final case class SubstitutionPair(what: Expression, repl: Expression) extends Product with Serializable
Representation of a substitution replacing what with repl uniformly, everywhere.
Data structure invariant: Only substitutable expressions will be accepted for what with compatible repl.
- what
the expression to be replaced.
whatcan have one of the following forms:- PredOf(p:Function, DotTerm/Nothing/Nested Pair of DotTerm)
- FuncOf(f:Function, DotTerm/Nothing/Nested Pair of DotTerm)
- ProgramConst or DifferentialProgramConst or SystemConst
- UnitPredicational
- UnitFunctional
- PredicationalOf(p:Function, DotFormula)
- DotTerm
- DotFormula
- repl
the expression to be used in place of
what.
- See also
Andre Platzer. A complete uniform substitution calculus for differential dynamic logic. Journal of Automated Reasoning, 59(2), pp. 219-266, 2017.
Andre Platzer. Uniform substitution at one fell swoop. In Pascal Fontaine, editor, International Conference on Automated Deduction, CADE'19, Natal, Brazil, Proceedings, volume 11716 of LNCS, pp. 425-441. Springer, 2019.
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Instance Constructors
-
new
SubstitutionPair(what: Expression, repl: Expression)
- what
the expression to be replaced.
whatcan have one of the following forms:- PredOf(p:Function, DotTerm/Nothing/Nested Pair of DotTerm)
- FuncOf(f:Function, DotTerm/Nothing/Nested Pair of DotTerm)
- ProgramConst or DifferentialProgramConst or SystemConst
- UnitPredicational
- UnitFunctional
- PredicationalOf(p:Function, DotFormula)
- DotTerm
- DotFormula
- repl
the expression to be used in place of
what.
Value Members
-
final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
final
def
##(): Int
- Definition Classes
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-
final
def
==(arg0: Any): Boolean
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final
def
asInstanceOf[T0]: T0
- Definition Classes
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lazy val
boundVars: SetLattice[Variable]
The (new) bound variables that this substitution introduces.
The (new) bound variables that this substitution introduces. That is the (new) bound variables introduces by this substitution, i.e. all bound variables of
repl. These are all the bound variables of the formula or programrepl(and irrelevant bottom for terms). -
def
clone(): AnyRef
- Attributes
- protected[java.lang]
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- @native() @throws( ... )
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final
def
eq(arg0: AnyRef): Boolean
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def
finalize(): scala.Unit
- Attributes
- protected[java.lang]
- Definition Classes
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- @throws( classOf[java.lang.Throwable] )
-
lazy val
freeVars: SetLattice[Variable]
The (new) free variables that this substitution introduces (without DotTerm/DotFormula arguments).
The (new) free variables that this substitution introduces (without DotTerm/DotFormula arguments). That is the (new) free variables introduced by this substitution, i.e. free variables of repl that are not bound as arguments in what.
- returns
essentially freeVars(repl) except for special handling of UnitFunctional and UnitPredicational arguments.
- See also
Definition 19 in Andre Platzer. A complete uniform substitution calculus for differential dynamic logic. Journal of Automated Reasoning, 59(2), pp. 219-266, 2017.
-
final
def
getClass(): Class[_]
- Definition Classes
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- Annotations
- @native()
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final
def
isInstanceOf[T0]: Boolean
- Definition Classes
- Any
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final
def
ne(arg0: AnyRef): Boolean
- Definition Classes
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final
def
notify(): scala.Unit
- Definition Classes
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final
def
notifyAll(): scala.Unit
- Definition Classes
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- Annotations
- @native()
- val repl: Expression
-
lazy val
signature: Set[NamedSymbol]
The signature of the replacement introduced by this substitution.
The signature of the replacement introduced by this substitution.
- Note
DotTerm and DotFormula arguments don't literally occur if bound by p(DotTerm) ~> DotTerm>5
-
final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
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-
def
toString(): String
- Definition Classes
- SubstitutionPair → AnyRef → Any
-
final
def
wait(): scala.Unit
- Definition Classes
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- @throws( ... )
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final
def
wait(arg0: Long, arg1: Int): scala.Unit
- Definition Classes
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final
def
wait(arg0: Long): scala.Unit
- Definition Classes
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- val what: Expression
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