object TaylorModelArith
Taylor model arithmetic
Here, a Taylor model is a data structure maintaining a proof that some term is element of the Taylor model.
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case class
TM(elem: Term, poly: PolynomialArithV2.Polynomial, lower: Term, upper: Term, prv: ProvableSig) extends Product with Serializable
data structure with certifying computations maintains the invariant prv: \exists err.
data structure with certifying computations maintains the invariant prv: \exists err. elem = poly + err & err \in [lower; upper]
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class
TemplateLemmas extends TaylorModel
generic lemmas for evolution of ODE ode with Taylor model approiximations of order order
Value Members
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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
Exact(elem: PolynomialArithV2.Polynomial, context: IndexedSeq[Formula]): TM
constructs a Taylor model with zero interval part
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def
TM(elem: Term, poly: PolynomialArithV2.Polynomial, lower: Term, upper: Term, context: IndexedSeq[Formula], be: BelleExpr): TM
constructs a Taylor model by proving the required certificate with a tactic
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final
def
asInstanceOf[T0]: T0
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def
clone(): AnyRef
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- @native() @throws( ... )
- def cutEq(prv: ProvableSig, eq: Formula): ProvableSig
- def cutSeq(prvs: Seq[ProvableSig], prv: ProvableSig): ProvableSig
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def
destTaylorModelFormula(fml: Formula): (Term, Term, Term, Term)
takes a formula encoding a Taylor model and returns (elem, poly, lower, upper)
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final
def
eq(arg0: AnyRef): Boolean
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def
equals(arg0: Any): Boolean
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def
evalFormula(fml: Formula, context: IndexedSeq[Formula], argumentMap: Map[Term, TM])(implicit options: TaylorModelOptions): Option[ProvableSig]
try to prove a formula using Taylor model arithmetic
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def
evalTerm(t: Term, context: IndexedSeq[Formula], argumentMap: Map[Term, TM])(implicit options: TaylorModelOptions): TM
evaluate a Term in Taylor model arithmetic
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def
finalize(): Unit
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final
def
getClass(): Class[_]
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def
hashCode(): Int
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- def hideAntes(hides: Seq[Int], prv: ProvableSig): ProvableSig
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def
identityPrecondition(x: Seq[TM], r: Seq[TM], prv: ProvableSig)(implicit options: TaylorModelOptions): (Seq[TM], Seq[TM], ProvableSig)
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lTM(r) ... (left) Taylor model with variables in r r
- returns
(x1, r1) ... x1 = r + c ... identity part of (lTM o rTM) r1 = rTM'(e, i) ... higher order terms + a fresh variable for symbolic remainders x = r + c r0 = TM(e, i)
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final
def
isInstanceOf[T0]: Boolean
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final
def
ne(arg0: AnyRef): Boolean
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final
def
notify(): Unit
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final
def
notifyAll(): Unit
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def
partition(xs: Seq[TM], newElems: Seq[Term], P: (BigDecimal, BigDecimal, PolynomialArithV2.PowerProduct) ⇒ Boolean)(implicit options: TaylorModelOptions): (Seq[TM], Seq[TM], Seq[Equal])
partition and weaken a sequence of Taylor models
- val refineConjunction: ProvableSig
- val refineLe1: ProvableSig
- val refineLe2: ProvableSig
- def refineTaylorModelFormula(fml: Formula, assms: IndexedSeq[Formula])(implicit options: TaylorModelOptions): (Exists, Exists, ProvableSig, (Term, PolynomialArithV2.Polynomial, Term, Term, ProvableSig))
- val refineTmExists: ProvableSig
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def
subtermOf(t: Term, fml: Formula): Boolean
t subterm of fml ?
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def
subtermOf(t: Term, s: Term): Boolean
t subterm of s ?
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final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
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- val taylorModelApproxPrv: ProvableSig
- val taylorModelCollectPrv: ProvableSig
- val taylorModelDivideExactPrv: ProvableSig
- val taylorModelEmptyIntervalPrv: ProvableSig
- val taylorModelEvalPrv: ProvableSig
- val taylorModelExactPrv: ProvableSig
- val taylorModelIntervalGe: ProvableSig
- val taylorModelIntervalGt: ProvableSig
- val taylorModelIntervalLe: ProvableSig
- val taylorModelIntervalLt: ProvableSig
- val taylorModelIntervalPrv: ProvableSig
- val taylorModelMinusPrv: ProvableSig
- val taylorModelNegPrv: ProvableSig
- val taylorModelPartitionPrv1: ProvableSig
- val taylorModelPartitionPrv2: ProvableSig
- val taylorModelPlusPrv: ProvableSig
- val taylorModelPowerEven: ProvableSig
- val taylorModelPowerOdd: ProvableSig
- val taylorModelPowerOne: ProvableSig
- val taylorModelSquarePrv: ProvableSig
- val taylorModelTimesPrv: ProvableSig
- val taylorModelTransElem: ProvableSig
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def
toString(): String
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def
trimContext(context: Seq[Formula], ts: Seq[Term]): Seq[Formula]
delete formulas that contain ts from context, written "context \ ts"
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final
def
wait(): Unit
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def
wait(arg0: Long, arg1: Int): Unit
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wait(arg0: Long): 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 (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