final case class URename(what: Variable, repl: Variable, semantic: Boolean = false) extends (Expression) ⇒ Expression with Product with Serializable
Uniformly rename all occurrences of what
and what'
to repl
and repl'
and vice versa, but clash for program constants etc.
Uniformly rename all occurrences of variable what
(and its associated DifferentialSymbol what'
) to
repl
(and its associated DifferentialSymbol repl'
) everywhere
and vice versa uniformly rename all occurrences of variable repl
(and its associated DifferentialSymbol) to what
(and what'
).
Uniform renaming, thus, is a transposition.
- what
What variable to replace (along with its associated DifferentialSymbol).
- repl
The target variable to replace
what
with and vice versa.- semantic
true
to allow semantic renaming of SpaceDependent symbols, which preserves local soundness when applied to inferences.
- Note
soundness-critical
- See also
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Instance Constructors
-
new
URename(what: Variable, repl: Variable, semantic: Boolean = false)
- what
What variable to replace (along with its associated DifferentialSymbol).
- repl
The target variable to replace
what
with and vice versa.- semantic
true
to allow semantic renaming of SpaceDependent symbols, which preserves local soundness when applied to inferences.
Value Members
-
final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
final
def
==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
def
andThen[A](g: (Expression) ⇒ A): (Expression) ⇒ A
- Definition Classes
- Function1
- Annotations
- @unspecialized()
-
def
apply(s: Sequent): Sequent
Apply uniform renaming everywhere in the sequent.
Apply uniform renaming everywhere in the sequent.
- Exceptions thrown
RenamingClashException
if this uniform renaming is not admissible for s (because a semantic symbol occurs despite !semantic).
-
def
apply(p: Program): Program
apply this uniform renaming everywhere in a program.
apply this uniform renaming everywhere in a program.
- Exceptions thrown
RenamingClashException
if this uniform renaming is not admissible for p (because a semantic symbol occurs despite !semantic).
-
def
apply(p: DifferentialProgram): DifferentialProgram
apply this uniform renaming everywhere in a differential program.
apply this uniform renaming everywhere in a differential program.
- Exceptions thrown
RenamingClashException
if this uniform renaming is not admissible for p (because a semantic symbol occurs despite !semantic).
-
def
apply(f: Formula): Formula
apply this uniform renaming everywhere in a formula.
apply this uniform renaming everywhere in a formula.
- Exceptions thrown
RenamingClashException
if this uniform renaming is not admissible for f (because a semantic symbol occurs despite !semantic).
-
def
apply(t: Term): Term
apply this uniform renaming everywhere in a term.
apply this uniform renaming everywhere in a term.
- Exceptions thrown
RenamingClashException
if this uniform renaming is not admissible for t (because a semantic symbol occurs despite !semantic).
-
def
apply(e: Expression): Expression
apply this uniform renaming everywhere in an expression, resulting in an expression of the same kind.
apply this uniform renaming everywhere in an expression, resulting in an expression of the same kind.
- Definition Classes
- URename → 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) ⇒ Expression): (A) ⇒ Expression
- Definition Classes
- Function1
- Annotations
- @unspecialized()
-
final
def
eq(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
def
finalize(): scala.Unit
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @throws( classOf[java.lang.Throwable] )
-
final
def
getClass(): Class[_]
- Definition Classes
- AnyRef → Any
- Annotations
- @native()
-
final
def
isInstanceOf[T0]: Boolean
- Definition Classes
- Any
-
final
def
ne(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
final
def
notify(): scala.Unit
- Definition Classes
- AnyRef
- Annotations
- @native()
-
final
def
notifyAll(): scala.Unit
- Definition Classes
- AnyRef
- Annotations
- @native()
- val repl: Variable
- val semantic: Boolean
-
final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
- AnyRef
-
def
toString(): String
- Definition Classes
- URename → Function1 → AnyRef → Any
-
final
def
wait(): scala.Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long, arg1: Int): scala.Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long): scala.Unit
- Definition Classes
- AnyRef
- Annotations
- @native() @throws( ... )
- val what: Variable
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