that has been parsed to check how it got parsed:

val parser = KeYmaeraXParser
val pp = KeYmaeraXPrettyPrinter
val input = "x^2>=0 & x<44 -> [x:=2;{x'=1&x<=10}]x>=1"
val parse = parser(input)
println("Parsed:   " + parse)
val print = pp(parse)
println("Printed:  " + print)
println("Original: " + input)
println("Can differ slightly by spacing and parentheses")

Recall that the default pretty printer uses compact parentheses and braces. That is, it only prints them when necessary to disambiguate. For storage purposes and disambiguation it can be better to use fully parenthesized printouts, which is what edu.cmu.cs.ls.keymaerax.parser.FullPrettyPrinter achieves:

val parser = KeYmaeraXParser
val pp = FullPrettyPrinter
val input = "x^2>=0 & x<44 -> [x:=2;{x'=1&x<=10}]x>=1"
val parse = parser(input)
println("Parsed:   " + parse)
val print = pp(parse)
// (((x)^(2)>=0)&(x < 44))->([{x:=2;}{{x'=1&x<=10}}](x>=1))
println("Printed:  " + print)
println("Original: " + input)

Pretty-Printing Sequents

Sequents can be pretty-printed using the default pretty printer via edu.cmu.cs.ls.keymaerax.core.Sequent.prettyString

val parser = KeYmaeraXParser
val sequent = Sequent(IndexedSeq(parse("x<22"), parse("x>0")), IndexedSeq(parse("[x:=x+1;]x<23")))
println("Printed:  " + sequent.prettyString)

Definition Classes: keymaerax
See also: Andre Platzer. A complete uniform substitution calculus for differential dynamic logic. Journal of Automated Reasoning, 59(2), pp. 219-266, 2017.
Andre Platzer. A uniform substitution calculus for differential dynamic logic. In Amy P. Felty and Aart Middeldorp, editors, International Conference on Automated Deduction, CADE'15, Berlin, Germany, Proceedings, LNCS. Springer, 2015. A uniform substitution calculus for differential dynamic logic. arXiv 1503.01981
Andre Platzer. Differential game logic. ACM Trans. Comput. Log. 17(1), 2015. arXiv 1408.1980
Nathan Fulton, Stefan Mitsch, Jan-David Quesel, Marcus Volp and Andre 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, LNCS. Springer, 2015.
Grammar of Differential Dynamic Logic
edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXParser
edu.cmu.cs.ls.keymaerax.parser.KeYmaeraXPrettyPrinter

ArchiveParser

AtomicBinaryFormat

AtomicFormat

AxiomFileMode

BasePrettyPrinter

BinaryNotation

BinaryOpSpec

Bottom

DLArchiveParser

DLAxiomParser

DLParser

DLParserUtils

Declaration

EOF

ExpressionMode

ExternalEvidence

FormatProvider

FullPrettyPrinter

FullPrettyPrinter0

InterpretedSymbols

KeYmaera3PrettyPrinter

KeYmaeraXArchiveParser

KeYmaeraXArchiveParserBase

KeYmaeraXArchivePrinter

KeYmaeraXAxiomParser

KeYmaeraXExtendedLemmaParser

KeYmaeraXLegacyArchivePrinter

KeYmaeraXLexer

KeYmaeraXNoContractPrettyPrinter

KeYmaeraXOmitInterpretationPrettyPrinter

KeYmaeraXParser

KeYmaeraXPrecedencePrinter

KeYmaeraXPrettierPrinter

KeYmaeraXPrettyPrinter

KeYmaeraXPrinter

KeYmaeraXSkipPrinter

KeYmaeraXStoredProvableParser

KeYmaeraXWeightedPrettyPrinter

LeftAssociative

LemmaFileMode

LexException

LexerMode

Location

MixedBinary

Name

NonAssociative

NoneFormatProvider

NullaryNotation

ODEToInterpreted

OpNotation

OpSpec

ParseException

ParsedArchiveEntry

Parser

ParserHelper

ParserInit

PostfixFormat

PrefixFormat

PrettierPrintFormatProvider

PrettyPrintFormatProvider

PrettyPrinter

ProblemFileMode

ProofEvidence

Region

ReservedSymbols

RightAssociative

SequentParser

Signature

Stack

StoredProvableMode

StringConverter

SubstitutionParser

SuffixRegion

TacticReservedSymbols

Terminal

TernaryNotation

TernaryOpSpec

TernaryPrefixFormat

TokenParser

UnaryNotation

UnaryOpSpec

UnificationSubstitutionParser

UnitOpSpec

UnknownLocation

edu.cmu.cs.ls.keymaerax.parser

OpSpec

Companion trait OpSpec

object OpSpec

Differential Dynamic Logic's concrete syntax: operator notation specifications.

Note: Subtleties: sPower right associative to ensure x²3 == x⁽²3) instead of (x²⁾3. sPower < sNeg to ensure -x^{2 == -(x}2) instead of (-x)^2. NUMBER lexer does not contain - sign to enable x-5 to be parsed. Parser will make up for this, respecting binary versus unary operators. sEquiv nonassociative to ensure that p()<->q()<->r() does not parse since binding unclear. sAnd and sOr are right-associative to simplify stable position ordering during sequent decomposition.

