object StaticSemantics
The static semantics of differential dynamic logic. This object defines the static semantics of differential dynamic logic in terms of the free variables and bound variables that expressions have as well as their signatures.
val fml = Imply(Greater(Variable("x",None,Real), Number(5)), Forall(Seq(Variable("y",None,Real)), GreaterEqual(Variable("x",None,Real), FuncOf(Function("f",None,Real,Real), Variable("y",None,Real))))) // determine the static semantics of the above formula val stat = StaticSemantics(fml) println("Free variables " + stat.fv) println("Bound variables " + stat.bv) // determine all function, predicate and program constants occurring in the above formula println("Signature " + StaticSemantics.signature(fml)) // determine all symbols occurring in the above formula println("Symbols " + StaticSemantics.symbols(fml))
 Note
soundnesscritical
 See also
Section 2.3 in Andre Platzer. A complete uniform substitution calculus for differential dynamic logic. Journal of Automated Reasoning, 59(2), pp. 219266, 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
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sealed
case class
VCF(fv: SetLattice[Variable], bv: SetLattice[Variable]) extends Product with Serializable
Variable Categories for Formulas: Structure recording which names are free or bound in a formula.
Variable Categories for Formulas: Structure recording which names are free or bound in a formula.
 fv
Free names (maybe read)
 bv
Bound names (maybe written)
 Note
The core does not uses bv.

sealed
case class
VCP(fv: SetLattice[Variable], bv: SetLattice[Variable], mbv: SetLattice[Variable]) extends Product with Serializable
Variable Categories for Programs: Structure recording which names are free, bound, or mustbound in a program.
Variable Categories for Programs: Structure recording which names are free, bound, or mustbound in a program.
 fv
Free names (maybe read)
 bv
Bound names (maybe written on some paths)
 mbv
Mustbound names (definitely written on all paths).
Value Members

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(a: Program): VCP
Compute the static semantics of program a, i.e., its set of free and bound and mustbound variables.

def
apply(f: Formula): VCF
Compute the static semantics of formula f, i.e., its set of free and bound variables.

def
apply(t: Term): SetLattice[Variable]
Compute the static semantics of term t, i.e., the set of its free variables.

final
def
asInstanceOf[T0]: T0
 Definition Classes
 Any

def
boundVars(s: Sequent): SetLattice[Variable]
The set BV(a) of bound variables of a sequent.

def
boundVars(a: Program): SetLattice[Variable]
The set BV(a) of bound variables of program a.

def
boundVars(f: Formula): SetLattice[Variable]
The set BV(f) of bound variables of formula f.

def
clone(): AnyRef
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final
def
eq(arg0: AnyRef): Boolean
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def
equals(arg0: Any): Boolean
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def
finalize(): scala.Unit
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def
freeVars(s: Sequent): SetLattice[Variable]
The set FV(a) of free variables of a sequent.

def
freeVars(e: Expression): SetLattice[Variable]
The set FV(e) of free variables of expression e.

def
freeVars(a: Program): SetLattice[Variable]
The set FV(a) of free variables of program a.

def
freeVars(f: Formula): SetLattice[Variable]
The set FV(f) of free variables of formula f.

def
freeVars(term: Term): SetLattice[Variable]
The set FV(term) of free variables of
term
. 
final
def
getClass(): Class[_]
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hashCode(): Int
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def
isDifferential(e: Expression): Boolean
Check whether expression e is literally a properly differential term/expression, i.e.
Check whether expression e is literally a properly differential term/expression, i.e. mentions differentials or differential symbols free.
 Note
Only verbatim mentions are counted, so not via indirect Space dependency.
,(5)' and (c())' will be considered as nondifferential terms on account of not mentioning variables, but (x+y)' is differential.
,AtomicODE uses isDifferential to ensure explicit differential equation x'=e has no primes in e.
,ODESystem uses isDifferential to ensure explicit differential equation x'=e&Q has no primes in Q.
,For proper terms (not using Anything), freeVars is finite so .symbols==.toSet, so checks for literally free DifferentialSymbols.

final
def
isInstanceOf[T0]: Boolean
 Definition Classes
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final
def
ne(arg0: AnyRef): Boolean
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final
def
notify(): scala.Unit
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 @native()

final
def
notifyAll(): scala.Unit
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def
signature(s: Sequent): Set[NamedSymbol]
The signature of a sequent.

def
signature(program: Program): Set[NamedSymbol]
The signature of a program, i.e., set of function, predicate, and atomic program symbols occurring in it.
The signature of a program, i.e., set of function, predicate, and atomic program symbols occurring in it.
 Note
Soundnesscritical in data structure invariant for interpreted functions.
,Not soundnesscritical otherwise since substitution only uses it in old USubstChurch not in new USubstOne.

def
signature(formula: Formula): Set[NamedSymbol]
The signature of a formula, i.e., set of (nonlogical) function, predicate, predicational, and atomic program symbols occurring in it.
The signature of a formula, i.e., set of (nonlogical) function, predicate, predicational, and atomic program symbols occurring in it.
 Note
Soundnesscritical in data structure invariant for interpreted functions.
,Not soundnesscritical otherwise since substitution only uses it in old USubstChurch not in new USubstOne.

def
signature(term: Term): Set[NamedSymbol]
The signature of a term, i.e., set of (nonlogical) function/functional symbols occurring in it.
The signature of a term, i.e., set of (nonlogical) function/functional symbols occurring in it. Disregarding number literals.
 Note
Soundnesscritical in data structure invariant for interpreted functions.
,Not soundnesscritical otherwise since substitution only uses it in old USubstChurch not in new USubstOne.

def
signature(e: Expression): Set[NamedSymbol]
The signature of expression e.
The signature of expression e.
signature(e).toList.sort // sorts by compare of NamedSymbol, by name and index signature(e).toList.sortBy(_.name) // sorts alphabetically by name, ignores indices
 Note
Result will not be order stable, so order could be different on different runs of the prover.
,Soundnesscritical in data structure invariant for interpreted functions.
,Not soundnesscritical otherwise since substitution only uses it in old USubstChurch not in new USubstOne.
Example: 
def
spaceVars(space: Space): SetLattice[Variable]
The variables and differential symbols that are in the given state space.
The variables and differential symbols that are in the given state space.
 space
The state space whose set (lattice) of variables and differential variables to compute.
AnyArg
returns the SetLattice.allVars.Except(taboo)
returns all variables except spaceTaboos(taboos), i.e. all variables and differential variables except the taboo x and x'.

def
symbols(s: Sequent): Set[NamedSymbol]
Any symbol occurring verbatim in a sequent, whether free or bound variable or function or predicate or program constant

def
symbols(p: Program): Set[NamedSymbol]
Any (nonlogical) symbol occurring verbatim in program, whether free or bound variable or function or predicate or program constant.

def
symbols(f: Formula): Set[NamedSymbol]
Any (nonlogical) symbol occurring verbatim in formula, whether free or bound variable or function or predicate or program constant.

def
symbols(t: Term): Set[NamedSymbol]
Any (nonlogical) symbol occurring verbatim in term, whether variable or function.

def
symbols(e: Expression): Set[NamedSymbol]
Any (nonlogical) symbols occurring verbatim in expression e, whether free or bound variable or function or predicate or program constant.

final
def
synchronized[T0](arg0: ⇒ T0): T0
 Definition Classes
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def
toString(): String
 Definition Classes
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def
vars(e: Expression): SetLattice[Variable]
The set var(e) of variables of expression e, whether free or bound.

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