sealed trait SetLattice[A] extends AnyRef
Lattice of sets, i.e. the lattice of sets that also includes bottom, top and near-top elements.
- A
Type of elements in the set
- Alphabetic
- By Inheritance
- SetLattice
- AnyRef
- Any
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Abstract Value Members
-
abstract
def
+(elem: A): SetLattice[A]
this+elem: Return the set lattice including the extra element
elem
-
abstract
def
++(other: GenTraversableOnce[A]): SetLattice[A]
this++other: Set union
-
abstract
def
-(elem: A): SetLattice[A]
this-elem: Return the set lattice with the element
elem
removed -
abstract
def
--(other: GenTraversableOnce[A]): SetLattice[A]
this--other: Set subtraction
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abstract
def
contains(elem: A): Boolean
True when this lattice contains the element
elem
. -
abstract
def
intersect(other: Set[A]): SetLattice[A]
this intersect other: Set intersection
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abstract
def
isEmpty: Boolean
True indicates that this lattice is the empty lattice bottom.
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abstract
def
isInfinite: Boolean
True indicates that this lattice represents an infinite set of elements, such as all variables.
- abstract def prettyString: String
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abstract
def
symbols[B >: A]: Set[B]
Set of verbatim occurring symbols in this (possibly top) SetLattice
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abstract
def
toSet[B >: A]: Set[B]
Convert to a (finite ordinary) Set if !isInfinite
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abstract
def
toSetLattice[B >: A]: SetLattice[B]
Convert to SetLattices of subtype parameters.
Concrete Value Members
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final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
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final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
def
++(other: SetLattice[A]): SetLattice[A]
this++other: Set union.
-
def
--(other: SetLattice[A]): SetLattice[A]
this--other: Set subtraction.
-
final
def
==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
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final
def
asInstanceOf[T0]: T0
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-
def
clone(): AnyRef
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final
def
eq(arg0: AnyRef): Boolean
- Definition Classes
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-
def
equals(arg0: Any): Boolean
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-
def
finalize(): scala.Unit
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final
def
getClass(): Class[_]
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- @native()
-
def
hashCode(): Int
- Definition Classes
- AnyRef → Any
- Annotations
- @native()
-
def
intersect(other: SetLattice[A]): SetLattice[A]
this intersect other: Set intersection.
-
final
def
isInstanceOf[T0]: Boolean
- Definition Classes
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final
def
ne(arg0: AnyRef): Boolean
- Definition Classes
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final
def
notify(): scala.Unit
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final
def
notifyAll(): scala.Unit
- Definition Classes
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- @native()
- def subsetOf(other: SetLattice[A]): Boolean
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final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
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def
toString(): String
- Definition Classes
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final
def
wait(): scala.Unit
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final
def
wait(arg0: Long, arg1: Int): scala.Unit
- Definition Classes
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final
def
wait(arg0: Long): scala.Unit
- Definition Classes
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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 (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.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 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-help
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
- 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