final case class USubstRenChurch(subsDefsInput: Seq[(Expression, Expression)]) extends (Expression) ⇒ Expression with Product with Serializable
Standalone Renaming Uniform Substitution operation, simultaneously combining URename and USubst. This implementation uses Church-style uniform substitution implementation a la USubstChurch. Liberal list of SubstitutionPair represented as merely a list of Pair, where the Variable~>Variable replacements are by uniform renaming, and the other replacements are by uniform substitution, simultaneously.
- Note
This implementation performs semantic renaming.
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- USubstRenChurch
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Instance Constructors
- new USubstRenChurch(subsDefsInput: Seq[(Expression, Expression)])
Value Members
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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 substitution renaming everywhere in the sequent.
-
def
apply(p: Program): Program
apply this uniform substitution renaming everywhere in a program
-
def
apply(p: DifferentialProgram): DifferentialProgram
apply this uniform substitution renaming everywhere in a program
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def
apply(f: Formula): Formula
apply this uniform substitution renaming everywhere in a formula
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def
apply(t: Term): Term
apply this uniform substitution renaming everywhere in a term
-
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
- USubstRenChurch → Function1
-
final
def
asInstanceOf[T0]: T0
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def
clone(): AnyRef
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- protected[java.lang]
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- @native() @throws( ... )
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def
compose[A](g: (A) ⇒ Expression): (A) ⇒ Expression
- Definition Classes
- Function1
- Annotations
- @unspecialized()
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final
def
eq(arg0: AnyRef): Boolean
- Definition Classes
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def
finalize(): Unit
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- @throws( classOf[java.lang.Throwable] )
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def
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 all repl that are not bound as arguments in what.
- returns
union of the freeVars of all our substitution pairs.
-
final
def
getClass(): Class[_]
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- @native()
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final
def
isInstanceOf[T0]: Boolean
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final
def
ne(arg0: AnyRef): Boolean
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final
def
notify(): Unit
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final
def
notifyAll(): Unit
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final
def
synchronized[T0](arg0: ⇒ T0): T0
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val
toCore: (Expression) ⇒ Expression
This USubstRen implemented strictly from the core.
-
def
toString(): String
- Definition Classes
- USubstRenChurch → Function1 → AnyRef → Any
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final
def
wait(): Unit
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final
def
wait(arg0: Long, arg1: Int): Unit
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final
def
wait(arg0: Long): Unit
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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