case class Function(name: String, index: Option[Int] = None, domain: Sort, sort: Sort, interp: Option[Formula] = None) extends NamedSymbol with Product with Serializable
Function symbol or predicate symbol or predicational symbol name_index:domain->sort
- domain
the sort of expected arguments.
- sort
the sort resulting when this function/predicate/predicational symbol has been applied to an argument.
- interp
if present, this function symbol
fhas a fixed interpretation defined byy = f(x) <-> P(y,x)where P is the interp formula, substituting (y,x) for the dot terms. That is
._0 = f(._1) <-> P(._0,._1)
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Instance Constructors
-
new
Function(name: String, index: Option[Int] = None, domain: Sort, sort: Sort, interp: Option[Formula] = None)
- domain
the sort of expected arguments.
- sort
the sort resulting when this function/predicate/predicational symbol has been applied to an argument.
- interp
if present, this function symbol
fhas a fixed interpretation defined byy = f(x) <-> P(y,x)where P is the interp formula, substituting (y,x) for the dot terms. That is
._0 = f(._1) <-> P(._0,._1)
Value Members
-
final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
def
<(that: NamedSymbol): Boolean
- Definition Classes
- Ordered
-
def
<=(that: NamedSymbol): Boolean
- Definition Classes
- Ordered
-
final
def
==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
def
>(that: NamedSymbol): Boolean
- Definition Classes
- Ordered
-
def
>=(that: NamedSymbol): Boolean
- Definition Classes
- Ordered
-
final
def
asInstanceOf[T0]: T0
- Definition Classes
- Any
-
def
asString: String
Get name with index of this NamedSymbol.
Get name with index of this NamedSymbol.
- Definition Classes
- Function → NamedSymbol
-
def
clone(): AnyRef
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @native() @throws( ... )
-
def
compare(other: NamedSymbol): Int
Compare named symbols lexicographically: by name, index, category.
Compare named symbols lexicographically: by name, index, category.
- returns
0 for equal symbols <0 if this is lexicographically before
other>0 if this is lexicographically afterother
- Definition Classes
- NamedSymbol → Ordered
- Note
not used in the core, so not soundness-critical, but breaks tactics if wrong.
-
def
compareTo(that: NamedSymbol): Int
- Definition Classes
- Ordered → Comparable
- val domain: Sort
-
final
def
eq(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
def
finalize(): scala.Unit
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @throws( classOf[java.lang.Throwable] )
-
def
fullString: String
Full string with names and full types
Full string with names and full types
- Definition Classes
- Function → NamedSymbol
-
final
def
getClass(): Class[_]
- Definition Classes
- AnyRef → Any
- Annotations
- @native()
-
val
index: Option[Int]
- Definition Classes
- Function → NamedSymbol
- val interp: Option[Formula]
- val interpreted: Boolean
-
final
def
isInstanceOf[T0]: Boolean
- Definition Classes
- Any
-
final
val
kind: Kind
What kind of an expression this is, e.g., TermKind, FormulaKind, ProgramKind.
What kind of an expression this is, e.g., TermKind, FormulaKind, ProgramKind.
- Definition Classes
- Function → Expression
-
val
name: String
- Definition Classes
- Function → NamedSymbol
-
final
def
ne(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
final
def
notify(): scala.Unit
- Definition Classes
- AnyRef
- Annotations
- @native()
-
final
def
notifyAll(): scala.Unit
- Definition Classes
- AnyRef
- Annotations
- @native()
-
def
prettyString: String
Pretty-printed string representing this expression
Pretty-printed string representing this expression
- Definition Classes
- Expression
- val realDomainDim: Option[Int]
-
val
sort: Sort
The sort of this expression, e.g., Real, Bool.
- Definition Classes
- Function → Expression
-
final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
- AnyRef
-
def
toString(): String
- Definition Classes
- NamedSymbol → Expression → AnyRef → Any
-
final
def
wait(): scala.Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long, arg1: Int): scala.Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long): scala.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,ProgramSequent- Sequents of formulasProvable- Proof certificates transformed by rules/axiomsRule- Proof rules as well asUSubstOnefor (one-pass) 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.kyxfilesDLParser- Combinator parser reading concrete KeYmaera X syntaxDLArchiveParser- Combinator parser reading KeYmaera X model and proof archive.kyxfilesedu.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 first-order logicHybridProgramCalculus- Hybrid Program Calculus for differential dynamic logicDifferentialEquationCalculus- Differential Equation Calculus for differential dynamic logicUnifyUSCalculus- 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 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 back-end solversMathematicaQETool- 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-helpto 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- 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