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final def getClass(): Class[_]

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final def isInstanceOf[T0]: Boolean

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val negativeNumber: Boolean
Whether to accept negative numbers as negative numbers as opposed to unary negation applied to a number.
final def notify(): Unit

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@native()
final def notifyAll(): Unit

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@native()
def op(expr: Expression): OpSpec
The operator notation of the top-level operator of expr with opcode, precedence and associativity
val sAnd: BinaryOpSpec[Formula]
val sAssign: BinaryOpSpec[Program]
val sAssignAny: UnaryOpSpec[Program]
val sAtomicODE: BinaryOpSpec[Program]
val sBox: BinaryOpSpec[Expression]
val sChoice: BinaryOpSpec[Program]
val sCompose: BinaryOpSpec[Program]
val sDChoice: BinaryOpSpec[Program]
val sDLoop: UnaryOpSpec[Program]
val sDiamond: BinaryOpSpec[Expression]
val sDifferential: UnaryOpSpec[Term]
val sDifferentialFormula: UnaryOpSpec[Formula]
val sDifferentialProduct: BinaryOpSpec[DifferentialProduct]
val sDifferentialProgramConst: UnitOpSpec
val sDifferentialSymbol: UnaryOpSpec[Term]
val sDivide: BinaryOpSpec[Term]
val sDotFormula: UnitOpSpec
val sDotTerm: UnitOpSpec
val sDual: UnaryOpSpec[Program]
val sEOF: UnitOpSpec
Parser needs a lookahead operator when actually already done, so don't dare constructing it
val sEqual: BinaryOpSpec[Equal]
val sEquiv: BinaryOpSpec[Formula]
val sExists: BinaryOpSpec[Expression]
val sFalse: UnitOpSpec
val sForall: BinaryOpSpec[Expression]
val sFuncOf: UnaryOpSpec[Term]
val sGreater: BinaryOpSpec[Greater]
val sGreaterEqual: BinaryOpSpec[GreaterEqual]
val sIfThenElse: TernaryOpSpec[Program]
val sImply: BinaryOpSpec[Formula]
val sLess: BinaryOpSpec[Less]
val sLessEqual: BinaryOpSpec[LessEqual]
val sLoop: UnaryOpSpec[Program]
val sMinus: BinaryOpSpec[Term]
val sNeg: UnaryOpSpec[Term]
val sNone: UnitOpSpec
val sNoneDone: UnitOpSpec
Parser needs a lookahead operator when actually already done, so don't dare constructing it
val sNoneUnfinished: UnitOpSpec
Parser needs a lookahead operator when actually already done, so don't dare constructing it
val sNot: UnaryOpSpec[Formula]
val sNotEqual: BinaryOpSpec[NotEqual]
val sNothing: UnitOpSpec
val sNumber: UnitOpSpec
val sODESystem: BinaryOpSpec[Expression]
val sOr: BinaryOpSpec[Formula]
val sPair: BinaryOpSpec[Term]
val sPlus: BinaryOpSpec[Term]
val sPower: BinaryOpSpec[Term]
val sPredOf: UnaryOpSpec[PredOf]
val sPredicationalOf: UnaryOpSpec[PredicationalOf]
val sProgramConst: UnitOpSpec
val sRevImply: BinaryOpSpec[Formula]
val sSystemConst: UnitOpSpec
val sTest: UnaryOpSpec[Program]
val sTimes: BinaryOpSpec[Term]
val sTrue: UnitOpSpec
val sUnitFunctional: UnitOpSpec
val sUnitPredicational: UnitOpSpec
val sVariable: UnitOpSpec
val statementSemicolon: Boolean
Whether to terminate atomic statements with a semicolon instead of separating sequential compositions by a semicolon.
final def synchronized[T0](arg0: ⇒ T0): T0

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final def wait(): Unit

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final def wait(arg0: Long): Unit

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@native() @throws( ... )
val weakNeg: Boolean
true has unary negation - bind weakly like binary subtraction.
true has unary negation - bind weakly like binary subtraction. false has unary negation - bind strong just shy of power ^.

Packages

KeYmaera X: An aXiomatic Tactical Theorem Prover

Package Structure

Entry Points

References

Usage Overview

Printing Differential Dynamic Logic

Printing Fully Parenthesized Forms

Parsing Differential Dynamic Logic

Parsing Only Formulas of Differential Dynamic Logic

Parsing Pretty-Printed Strings

Pretty-Printing Parsed Strings

Pretty-Printing Sequents

OpSpec

Companion trait OpSpec

object OpSpec

Value Members

Inherited from AnyRef

Inherited from Any

Ungrouped

Packages

KeYmaera X: An aXiomatic Tactical Theorem Prover

Package Structure

Entry Points

References

Usage Overview

Printing Differential Dynamic Logic

Printing Fully Parenthesized Forms

Parsing Differential Dynamic Logic

Parsing Only Formulas of Differential Dynamic Logic

Parsing Pretty-Printed Strings

Pretty-Printing Parsed Strings

Pretty-Printing Sequents

OpSpec 

Companion trait OpSpec

object OpSpec

Value Members

Inherited from AnyRef

Inherited from Any

Ungrouped

OpSpec