class RingsLibrary extends AnyRef
A link to Rings library for its algebra tools
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##(): Int
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def
distributive(t: Ring, xs: Seq[Term]): Map[List[Int], Term]
a distributive representation w.r.t.
a distributive representation w.r.t. Variables in
xs
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eq(arg0: AnyRef): Boolean
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finalize(): Unit
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- def fromRing(p: Ring): Term
- def fromRing(m: Monomial[Rational[BigInteger]]): Term
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getClass(): Class[_]
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- def groebnerBasis(polynomials: List[Term]): List[Term]
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def
integrate(i: Int)(t: Ring): Ring
integral/primitive of
t
w.r.t.integral/primitive of
t
w.r.t. Variablei
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def
internalQuotientRemainder(a: Ring, b: Ring): Option[(Ring, Ring)]
return None if b does not divide a, otherwise Some (quotient, remainder)
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isInstanceOf[T0]: Boolean
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def
lieDerivative(ode: (Variable) ⇒ Option[Ring])(t: Term): Ring
compute the Lie Derivative
- val mapper: Map[NamedSymbol, String]
- val names: List[NamedSymbol]
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ne(arg0: AnyRef): Boolean
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def
normalizeLessEquals(qeTac: BelleExpr): DependentPositionTactic
a<=b to 0<=a-b, with the standard representation of a-b (distributive and according to the variable order of ring, also distributes over conjunctions...
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- def polynomialReduce(polynomial: Term, GB: List[Term]): (List[Term], Term)
- def quotientRemainder(term: Term, div: Term, x: Variable): (Term, Term)
- implicit val ring: MultivariateRing[Rational[BigInteger]]
- val ringsNames: List[(NamedSymbol, String)]
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def
splitInternal(p: Ring, order: Int, vars: Seq[Int], drop: Seq[Int]): (Ring, Ring)
split polynomial according to order w.r.t variables in list
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def
substitutes(subst: (Variable) ⇒ Option[Ring])(t: Term): Ring
Substitute into a polynomial: simultaneously replace Variables v
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synchronized[T0](arg0: ⇒ T0): T0
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def
toHorner(t: Ring, xs: List[Ring]): Term
rewrite t to Horner Form (w.r.t.
rewrite t to Horner Form (w.r.t. "Variables" xs)
we allow arbitrary terms as "variables" in order to factor out e.g., squares.
- def toRing(term: Term): Ring
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toString(): String
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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