final case class USubstOne(subsDefsInput: Seq[SubstitutionPair]) extends (Expression) ⇒ Expression with Product with Serializable
A Uniform Substitution with its one-pass application mechanism. A Uniform Substitution uniformly replaces all occurrences of a given predicate p(.) by a formula in (.). It can also replace all occurrences of a function symbol f(.) by a term in (.) and all occurrences of a predicational / quantifier symbols C(-) by a formula in (-) and all occurrences of program constant symbol b by a hybrid program.
This type implements the application of uniform substitutions to terms, formulas, programs, and sequents.
- Since
4.7
- Note
Implements the one-pass version that checks admissibility on the fly and checking upon occurrence. Faster than the alternative USubstChurch. Main ingredient of prover core.
,soundness-critical
- See also
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.
edu.cmu.cs.ls.keymaerax.core.Provable.apply(subst:edu\.cmu\.cs\.ls\.keymaerax\.core\.USubstOne):edu\.cmu\.cs\.ls\.keymaerax\.core\.Provable*
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- USubstOne
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Instance Constructors
- new USubstOne(subsDefsInput: Seq[SubstitutionPair])
Value Members
-
final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
def
++(other: USubstOne): USubstOne
Union of uniform substitutions, i.e., both replacement lists merged.
Union of uniform substitutions, i.e., both replacement lists merged.
- Note
Convenience method not used in the core, but used for stapling uniform substitutions together during unification etc.
-
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 everywhere in the sequent.
Apply uniform substitution everywhere in the sequent.
- Exceptions thrown
SubstitutionClashException
if this substitution is not admissible for s.
-
def
apply(p: DifferentialProgram): DifferentialProgram
apply this uniform substitution everywhere in a differential program
apply this uniform substitution everywhere in a differential program
- Exceptions thrown
SubstitutionClashException
if this substitution is not admissible for p.
-
def
apply(p: Program): Program
apply this uniform substitution everywhere in a program
apply this uniform substitution everywhere in a program
- Exceptions thrown
SubstitutionClashException
if this substitution is not admissible for p.
-
def
apply(f: Formula): Formula
apply this uniform substitution everywhere in a formula
apply this uniform substitution everywhere in a formula
- Exceptions thrown
SubstitutionClashException
if this substitution is not admissible for f.
-
def
apply(t: Term): Term
apply this uniform substitution everywhere in a term
apply this uniform substitution everywhere in a term
- Exceptions thrown
SubstitutionClashException
if this substitution is not admissible for t.
-
def
apply(e: Expression): Expression
- Definition Classes
- USubstOne → Function1
-
def
applyAllTaboo(s: Sequent): Sequent
Apply uniform substitution everywhere in the sequent with SetLattice.allVars as taboos.
Apply uniform substitution everywhere in the sequent with SetLattice.allVars as taboos.
- Exceptions thrown
SubstitutionClashException
if this substitution is not admissible for s (e.g. because it introduces any free variables).
-
def
applyAllTaboo(f: Formula): Formula
apply this uniform substitution everywhere in a formula with SetLattice.allVars as taboos.
apply this uniform substitution everywhere in a formula with SetLattice.allVars as taboos.
- Exceptions thrown
SubstitutionClashException
if this substitution is not admissible for f (e.g. because it introduces any free variables).
-
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] )
-
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 all repl that are not bound as arguments in what.
- returns
union of the freeVars of all our substitution pairs.
- Note
unused
-
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 subsDefsInput: Seq[SubstitutionPair]
-
final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
- AnyRef
-
def
toString(): String
- Definition Classes
- USubstOne → 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( ... )
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