case class AppliedBuiltinTwoPositionTactic(positionTactic: BuiltInTwoPositionTactic, posOne: Position, posTwo: Position) extends BelleExpr with (ProvableSig) ⇒ ProvableSig with Product with Serializable
Motivation is similar to AppliedPositionTactic, but for BuiltInTwoPositionTactic
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Instance Constructors
- new AppliedBuiltinTwoPositionTactic(positionTactic: BuiltInTwoPositionTactic, posOne: Position, posTwo: Position)
Value Members
-
final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
def
&(other: BelleExpr): BelleExpr
this & other: sequential composition this ; other executes other on the output of this, failing if either fail.
this & other: sequential composition this ; other executes other on the output of this, failing if either fail.
- Definition Classes
- BelleExpr
-
def
*(n: Int): BelleExpr
this*n: bounded repetition executes this tactic to
timesnumber of times, failing if any of those repetitions fail.this*n: bounded repetition executes this tactic to
timesnumber of times, failing if any of those repetitions fail.- Definition Classes
- BelleExpr
-
def
<(children: BelleExpr*): BelleExpr
<(e1,...,en): branching to run tactic
eion branchi, failing if any of them fail or if there are not exactlynbranches.<(e1,...,en): branching to run tactic
eion branchi, failing if any of them fail or if there are not exactlynbranches.- Definition Classes
- BelleExpr
- Note
Equivalent to
a & Idioms.<(b,c)
-
final
def
==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
def
U(p: (SequentType, (RenUSubst) ⇒ BelleExpr)*): BelleExpr
case _ of {fi => ei} uniform substitution case pattern applies the first ei such that fi uniformly substitutes to current provable for which ei does not fail, fails if the ei of all matching fi fail.
case _ of {fi => ei} uniform substitution case pattern applies the first ei such that fi uniformly substitutes to current provable for which ei does not fail, fails if the ei of all matching fi fail.
- Definition Classes
- BelleExpr
-
def
andThen[A](g: (ProvableSig) ⇒ A): (ProvableSig) ⇒ A
- Definition Classes
- Function1
- Annotations
- @unspecialized()
-
def
apply(provable: ProvableSig): ProvableSig
- Definition Classes
- AppliedBuiltinTwoPositionTactic → Function1
-
final
def
asInstanceOf[T0]: T0
- Definition Classes
- Any
-
def
clone(): AnyRef
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @native() @throws( ... )
-
def
compose[A](g: (A) ⇒ ProvableSig): (A) ⇒ ProvableSig
- Definition Classes
- Function1
- Annotations
- @unspecialized()
- final def computeResult(provable: ProvableSig): ProvableSig
-
final
def
eq(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
def
finalize(): Unit
- Attributes
- protected[java.lang]
- Definition Classes
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- @throws( classOf[java.lang.Throwable] )
-
final
def
getClass(): Class[_]
- Definition Classes
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- Annotations
- @native()
-
def
getLocation: Location
Get the location where this tactic stems from.
Get the location where this tactic stems from.
- Definition Classes
- BelleExpr
-
final
def
isInstanceOf[T0]: Boolean
- Definition Classes
- Any
-
final
def
ne(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
final
def
notify(): Unit
- Definition Classes
- AnyRef
- Annotations
- @native()
-
final
def
notifyAll(): Unit
- Definition Classes
- AnyRef
- Annotations
- @native()
- val posOne: Position
- val posTwo: Position
- val positionTactic: BuiltInTwoPositionTactic
-
def
prettyString: String
pretty-printed form of this Bellerophon tactic expression
pretty-printed form of this Bellerophon tactic expression
- Definition Classes
- AppliedBuiltinTwoPositionTactic → BelleExpr
-
def
setLocation(newLocation: Location): Unit
- Definition Classes
- BelleExpr
- Note
location is private so that it's not something that effects case class quality, and mutable so that it can be ignored when building up custom tactics.
-
def
switch(children: (BelleLabel, BelleExpr)*): BelleExpr
- Definition Classes
- BelleExpr
-
final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
- AnyRef
-
def
toString(): String
- Definition Classes
- Function1 → AnyRef → Any
-
final
def
wait(): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long, arg1: Int): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long): Unit
- Definition Classes
- AnyRef
- Annotations
- @native() @throws( ... )
-
def
|(other: BelleExpr): BelleExpr
this | other: alternative composition executes other if applying this fails, failing if both fail.
this | other: alternative composition executes other if applying this fails, failing if both fail.
- Definition Classes
- BelleExpr
-
def
|!(other: BelleExpr): BelleExpr
this |! other: alternative composition executes other if applying this fails (even critically), failing if both fail.
this |! other: alternative composition executes other if applying this fails (even critically), failing if both fail.
- Definition Classes
- BelleExpr
-
def
||(other: BelleExpr): BelleExpr
- Definition Classes
- BelleExpr
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