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final
def
!=(arg0: Any): Boolean
- Definition Classes
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-
final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
def
<(s1: (BelleLabel, BelleExpr), spec: (BelleLabel, BelleExpr)*): BelleExpr
Execute different tactics depending on branch label, fall back to branch order if branches come without labels.
Execute different tactics depending on branch label, fall back to branch order if branches come without labels. <((lbl1,t1), (lbl2,t2)) uses tactic t1 on branch labelled lbl1 and uses t2 on lbl2.
- See also
-
def
<(t: BelleExpr*): BelleExpr
Execute ts by branch order.
-
final
def
==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
def
?(t: BelleExpr): BelleExpr
Optional tactic
-
def
NamedTactic(name: String, tactic: ⇒ BelleExpr): DependentTactic
Gives a name to a tactic to a definable tactic.
-
final
def
asInstanceOf[T0]: T0
- Definition Classes
- Any
- def cases(c1: (Case, BelleExpr), cs: (Case, BelleExpr)*): BelleExpr
- def cases(exhaustive: BelleExpr = TactixLibrary.master())(c1: (Case, BelleExpr), cs: (Case, BelleExpr)*): BelleExpr
-
def
clone(): AnyRef
- Attributes
- protected[java.lang]
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- @native() @throws( ... )
-
def
doIf(condition: (ProvableSig) ⇒ Boolean)(t: ⇒ BelleExpr): DependentTactic
Executes t if condition is true.
-
def
doIfElse(condition: (ProvableSig) ⇒ Boolean)(t: ⇒ BelleExpr, f: ⇒ BelleExpr): DependentTactic
Executes t if condition is true and f otherwise.
- def doIfElseFw(condition: (ProvableSig) ⇒ Boolean)(t: ⇒ (ProvableSig) ⇒ ProvableSig, f: ⇒ (ProvableSig) ⇒ ProvableSig): BuiltInTactic
- def doIfFw(condition: (ProvableSig) ⇒ Boolean)(t: ⇒ (ProvableSig) ⇒ ProvableSig): BuiltInTactic
-
final
def
eq(arg0: AnyRef): Boolean
- Definition Classes
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-
def
equals(arg0: Any): Boolean
- Definition Classes
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-
def
finalize(): Unit
- Attributes
- protected[java.lang]
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- @throws( classOf[java.lang.Throwable] )
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final
def
getClass(): Class[_]
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- @native()
-
def
hashCode(): Int
- Definition Classes
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lazy val
ident: BuiltInTactic
no-op nil
-
final
def
isInstanceOf[T0]: Boolean
- Definition Classes
- Any
-
def
mapSubpositions[T](pos: Position, sequent: Sequent, trafo: (Expression, Position) ⇒ Option[T]): List[T]
Map sub-positions of
pos
to Ts that fit to the expressions at those sub-positions pertrafo
. -
def
must(t: BelleExpr, msg: Option[String] = None): BelleExpr
must(t) runs tactic
t
but only ift
actually changed the goal. -
final
def
ne(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
- lazy val nil: BuiltInTactic
-
final
def
notify(): Unit
- Definition Classes
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- Annotations
- @native()
-
final
def
notifyAll(): Unit
- Definition Classes
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- Annotations
- @native()
- def opt(t: (ProvableSig) ⇒ ProvableSig): (ProvableSig) ⇒ ProvableSig
-
def
or(s: (ProvableSig) ⇒ ProvableSig, t: (ProvableSig) ⇒ ProvableSig): (ProvableSig) ⇒ ProvableSig
Try
s
and recover from proof search control failure witht
. -
def
rememberAs(lemmaName: String)(implicit lemmaDB: LemmaDB): BelleExpr
Stores a lemma
lemmaName
if the current provable is proved. -
def
rememberAs(lemmaName: String, t: BelleExpr)(implicit lemmaDB: LemmaDB, lookupRememberedLemmas: Boolean = true): BelleExpr
Proves by lemma, if lemma
lemmaName
exists, else by tactict
and stores the proof found byt
as lemmalemmaName
.Proves by lemma, if lemma
lemmaName
exists, else by tactict
and stores the proof found byt
as lemmalemmaName
. Must be used on a provable with exactly 1 subgoal (e.g., as created in a Case). Lemma lookup can be disabled withdoLemmaLookup
(lemma lookup enabled by default). -
def
repeatWhile(condition: (Expression) ⇒ Boolean)(t: BelleExpr): DependentPositionTactic
Repeats t while condition at position is true.
-
def
saturate(t: (ProvableSig) ⇒ ProvableSig): (ProvableSig) ⇒ ProvableSig
Saturate tactic
t
until no longer applicable. -
def
searchApplyIn(f: Formula, t: DependentPositionTactic, in: PosInExpr): DependentTactic
Search for formula
f
in the sequent and apply tactict
at subpositionin
of the found position. -
def
shift[T <: BelleExpr](child: PosInExpr, t: AtPosition[T]): DependentPositionTactic
shift(child, t) does t to positions shifted by child
-
def
shift[T <: BelleExpr](shift: (PosInExpr) ⇒ PosInExpr, t: AtPosition[T]): DependentPositionTactic
shift(shift, t) does t shifted from position p to shift(p)
-
final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
- AnyRef
-
def
times(t: (ProvableSig) ⇒ ProvableSig, n: Int): (ProvableSig) ⇒ ProvableSig
Executes tactic
t
n
times. -
def
toString(): String
- Definition Classes
- AnyRef → Any
-
lazy val
todo: BuiltInTactic
no-op nil
-
final
def
wait(): Unit
- Definition Classes
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- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long, arg1: Int): Unit
- Definition Classes
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- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long): 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
,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